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** " multivariate" Definitions**

- having or involving a number of independent mathematical or statistical variables

** " multivariate" Synonyms**

bivariate heptvariate hexvariate octvariate pentavariate tetravariate trivariate

*How to use multivariate in a sentence? Find typical usage patterns (collocations)/phrases/context for "multivariate" and check conjugation/comparative form for "multivariate". Mastering all the usages of "multivariate" from sentence examples published by news publications.*

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What might a multivariate solution to a problem like obesity look like? | |

In multivariate data, many variables are considered simultaneously rather than one at a time. | |

AI will allow marketplaces to process richer data, and to therefore understand complex multivariate needs. | |

Before multivariate creditscoring, a banker couldn't tell two neighbors apart when pricing a mortgage. | |

Can you give me an example of a "multivariate" approach to health that exists today? | |

At this point you might pause and wonder what all these multivariate works have in common. | |

Enabled by sensor technologies and AI, marketplaces will be able to understand and satisfy complex multivariate needs. | |

A/B and multivariate tests can be run to squeeze out every possible increase in conversion rates. | |

Their standard professional techniques – think multivariate regression analysis and complex specifications of indifference curves – are not relevant. | |

"Hypotheses on the positive effects of monthly startup reports were tested, using several multivariate regressions," write the paper's authors. | |

Credit will become more multivariate, using machine learning and breaking apart the contributing factors and weights that make up FICO. | |

We then developed two multivariate models for each state to deduce which election-related terms correlated against anxiety and depression. | |

The study used multivariate regression analysis to control for education levels and pitch quality to conclude that gender was a statistically significant factor. | |

His most recent album, "Heart of Brazil: A Tribute to Egberto Gismonti," honors a composer whose music is as multivariate as Brazil's culture. | |

And yet a simple multivariate statistical test* of these patterns finds no clear relationship between state partisanship, seniority, and support for the legislative filibuster. | |

"We are, make no mistake … in the middle of an enormous multivariate kind of Ponzi scheme," Palihapitiya said at a San Francisco conference three weeks ago. | |

"We are, make no mistake … in the middle of an enormous multivariate kind of Ponzi scheme," said Palihapitiya, at the Launch Scale conference in San Francisco. | |

Cryptographers are debating the relative merits of such mathematical curiosities as supersingular isogenies, structured and unstructured lattices, and multivariate polynomials as foundations for quantum-proof cryptography. | |

If we model the distribution of font vectors as a multivariate normal, we can sample random vectors from it and look at the fonts they generate. | |

"You can expect a lot more from us in the same vein, trying to help people [sleep] across the board, in a multivariate way," Parikh said. | |

We then used multivariate linear regression to determine which, if any, indicators were statistically significant determinants of state-by-state over-representation ratio—the outcome of interest. | |

In a city notorious for crime, he implemented policing measures that were popular in the United States — think "zero tolerance" and "broken windows" and "compstat" multivariate analysis. | |

Combine that with quantitative data from analytics and start testing variations of the interface utilizing A/B and multivariate testing software to determine the optimal design and experience. | |

"There's this never-ending hunt for new and better data sources," said Max Wolff, a former hedge fund executive who is now a managing director at Multivariate, a data-driven consulting firm. | |

Professor Anderson's book "An Introduction to Multivariate Statistical Analysis" (1958) remains a classic in the field, educating generations of statisticians in the conceptual underpinnings of a particularly challenging kind of data analysis. | |

I used multivariate regression to predict the Republican vote increase in every state based on the county-level percentage of white residents and whether the county was a high-education one or not. | |

Using what is called "multivariate regression analysis," they found that belief in the fake news statements still accounted for a significant portion of the defections when other factors were considered, including the extent to which voters disliked Clinton or liked Trump. | |

Drawing on an array of influences that includes Chicago footwork, Japanese synth-pop pioneers Yellow Magic Orchestra, and the frenetic BPMs of South African Shangaan electro, the ten songs showcase Lanza's multivariate production skills, along with those of collaborator Junior Boys' Jeremy Greenspan. | |

From the Phosphorus website: While prior studies examined one or two genetic factors in relation to fertility, FertilityMap takes a large-scale, multivariate approach: analyzing hundreds of clinical variables and thousands of genetic variants in the context of each participant's personal, pregnancy and family medical history. | |

Current projections, which I have commissioned from leading Silicon Valley firms, suggest that the process of aggregating these multivariate emotional states across the U.S. will be only moderately less humane than analog voting, as well as require minimal force projection from the S.M.I.L.E. Squads to complete. | |

The solution lies in the development ofquantum-safe cryptography, consisting of information theoretically secure schemes, hash-based cryptography, code-based cryptography and exotic-sounding technologies like lattice-based cryptography, multivariate cryptography (like the "Unbalanced Oil and Vinegar scheme"), and even supersingular elliptic curve isogeny cryptography. | |

Mailchimp itself has a big marketing presence already: it says that daily, more than1.25 million e-commerce orders are generated through Mailchimp campaigns; over 450 million e-commerce orders were made through Mailchimp campaigns in 2018; and itscustomers have sold over $250 million in goods through multivariate + A/B campaigns run through Mailchimp. | |

"Dark patterns tend to perform very well in A/B and multivariate tests simply because a design that tricks users into doing something is likely to achieve more conversions than one that allows users to make an informed decision," wrote Brignull in 2011— highlighting exactly why web designers were skewing towards being so tricksy: Superficially it works. | |

The multivariate stable distribution is a multivariate probability distribution that is a multivariate generalisation of the univariate stable distribution. The multivariate stable distribution defines linear relations between stable distribution marginals. In the same way as for the univariate case, the distribution is defined in terms of its characteristic function. The multivariate stable distribution can also be thought as an extension of the multivariate normal distribution. | |

Fang has contributed to the theory of elliptical distributions, like bivariate normal distribution (pictured), which have elliptical contours. In mathematical statistics, Fang has published textbooks and monographs in multivariate analysis. In particular, his books have extended classical multivariate analysis beyond the multivariate normal distribution to a generalized multivariate analysis using more general elliptical distributions, which have elliptically contoured distributions. His book on Generalized multivariate analysis (with Zhang) has extensive results on multivariate analysis for elliptical distributions, to which T. W. Anderson refers readers of his An introduction to multivariate statistical analysis (3rd ed. | |

Named joint distributions that arise frequently in statistics include the multivariate normal distribution, the multivariate stable distribution, the multinomial distribution, the negative multinomial distribution, the multivariate hypergeometric distribution, and the elliptical distribution. | |

The Society of Multivariate Experimental Psychology (SMEP) is a small academic organization of research psychologists who have interests in multivariate statistical models for advancing psychological knowledge. It publishes a journal, Multivariate Behavioral Research. | |

The partial correlation coincides with the conditional correlation if the random variables are jointly distributed as the multivariate normal, other elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial or Dirichlet distribution, but not in general otherwise. | |

In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests involving individual dependent variables separately.Stevens, J. P. (2002). Applied multivariate statistics for the social sciences. | |

In statistics, a multivariate Pareto distribution is a multivariate extension of a univariate Pareto distribution. There are several different types of univariate Pareto distributions including Pareto Types I−IV and Feller−Pareto. Chapter 3. Multivariate Pareto distributions have been defined for many of these types. | |

A series of independent measurements is used to estimate the full spectrum of the mixture, and the spectrometer renders a measurement of the spectral intensity at many wavelengths. Multivariate statistics can then be applied to the spectrum produced. In contrast, when using multivariate optical computing, the light entering the instrument strikes an application specific multivariate optical element, which is uniquely tuned to the pattern that needs to be measured using multivariate analysis. This system can produce the same result that multivariate analysis of a spectrum would produce. | |

In marketing, multivariate testing or multi-variable testing techniques apply statistical hypothesis testing on multi-variable systems, typically consumers on websites. Techniques of multivariate statistics are used. Companies that use multivariate testing include Optimizely (for landing page optimization), Mailchimp (for email), and Marpipe (for ad creative). | |

These processes are used for regression, prediction, Bayesian optimization and related problems. For multivariate regression and multi- output prediction, the multivariate Student t-processes are introduced and used. | |

Multivariate cryptography is the generic term for asymmetric cryptographic primitives based on multivariate polynomials over a finite field F. In certain cases those polynomials could be defined over both a ground and an extension field. If the polynomials have the degree two, we talk about multivariate quadratics. Solving systems of multivariate polynomial equations is proven to be NP-complete. That's why those schemes are often considered to be good candidates for post-quantum cryptography. | |

Multivariate optical computing, also known as molecular factor computing, is an approach to the development of compressed sensing spectroscopic instruments, particularly for industrial applications such as process analytical support. "Conventional" spectroscopic methods often employ multivariate and chemometric methods, such as multivariate calibration, pattern recognition, and classification, to extract analytical information (including concentration) from data collected at many different wavelengths. Multivariate optical computing uses an optical computer to analyze the data as it is collected. The goal of this approach is to produce instruments which are simple and rugged, yet retain the benefits of multivariate techniques for the accuracy and precision of the result. | |

In statistics, the multivariate normal distribution (of Gauss) is used in classical multivariate analysis, in which most methods for estimation and hypothesis-testing are motivated for the normal distribution. In contrast to classical multivariate analysis, generalized multivariate analysis refers to research on elliptical distributions without the restriction of normality. For suitable elliptical distributions, some classical methods continue to have good properties. Under finite-variance assumptions, an extension of Cochran's theorem (on the distribution of quadratic forms) holds. | |

Fang Kaitai (; born 1940), also known as Kai-Tai Fang, is a Chinese mathematician and statistician who has helped to develop generalized multivariate analysis, which extends classical multivariate analysis beyond the multivariate normal distribution to more general elliptical distributions.OCLC listing of publications by K'ai-T'ai Fang. Retrieved 4 September 2016. He has also contributed to the design of experiments. | |

In statistics, Wilks' lambda distribution (named for Samuel S. Wilks), is a probability distribution used in multivariate hypothesis testing, especially with regard to the likelihood-ratio test and multivariate analysis of variance (MANOVA). | |

Taylor's theorem also generalizes to multivariate and vector valued functions. | |

This includes cryptographic systems such as the Rainbow (Unbalanced Oil and Vinegar) scheme which is based on the difficulty of solving systems of multivariate equations. Various attempts to build secure multivariate equation encryption schemes have failed. However, multivariate signature schemes like Rainbow could provide the basis for a quantum secure digital signature. There is a patent on the Rainbow Signature Scheme. | |

Multivariate cryptography has been very productive in terms of design and cryptanalysis. Overall, the situation is now more stable and the strongest schemes have withstood the test of time. It is commonly admitted that Multivariate cryptography turned out to be more successful as an approach to build signature schemes primarily because multivariate schemes provide the shortest signature among post-quantum algorithms. | |

The new superimposed landmarks can now be analyzed in multivariate statistical analyses. | |

The univariate Pareto distribution has been extended to a multivariate Pareto distribution. | |

Similarly they came up with the technique of Sequential Regression Multivariate Imputation. | |

By construction, the marginal distribution over \boldsymbol\Lambda is a Wishart distribution, and the conditional distribution over \boldsymbol\mu given \boldsymbol\Lambda is a multivariate normal distribution. The marginal distribution over \boldsymbol\mu is a multivariate t-distribution. | |

Rao, C.R. (1952) Advanced Statistical Methods in Multivariate Analysis, Wiley. (Section 9c) Later work for the multivariate normal distribution allowed the classifier to be nonlinear:Anderson, T.W. (1958) An Introduction to Multivariate Statistical Analysis, Wiley. several classification rules can be derived based on different adjustments of the Mahalanobis distance, with a new observation being assigned to the group whose centre has the lowest adjusted distance from the observation. | |

Kernel density estimators were first introduced in the scientific literature for univariate data in the 1950s and 1960s and subsequently have been widely adopted. It was soon recognised that analogous estimators for multivariate data would be an important addition to multivariate statistics. Based on research carried out in the 1990s and 2000s, multivariate kernel density estimation has reached a level of maturity comparable to its univariate counterparts. | |

Multivariate Behavioral Research is a peer-reviewed academic journal published by Taylor & Francis Group on behalf of the Society of Multivariate Experimental Psychology. The editor-in-chief is Peter Molenaar (Pennsylvania State University). Its 2017 impact factor is 3.691. | |

Multivariate data analysis (8th ed.). Cengage.Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2009). Multivariate data analysis (7th ed.). Pearson.Hair, J. F., Black, W. C., Babin, B. J., Anderson, R. E., & Tatham, R. L. (2005). | |

By construction, the marginal distribution over \boldsymbol\Sigma is an inverse Wishart distribution, and the conditional distribution over \boldsymbol\mu given \boldsymbol\Sigma is a multivariate normal distribution. The marginal distribution over \boldsymbol\mu is a multivariate t-distribution. | |

Multivariate Statistics for Wildlife and Ecology Research. New York, New York, USA: Springer. | |

Fundamental to process analytical technology (PAT) initiatives are the basics of multivariate analysis (MVDA) and design of experiments (DoE). This is because analysis of the process data is a key to understand the process and keep it under multivariate statistical control. | |

The SMEP journal, Multivariate Behavioral Research (MBR), publishes research articles on multivariate methodology and its use in psychological research. The 2014 Editor of MBR is Keith Widaman, and the journal is published by Taylor & Francis Group. SMEP and Taylor & Francis also cooperate in the publication of a series of books on applications of multivariate quantitative methods to important substantive research issues. The book series is edited by Lisa Harlow. | |

Within the tangent space, conventional multivariate statistical methods such as multivariate analysis of variance and multivariate regression, can be used to test statistical hypotheses about shape. Procrustes-based analyses have some limitations. One is that the Procrustes superimposition uses a least-squares criterion to find the optimal rotation; consequently, variation that is localized to a single landmark will be smeared out across many. This is called the 'Pinocchio effect'. | |

Niinimaa, A., and H. Oja. "Multivariate median." Encyclopedia of statistical sciences (1999).Mosler, Karl. | |

Elliptical distributions are also used in robust statistics to evaluate proposed multivariate-statistical procedures. | |

Exploratory Multivariate Analysis by Example Using R. Chapman & Hall/CRC The R Series, London. | |

There are many different models, each with its own type of analysis: # Multivariate analysis of variance (MANOVA) extends the analysis of variance to cover cases where there is more than one dependent variable to be analyzed simultaneously; see also Multivariate analysis of covariance (MANCOVA). #Multivariate regression attempts to determine a formula that can describe how elements in a vector of variables respond simultaneously to changes in others. For linear relations, regression analyses here are based on forms of the general linear model. Some suggest that multivariate regression is distinct from multivariable regression, however, that is debated and not consistently true across scientific fields. | |

With Stephen Fienberg and Paul W. Holland, Bishop became the author of a book on multivariate statistics, Discrete Multivariate Analysis: Theory and Practice (MIT Press, 1975; Springer, 2007). By 1980 the book had already become regarded as a "classic" in the field. | |

He was also the editor of the psychological journal Multivariate Behavioral Research for 8 years. | |

It uses high resolution imaging. All the instruments are coupled with multivariate statistics data processing. | |

Besides, recently an estimation method accounting for continuous and multivariate outputs, Y, was proposed in . | |

A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector—a set of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution. | |

It has parameter, α, which is defined over the range 0 < α ≤ 2, and where the case α = 2 is equivalent to the multivariate normal distribution. It has an additional skew parameter that allows for non-symmetric distributions, where the multivariate normal distribution is symmetric. | |

The algorithm illustrated above can be generalized for mixtures of more than two multivariate normal distributions. | |

The GIG and GNIG distributions are the basis for the exact and near-exact distributions of a large number of likelihood ratio test statistics and related statistics used in multivariate analysis. Bilodeau, M., Brenner, D. (1999) "Theory of Multivariate Statistics". Springer, New York [Ch. 11, sec. | |

Matrix differential calculus is used in statistics, particularly for the statistical analysis of multivariate distributions, especially the multivariate normal distribution and other elliptical distributions. It is used in regression analysis to compute, for example, the ordinary least squares regression formula for the case of multiple explanatory variables. | |

Multivariate Dispersion, Central Regions, and Depth: The Lift Zonoid Approach. Vol. 165. Springer Science & Business Media, 2012. | |

Multivariate local characteristic tables and detailed themes around: economic activity; country of birth; occupation; and unpaid care. | |

The distribution arises in multivariate statistics in undertaking tests of the differences between the (multivariate) means of different populations, where tests for univariate problems would make use of a t-test. The distribution is named for Harold Hotelling, who developed it as a generalization of Student's t-distribution. | |

From 1938 to 1945, Hsu published several papers in the forefront of the development of the theory of multivariate analysis. He obtained several exact or asymptotic distributions of important statistics in the theory of multivariate analysis.Academic Achievements of Professor P.L. Hsu Lehmann, E. L. (1979). Hsu's work on inference. | |

Rafsky invented the Friedman-Rafsky Test, along with Jerome H. Friedman, now a fundamental procedure in multivariate data. This Multivariate normality test checks a given set of data for goodness-of-fit to the multivariate normal distribution. The null hypothesis is that the data set is a sample from the normal distribution, therefore a sufficiently small p-value indicates non-normal data. In 1981, Rafsky outlined this algorithm in a study published by the Journal of the American Statistical Association. | |

A multivariate version of the Fisher's distribution is used if there are more than two colors of balls. | |

In statistics, Bayesian multivariate linear regression is a Bayesian approach to multivariate linear regression, i.e. linear regression where the predicted outcome is a vector of correlated random variables rather than a single scalar random variable. A more general treatment of this approach can be found in the article MMSE estimator. | |

He made significant contributions to multivariate aging notions and multivariate life distributions, as well as to accelerated life tests (inference, non- parametric approach and goodness of fit). Moshe Shaked was born in Jerusalem to a Polish immigrant family. He was a knowledgeable ancient-coin enthusiast and a passionate museum-goer. | |

One generalisation of the problem involves multivariate normal distributions with unknown covariance matrices, and is known as the multivariate Behrens–Fisher problem.Belloni & Didier (2008) The nonparametric Behrens–Fisher problem does not assume that the distributions are normal. Tests include the Cucconi test of 1968 and the Lepage test of 1971. | |

His awards included the Society for Multivariate Experimental Psychology's Raymond Cattell Award (1993) and Herb Eber Award (2001), as well as three Tanaka Awards (1995, 1999, and 2005) for work in Multivariate Behavioral Research. Millsap died suddenly on May 9, 2014. He was survived by his wife and four children. | |

Mahalanobis distance (D) is a multivariate generalization of Cohen's d, which takes into account the relationships between the variables. | |

When you are dealing with a multivariate system such as the brain, you cannot characterize it using univariant measures. | |

The Journal of Multivariate Analysis is a monthly peer-reviewed scientific journal that covers applications and research in the field of multivariate statistical analysis. The journal's scope includes theoretical results as well as applications of new theoretical methods in the field. Some of the research areas covered include copula modeling, functional data analysis, graphical modeling, high-dimensional data analysis, image analysis, multivariate extreme-value theory, sparse modeling, and spatial statistics. According to the Journal Citation Reports, the journal has a 2017 impact factor of 1.009. | |

The study of heteroscedasticity has been generalized to the multivariate case, which deals with the covariances of vector observations instead of the variance of scalar observations. One version of this is to use covariance matrices as the multivariate measure of dispersion. Several authors have considered tests in this context, for both regression and grouped-data situations. Bartlett's test for heteroscedasticity between grouped data, used most commonly in the univariate case, has also been extended for the multivariate case, but a tractable solution only exists for 2 groups. | |

The Unscrambler X is a commercial software product for multivariate data analysis, used for calibration of multivariate data which is often in the application of analytical data such as near infrared spectroscopy and Raman spectroscopy, and development of predictive models for use in real-time spectroscopic analysis of materials. The software was originally developed in 1986 by Harald MartensHarald Martens, Terje Karstang, Tormod Næs (1987) Improved selectivity in spectroscopy by multivariate calibration Journal of Chemometrics 1(4):201-219 and later by CAMO Software. | |

In probability theory and statistics, the generalized multivariate log-gamma (G-MVLG) distribution is a multivariate distribution introduced by Demirhan and Hamurkaroglu in 2011. The G-MVLG is a flexible distribution. Skewness and kurtosis are well controlled by the parameters of the distribution. This enables one to control dispersion of the distribution. | |

Key and Quick statistics Part 3 – Multivariate, Local and Detailed Characteristics tables for: Living arrangements; Household composition; and Accommodation type. | |

In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint distribution forms an ellipse and an ellipsoid, respectively, in iso-density plots. In statistics, the normal distribution is used in classical multivariate analysis, while elliptical distributions are used in generalized multivariate analysis, for the study of symmetric distributions with tails that are heavy, like the multivariate t-distribution, or light (in comparison with the normal distribution). Some statistical methods that were originally motivated by the study of the normal distribution have good performance for general elliptical distributions (with finite variance), particularly for spherical distributions (which are defined below). | |

An instrument which implements this approach may be described as a multivariate optical computer. Since it describes an approach, rather than any specific wavelength range, multivariate optical computers may be built using a variety of different instruments (including Fourier Transform Infrared (FTIR)1 and Raman). The "software" in multivariate optical computing is encoded directly into an optical element spectral calculation engine such as an interference filter based multivariate optical element (MOE), holographic grating, liquid crystal tunable filter, spatial light modulator (SLM), or digital micromirror device (DMD) and is specific to the particular application. The optical pattern for the spectral calculation engine is designed for the specific purpose of measuring the magnitude of that multi-wavelength pattern in the spectrum of a sample, without actually measuring a spectrum. | |

In statistics, the multivariate Behrens–Fisher problem is the problem of testing for the equality of means from two multivariate normal distributions when the covariance matrices are unknown and possibly not equal. Since this is a generalization of the univariate Behrens-Fisher problem, it inherits all of the difficulties that arise in the univariate problem. | |

In probability theory and statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix).Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. | |

Graphical models can still be used when the variables of choice are continuous. In these cases, the probability distribution is represented as a multivariate probability distribution over continuous variables. Each family of distribution will then impose certain properties on the graphical model. Multivariate Gaussian distribution is one of the most convenient distributions in this problem. | |

Each time period t there is a binary mixing variable b(t). If b(t)=0 then the factor return in that period is drawn from the normal distribution and if b(t)=1 it drawn from the jump distribution. Torre found that simultaneous jumps occur in factors. Accordingly, in the multivariate case it is necessary to introduce a multivariate shock vector w(i,t) where w(i,t)=0 if the multivariate mixing variable b(i,t)=0 and w(i,t) is drawn from the ith jump distribution if b(i,t)=1. | |

Careful development of a set of calibration samples and application of multivariate calibration techniques is essential for near-infrared analytical methods. | |

It looks for correlation of income with lifestyle, by comparing with multivariate statistical models; outliers from expected variance will be investigated. | |

J. Statist. Plan. Infer. 144, 3–18., Multivariate analysis and Combinatorial mathematics. Srivastava was a Fellow of Institute of Mathematical Statistics. | |

Multivariate kurtosis can be problematic by either inflating or deflating standard errors depending on whether the distribution is leptokurtic or platykurtic. | |

Multivariate kurtosis can be problematic by either inflating or deflating standard errors depending on whether the distribution is leptokurtic or platykurtic. | |

Multivariate adaptive regression splines (MARS) is a non-parametric technique that builds flexible models by fitting piecewise linear regressions. Multivariate and adaptive regression spline approach deliberately overfits the model and then prunes to get to the optimal model. The algorithm is computationally very intensive, and in practice an upper limit on the number of basis functions is specified. | |

The third market model assumes that the logarithm of the return, or, log-return, of any risk factor typically follows a normal distribution. Collectively, the log-returns of the risk factors are multivariate normal. Monte Carlo algorithm simulation generates random market scenarios drawn from that multivariate normal distribution. For each scenario, the profit (loss) of the portfolio is computed. | |

He also framed the Anderson–Bahadur algorithmClassification into two multivariate normal distributions with different covariance matrices (1962), T W Anderson, R R Bahadur, Annals of Mathematical Statistics along with Raghu Raj Bahadur which is used in statistics and engineering for solving binary classification problems when the underlying data have multivariate normal distributions with different covariance matrices. | |

In 1968, he received an invitation to the University of California, Berkeley, which he accepted and where he retired in 1984. Kaiser provided fundamental contributions to psychometrics and statistical psychology. His contributions to factor analysis were central. Kaiser was president of the Psychometric Society and the Society for Multivariate Experimental Psychology and publisher of the journal Multivariate Behavioral Research. | |

Natural language modelling and text processing. His specific areas of interest are natural language understanding systems and multivariate analysis of text corpora. | |

In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions. | |

When the measurement model is multivariate, that is, it has any number of output quantities, the above concepts can be extended. The output quantities are now described by a joint probability distribution, the coverage interval becomes a coverage region, the law of propagation of uncertainty has a natural generalization, and a calculation procedure that implements a multivariate Monte Carlo method is available. | |

An example of a Motion Chart illustrating the relationships between six variables. A motion chart is a dynamic bubble chart which allows efficient and interactive exploration and visualization of longitudinal multivariate Data.Al-Aziz, J, Christou, N, Dinov, I D (2010). SOCR Motion Charts: An Efficient, Open-Source, Interactive and Dynamic Applet for Visualizing Longitudinal Multivariate Data, JSE, 18(3), 1-29. | |

The Wishart distribution arises as the distribution of the sample covariance matrix for a sample from a multivariate normal distribution. It occurs frequently in likelihood-ratio tests in multivariate statistical analysis. It also arises in the spectral theory of random matrices and in multidimensional Bayesian analysis. It is also encountered in wireless communications, while analyzing the performance of Rayleigh fading MIMO wireless channels . | |

A Diophantine equation is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. Algebraic geometry is the study of the solutions in an algebraically closed field of multivariate polynomial equations. Two equations are equivalent if they have the same set of solutions. In particular the equation P = Q is equivalent to P-Q = 0. | |

Freeman, P. W. (1981). A multivariate study of the family Molossidae (Mammalia, Chiroptera): morphology, ecology, evolution. Mammalogy papers: University of Nebraska State Museum, 26. | |

M-estimators can be constructed for location parameters and scale parameters in univariate and multivariate settings, as well as being used in robust regression. | |

This aspect should be carefully considered when interpretation of the results is a key point, such in the multivariate processing of chemical data (chemometrics). | |

Villani and Larsson propose to use either maximum likelihood method or bayesian estimation and provide some analytical results for either univariate and multivariate case. | |

PCA of a multivariate Gaussian distribution. The vectors shown are the first (longer vector) and second principal components, which indicate the directions of maximum variance. | |

Mulltivariate data analysis (6th ed.). Pearson.Hair, J. F., Tatham, R. L., Anderson, R. E., & Black, W. C. (1998). Multivariate data analysis (5th ed.). Prentice Hall. | |

Paolella, M. S. (2007) "Intermediate Probability - A Computational Approach". J. Wiley & Sons, New York [Ch. 2, sec. 2.2]Timm, N. H. (2002) "Applied Multivariate Analysis". | |

This is equivalent to a search problem in an almost certainly, multidimensional (multivariate), multi-modal space with a single (or weighted) objective or multiple objectives. | |

A method to visualize and utilize complex multivariate data is needed, with the ultimate goal of identifying predictive patterns to protocolize and guide medical care. | |

In statistics, the graphical lasso is a sparse penalized maximum likelihood estimator for the concentration or precision matrix (inverse of covariance matrix) of a multivariate elliptical distribution. The original variant was formulated to solve Dempster's covariance selection problem for the multivariate Gaussian distribution when observations were limited. Subsequently, the optimization algorithms to solve this problem were improved and extended to other types of estimators and distributions. | |

In other words, the problem is an exercise in multivariate analysis rather than the univariate approach of most of the traditional methods of estimating missing values and outliers; a multivariate model will therefore be more representative than a univariate one for predicting missing values. The Kohonen self organising map (KSOM) offers a simple and robust multivariate model for data analysis, thus providing good possibilities to estimate missing values, taking into account its relationship or correlation with other pertinent variables in the data record. Standard Kalman filters are not robust to outliers. To this end have recently shown that a modification of Masreliez's theorem can deal with outliers. | |

Similar pdfs can be constructed for other variables in the family tree shown above, simply by placing an M in front of each pdf name and finding the appropriate limiting and special cases of the MGB as indicated by the constraints and limits of the univariate distribution. Additional multivariate pdfs in the literature include the Dirichlet distribution (standard form) given by MGB1(y; a = 1, b = 1, p, q), the multivariate inverted beta and inverted Dirichlet (Dirichlet type 2) distribution given by MGB2(y; a = 1, b = 1, p, q), and the multivariate Burr distribution given by MGB2(y; a, b, p, q = 1). | |

Due to his work with the American Psychological Association (APA) in raising awareness for ethnic minority issues in academia (Tanaka was an ethnic minority himself), the APA named their Jeffrey S. Tanaka Dissertation Award in his memory. In 1993, the Journal of Personality started to run a series of papers titled The Jeffrey S. Tanaka Occasional Papers in Quantitative Methods for Personality in Tanaka's memory. As of 2011, papers were still being written for the series. In 1994, the Society of Multivariate Experimental Psychology introduced the Tanaka Award for Best Article in Multivariate Behavioral Research, given annually to the authors of the most outstanding paper in the Multivariate Behavioral Research journal. | |

Ansiedad y Estrés (Anxiety and Stress), 3, i-ii.Boyle, G.J. (2000). Obituaries: Raymond B. Cattell and Hans J. Eysenck. Multivariate Experimental Clinical Research, 12, i-vi. | |

Experimentation-based LPO can be achieved using A/B testing, multivariate LPO, and total-experience testing. These methodologies are applicable to both closed- and open-ended experimentation. | |

Throughout her academic career, Hamaker has served as Associate Editor of Multivariate Behavioral Research. In 2019 she was appointed a fellow of the Association for Psychological Science. | |

In Bayesian statistics, in the context of the multivariate normal distribution, the Wishart distribution is the conjugate prior to the precision matrix , where is the covariance matrix. | |

More than two sets of data leads to multivariate mapping. For example, a single map might show population density in addition to annual rainfall and cancer rates. | |

Certain deterministic recursive multivariate models which include threshold effects have been shown to produce fractal effects.Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach, OUP. | |

Table of height and weight for boys over time. The growth curve model (also known as GMANOVA) is used to analyze data such as this, where multiple observations are made on collections of individuals over time. The growth curve model in statistics is a specific multivariate linear model, also known as GMANOVA (Generalized Multivariate Analysis-Of-Variance). It generalizes MANOVA by allowing post-matrices, as seen in the definition. | |

Quantitative psychology is served by several scientific organizations. These include the Psychometric Society, Division 5 of the American Psychological Association (Evaluation, Measurement and Statistics), the Society of Multivariate Experimental Psychology, and the European Society for Methodology. Associated disciplines include statistics, mathematics, educational measurement, educational statistics, sociology, and political science. Several scholarly journals reflect the efforts of scientists in these areas, notably Psychometrika, Multivariate Behavioral Research, Structural Equation Modeling and Psychological Methods. | |

Astrostatistics is a discipline which spans astrophysics, statistical analysis and data mining. Astrostatistics and Astroinformatics Portal It is used to process the vast amount of data produced by automated scanning of the cosmos, to characterize complex datasets, and to link astronomical data to astrophysical theory. Many branches of statistics are involved in astronomical analysis including nonparametrics, multivariate regression and multivariate classification, time series analysis, and especially Bayesian inference. | |

The GRBD has a real-number response. For vector responses, multivariate analysis considers similar two-way models with main effects and with interactions or errors. Without replicates, error terms are confounded with interaction, and only error is estimated. With replicates, interaction can be tested with the multivariate analysis of variance and coefficients in the linear model can be estimated without bias and with minimum variance (by using the least-squares method). | |

Moshe Shaked was a Fellow of the Institute of Mathematical Statistics. Shaked was a leading figure in stochastic order and distribution theory. He published widely in applied probability and statistics. He became most celebrated internationally for his collection of influential papers on stochastic order and multivariate dependence. Shaked’s contribution also includes pioneering studies on stochastic convexity and on multivariate phase-type distributions, with important applications in reliability modelling and queueing analysis. | |

Glymour, Clark; Scheines, Richard; Spirtes, Peter; Kelly, Kevin. "TETRAD: Discovering Causal Structure" Multivariate Behavioral Research 23.2 (1988). 10 July 2010. . . Glymour earned undergraduate degrees in chemistry and philosophy. | |

CoCoA (COmputations in COmmutative Algebra) is open-source software used for computing multivariate polynomials and initiated in 1987. Originally written in Pascal, CoCoA was later translated into C. | |

In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of characteristic set. | |

A multivariate optical element (MOE), is the key part of a multivariate optical computer; an alternative to conventional spectrometry for the chemical analysis of materials. It is helpful to understand how light is processed in a multivariate optical computer, as compared to how it is processed in a spectrometer. For example, if we are studying the composition of a powder mixture using diffuse reflectance, a suitable light source is directed at the powder mixture and light is collected, usually with a lens, after it has scattered from the powder surface. Light entering a spectrometer first strikes a device (either a grating or interferometer) that separates light of different wavelengths to be measured. | |

McGue, M.; Bouchard Jr., T. J.; Iacono, W. G. and Lykken, D. T. (1993) "Behavioral Genetics of Cognitive Ability: A Life-Span Perspective", in Nature, Nurture, and Psychology, by R. Plomin & G. E. McClearn (Eds.) Washington, DC: American Psychological Association Multivariate genetic analysis examines the genetic contribution to several traits that vary together. For example, multivariate genetic analysis has demonstrated that the genetic determinants of all specific cognitive abilities (e.g., memory, spatial reasoning, processing speed) overlap greatly, such that the genes associated with any specific cognitive ability will affect all others. Similarly, multivariate genetic analysis has found that genes that affect scholastic achievement completely overlap with the genes that affect cognitive ability. | |

Local Characteristics tables. Multivariate statistics, comprising a combination of; age, sex, resident type, ethnic group, economic activity, general health and provision of unpaid care, country of birth, and occupation. | |

Determination of the mineral composition of Caigua (Cyclanthera pedata) and evaluation using multivariate analysis. Food Chemistry, 152, 619–623. Fruit flavor might be similar to cucumber or otherwise tasteless. | |

In 1949 he accepted the second chair of statistics at the London School of Economics, University of London. Here he worked part-time as the director of the new Research Techniques Division. From 1952 to 1957 he edited a two-volume work on Statistical Sources in the United Kingdom, which was a standard reference until the mid-1970s. In the 1950s he also worked on multivariate analysis, and developed the text Multivariate Analysis in 1957. | |

An open source Matlab implementation is freely available at the authors web page. Kumal et al. extended the algorithm to incorporate local invariances to multivariate polynomial transformations and improved regularization. | |

In multivariate analysis, DDE but not PCB was linked with a lowered age of menarche. Limitations of the study included indirect measurement of the exposure and self reporting of menarche. | |

In multivariate calculus, a differential is said to be exact or perfect, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q. | |

Multivariate landing page optimization is based on experimental design (e.g., discrete choice, conjoint analysis, Taguchi methods, IDDEA, etc.), which tests a structured combination of webpage elements. Some vendors (e.g., Memetrics. | |

Danilo Mandic from the Imperial College London, London, UK was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2013 for contributions to multivariate and nonlinear learning systems. | |

Reiss, I.L., Anderson, R.E., & Sponaugle, G.C. (1980). A Multivariate Model of the Determinants of Extramarital Sexual Permissiveness. Journal of Marriage and the Family, 42, 395–411.Weis, D.L., & Jurich, J. (1985). | |

The Wishart distribution is the sampling distribution of the maximum-likelihood estimator (MLE) of the covariance matrix of a multivariate normal distribution. A derivation of the MLE uses the spectral theorem. | |

In statistics, particularly in hypothesis testing, the Hotelling's T-squared distribution (T2), proposed by Harold Hotelling, is a multivariate probability distribution that is tightly related to the F-distribution and is most notable for arising as the distribution of a set of sample statistics that are natural generalizations of the statistics underlying the Student's t-distribution. The Hotelling's t-squared statistic (t2) is a generalization of Student's t-statistic that is used in multivariate hypothesis testing. | |

For instance, a multivariate Thurstonian model for preferential choice with a general variance-covariance structure is discussed in chapter 6 of Ennis (2016) that was based on papers published in 1993 and 1994. Even earlier, a closed form for a Thurstonian multivariate model of similarity with arbitrary covariance matrices was published in 1988 as discussed in Chapter 7 of Ennis (2016). This model has numerous applications and is not limited to any particular number of items or individuals. | |

A distribution-free multivariate Kolmogorov–Smirnov goodness of fit test has been proposed by Justel, Peña and Zamar (1997). The test uses a statistic which is built using Rosenblatt's transformation, and an algorithm is developed to compute it in the bivariate case. An approximate test that can be easily computed in any dimension is also presented. The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to multivariate data. | |

SMEP was founded in 1960 by Raymond Cattell and others as an organization of scientific researchers interested in applying complex multivariate quantitative methods to substantive problems in psychology. The two main functions of the society are to hold an annual meeting of scientific or quantitative psychology specialists and to publish a journal, Multivariate Behavioral Research. The first meeting of the Society was held in Chicago in the fall of 1961. Beginning in 1993, the meeting has been held annually. | |

The concept of reduction, also called multivariate division or normal form computation, is central to Gröbner basis theory. It is a multivariate generalization of the Euclidean division of univariate polynomials. In this section we suppose a fixed monomial ordering, which will not be defined explicitly. Given two polynomials f and g, one says that f is reducible by g if some monomial m in f is a multiple of the leading monomial lm(g) of g. | |

Brushing dependent variables and watching the connection to other dependent variables is called multivariate analysis. This could for example be used to find out if high temperatures are correlated with pressure by brushing high temperatures and watching a linked view of pressure distributions. Since each of the linked views usually has two or more dimensions, multivariate analysis can implicitly uncover higher-dimensional features of the data which would not be readily apparent from e.g. a simple scatterplot. | |

The measures based on Granger causality principle are: Granger Causality Index (GCI), Directed Transfer Function (DTF) and Partial Directed Coherence (PDC). These measures are defined in the framework of Multivariate Autoregressive Model. | |

An additional advantage of the method in comparison to other multivariate methods is that it gives a quantification of the treatment response of individual species that are present in the different groups. | |

Guerra-Garcia, J. M., Gonzalez-Vila, F. J. & Garcia-Gomez, J. C. Aliphatic hydrocarbon pollution and macrobenthic assemblages in Ceuta harbour: a multivariate approach. Marine Ecology Progress Series 263, 127-138 (2003). | |

Recently, a version of the algorithm that accounts for continuous and multivariate outputs was proposed with applications in cellular signaling . There exists also a version of Blahut–Arimoto algorithm for directed information. | |

Belcher, J., P.A. Keddy and P.F.C. Catling. (1992). "Alvar vegetation in Canada: a multivariate description at two scales." Canadian Journal of Botany 70: 1279-1291.Belcher, J. W. and P. A. Keddy. (1992). | |

This identify is useful in developing a Bayes estimator for multivariate Gaussian distributions. The identity also finds applications in random matrix theory by relating determinants of large matrices to determinants of smaller ones. | |

In the mathematical fields of numerical analysis and approximation theory, box splines are piecewise polynomial functions of several variables. Box splines are considered as a multivariate generalization of basis splines (B-splines) and are generally used for multivariate approximation/interpolation. Geometrically, a box spline is the shadow (X-ray) of a hypercube projected down to a lower-dimensional space. Box splines and simplex splines are well studied special cases of polyhedral splines which are defined as shadows of general polytopes. | |

Multi-channel, Multivariate SSA (or M-SSA) is a natural extension of SSA to for analyzing multivariate time series, where the size of different univariate series does not have to be the same. The trajectory matrix of multi-channel time series consists of linked trajectory matrices of separate times series. The rest of the algorithm is the same as in the univariate case. System of series can be forecasted analogously to SSA recurrent and vector algorithms (Golyandina and Stepanov, 2005). | |

There is no universal answer to this question. Another issue in the multivariate case is that the limiting model is not as fully prescribed as in the univariate case. In the univariate case, the model (GEV distribution) contains three parameters whose values are not predicted by the theory and must be obtained by fitting the distribution to the data. In the multivariate case, the model not only contains unknown parameters, but also a function whose exact form is not prescribed by the theory. | |

One of the most commonly reported effect size statistics for rANOVA is partial eta- squared (ηp2). It is also common to use the multivariate η2 when the assumption of sphericity has been violated, and the multivariate test statistic is reported. A third effect size statistic that is reported is the generalized η2, which is comparable to ηp2 in a one-way repeated measures ANOVA. It has been shown to be a better estimate of effect size with other within-subjects tests. | |

In practical applications, Gaussian process models are often evaluated on a grid leading to multivariate normal distributions. Using these models for prediction or parameter estimation using maximum likelihood requires evaluating a multivariate Gaussian density, which involves calculating the determinant and the inverse of the covariance matrix. Both of these operations have cubic computational complexity which means that even for grids of modest sizes, both operations can have a prohibitive computational cost. This drawback led to the development of multiple approximation methods. | |

Copulas are multivariate distributions with uniform univariate margins. Representing a joint distribution as univariate margins plus copulas allows the separation of the problems of estimating univariate distributions from the problems of estimating dependence. This is handy in as much as univariate distributions in many cases can be adequately estimated from data, whereas dependence information is rough known, involving summary indicators and judgment. Although the number of parametric multivariate copula families with flexible dependence is limited, there are many parametric families of bivariate copulas. | |

Haym Benaroya, Seon Mi Han, and Mark Nagurka. Probability Models in Engineering and Science. CRC Press, 2005. It is possible to generalize this to functions of more than one variable using multivariate Taylor expansions. | |

Some studiesHerremans, M. (1990). Taxonomy and evolution in Redpolls Carduelis flammea – hornemanni; a multivariate study of their biometry. Ardea 78(3): 441–458. HTML abstract favour three species, but this is certainly not definite. | |

In general, total sum of squares = explained sum of squares + residual sum of squares. For a proof of this in the multivariate ordinary least squares (OLS) case, see partitioning in the general OLS model. | |

Noble, in addition to her work on the wine aroma wheel, also did research on multivariate statistics of sensory data and its applications. She also published over 150 research papers in her time there. | |

For 3 variables, Brenner et al. applied multivariate mutual information to neural coding and called its negativity "synergy" and Watkinson et al. applied it to genetic expression . For arbitrary k variables, Tapia et al. | |

In multivariate statistics, Wold contributed the methods of partial least squares (PLS) and graphical models. Wold's work on causal inference from observational studies was decades ahead of its time, according to Judea Pearl.Causality, 2nd ed. | |

He is noted for his contributions to Marketing Research and Multivariate Data Analysis. In 2018 Clarivate Analytics recognized Dr. Hair as part of the top 1% of all Business and Economics professors in the world. | |

Dalal, R. S. & Hulin, C. L. (2008). Motivation for what? A multivariate dynamic perspective of the criterion. In R. Kanfer, G. Chen, & R. D. Pritchard (Eds.), Work motivation: Past, present, and future (pp. 63-100). | |

Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one. | |

Claridge et al.:404. It differs from the morphological species concept in including a numerical measure of distance or similarity to cluster entities based on multivariate comparisons of a reasonably large number of phenotypic traits. | |

If we make a vector of the values of f at N points, x1, ..., xN, in the D-dimensional space, then the vector (f(x1), ..., f(xN)) will always be distributed as a multivariate Gaussian. | |

In numerical analysis, multivariate interpolation or spatial interpolation is interpolation on functions of more than one variable. The function to be interpolated is known at given points (x_i, y_i, z_i, \dots) and the interpolation problem consist of yielding values at arbitrary points (x,y,z,\dots). Multivariate interpolation is particularly important in geostatistics, where it is used to create a digital elevation model from a set of points on the Earth's surface (for example, spot heights in a topographic survey or depths in a hydrographic survey). | |

There are a number of methods with distinct names and uses that share a common relationship. Cluster analysis is, like LCA, used to discover taxon-like groups of cases in data. Multivariate mixture estimation (MME) is applicable to continuous data, and assumes that such data arise from a mixture of distributions: imagine a set of heights arising from a mixture of men and women. If a multivariate mixture estimation is constrained so that measures must be uncorrelated within each distribution it is termed latent profile analysis. | |

Univariate functions can be applied point-wise to multivariate data to modify their marginal distributions. It is also possible to modify some attributes of a multivariate distribution using an appropriately constructed transformation. For example, when working with time series and other types of sequential data, it is common to difference the data to improve stationarity. If data generated by a random vector X are observed as vectors Xi of observations with covariance matrix Σ, a linear transformation can be used to decorrelate the data. | |

Multivariate optical computing allows instruments to be made with the mathematics of pattern recognition designed directly into an optical computer, which extracts information from light without recording a spectrum. This makes it possible to achieve the speed, dependability, and ruggedness necessary for real time, in-line process control instruments. Multivariate optical computing encodes an analog optical regression vector of a transmission function for an optical element. Light which emanates from a sample contains the spectral information of that sample, whether the spectrum is discovered or not. | |

In the same period began the algebraization of the algebraic geometry through commutative algebra. The prominent results in this direction are Hilbert's basis theorem and Hilbert's Nullstellensatz, which are the basis of the connexion between algebraic geometry and commutative algebra, and Macaulay's multivariate resultant, which is the basis of elimination theory. Probably because of the size of the computation which is implied by multivariate resultants, elimination theory was forgotten during the middle of the 20th century until it was renewed by singularity theory and computational algebraic geometry. | |

In statistics and econometrics, the multivariate probit model is a generalization of the probit model used to estimate several correlated binary outcomes jointly. For example, if it is believed that the decisions of sending at least one child to public school and that of voting in favor of a school budget are correlated (both decisions are binary), then the multivariate probit model would be appropriate for jointly predicting these two choices on an individual-specific basis. This approach was initially developed by Siddhartha Chib and Edward Greenberg. | |

Permutational multivariate analysis of variance (PERMANOVA), is a non- parametric multivariate statistical test. PERMANOVA is used to compare groups of objects and test the null hypothesis that the centroids and dispersion of the groups as defined by measure space are equivalent for all groups. A rejection of the null hypothesis means that either the centroid and/or the spread of the objects is different between the groups. Hence the test is based on the prior calculation of the distance between any two objects included to the experiment. | |

In multivariate statistics, if \varepsilon is a vector of n random variables, and \Lambda is an n-dimensional symmetric matrix, then the scalar quantity \varepsilon^T\Lambda\varepsilon is known as a quadratic form in \varepsilon. | |

From September 2015 to May 2019, he was editor in chief of the Journal of Multivariate Analysis. His many contributions earned him the Distinguished Service Award from the Statistical Society of Canada as early as 1997. | |

Pietrusewsky, Michael. (1992). Japan, Asia and the Pacific: A multivariate craniometric investigation. In book: Japanese as a member of the Asian and Pacific populations, Publisher: Kyoto: International Research Center for Japanese Studies. International Symposium No. 4. | |

This analysis (which also covers several multivariate public-key schemes as well as the QUAD stream cipher) studied in part the impact of changing the size of the field on the performances without considering the security aspect. | |

Other multivariate extension is 2D-SSA that can be applied to two-dimensional data like digital images (Golyandina and Usevich, 2010). The analogue of trajectory matrix is constructed by moving 2D windows of size L_x \times L_y. | |

Collocation methods for the solution of differential and integral equations are based on polynomial interpolation. The technique of rational function modeling is a generalization that considers ratios of polynomial functions. At last, multivariate interpolation for higher dimensions. | |

Computed (calculated) ABC analysis delivers a precise mathematical calculation of the limits for the ABC classes.Ultsch, Alfred, Jörn Lötsch. "Computed ABC analysis for rational selection of most informative variables in multivariate data." PLOS One 10.6 (2015): e0129767. | |

Many research areas see increasingly large numbers of variables in only few samples. The low sample to variable ratio creates problems known as multicollinearity and singularity. Because of this, most traditional multivariate statistical methods cannot be applied. | |

Gaussian processes are thus useful as a powerful non- linear multivariate interpolation tool. Gaussian process regression can be further extended to address learning tasks in both supervised (e.g. probabilistic classification) and unsupervised (e.g. manifold learning) learning frameworks. | |

The multivariate generalization of the split normal distribution was proposed by Villani and Larsson. They assume that each of the principal components has univariate split normal distribution with a different set of parameters μ, σ2 and σ1. | |

Appearance event ordination or AEO is a scientific method for biochronology through the ordering of the appearance of fossil mammal genera by multivariate analysis, using conjunctional (overlapping) and disconjunctional (nonoverlapping) range distributions in large sets of data. | |

Multiple or multivariate regression is an approach to look at the relationship between several independent or predictor variables and a dependent or influential variable. It is best used in geometric morphometrics when analyzing shape variables based on an external influence. For example, it can be used in studies with attached functional or environmental variables like age or the development over time in certain environments. The multivariate regression of shape based on the logarithm of centroid size (square root of the sum of squared distances of landmarks) is ideal for allometric studies. | |

ROCCET’s ROC curve generation and analysis is specifically tailored for metabolomics datasets. Metabolomics data sets produced by high throughput analytical chemistry techniques typically consist of large matrices containing multiple values for multiple samples. The comparison between groups or subsets of samples within the data usually involves statistical procedures employing univariate analysis and multivariate analysis such as Partial Least Squares - Discriminant Analysis (PLS-DA) or machine learning classification procedures such as Support Vector Machine (SVM). As a result, ROCCET offers two different kinds of analytical modules – a univariate module and a multivariate module. | |

Second, linear correlation measures are only natural dependence measures if the joint distribution of the variables is elliptical. However, only few financial distributions such as the multivariate normal distribution and the multivariate student-t distribution are special cases of elliptical distributions, for which the linear correlation measure can be meaningfully interpreted. Third, a zero Pearson product-moment correlation coefficient does not necessarily mean independence, because only the two first moments are considered. For example, Y = X^2 (y ≠ 0) will lead to Pearson correlation coefficient of zero, which is arguably misleading. | |

Early work on statistical classification was undertaken by Fisher, in the context of two-group problems, leading to Fisher's linear discriminant function as the rule for assigning a group to a new observation.Gnanadesikan, R. (1977) Methods for Statistical Data Analysis of Multivariate Observations, Wiley. (p. 83-86) This early work assumed that data-values within each of the two groups had a multivariate normal distribution. The extension of this same context to more than two-groups has also been considered with a restriction imposed that the classification rule should be linear. | |

He published numerous papers Bahadur's CV hosted at University of Chicago and is best known for the concepts of "Bahadur efficiency" A paper about Bahadur efficiency and the Bahadur–Ghosh–Kiefer representation (with J. K. Ghosh and Jack Kiefer). He also framed the Anderson–Bahadur algorithmClassification into two multivariate normal distributions with different covariance matrices (1962), T W Anderson, R R Bahadur, Annals of Mathematical Statistics along with Theodore Wilbur Anderson which is used in statistics and engineering for solving binary classification problems when the underlying data have multivariate normal distributions with different covariance matrices. | |

Stephen Gano West (born 1946) is an American quantitative psychologist and professor of psychology at Arizona State University. He was the editor-in- chief of the Journal of Personality from 1986 to 1991, of Psychological Methods from 2001 to 2007, and of Multivariate Behavioral Research in 2015. He was also the president of the Society of Multivariate Experimental Psychology from 2007 to 2008. He was educated at Cornell University and the University of Texas at Austin, and received the Society for Personality and Social Psychology's Murray Award in 2000. | |

In 2008, Ali received the Qazi Motahar Husain Gold Medal Award in recognition of his contributions to statistics. Ali's research interests in statistics and mathematics included order statistics, distribution theory, characterizations, spherically symmetric and elliptically contoured distributions, multivariate statistics, and n-dimensional geometry. He published articles in well-known statistical journals, such as the Annals of Mathematical Statistics, the Journal of the Royal Statistical Society, the Journal of Multivariate Analysis, and Biometrika. Two of his most highly rated papers are in geometry, and appeared in the Pacific Journal of Mathematics. | |

H(x,y) is everything but green). The joint entropy H(x,y,z) of all three variables is the union of all three circles. It is partitioned into 7 pieces, red, blue, and green being the conditional entropies H(xy,z), H(yx,z), H(zx,y) respectively, yellow, magenta and cyan being the conditional mutual informations I(x;zy), I(y;zx) and I(x;yz) respectively, and gray being the multivariate mutual information I(x;y;z). The multivariate mutual information is the only one of all that may be negative. | |

Over the years, researchers of ISI made fundamental contributions in various fields of Statistics such as Design of Experiments, Sample Survey, Multivariate statistics and Computer Science. Mahalanobis introduced the measure Mahalanobis distance which is used in multivariate statistics and other related fields. Raj Chandra Bose, who is known for his contributions in coding theory, worked on Design of Experiments during his tenure at ISI, and was one of the three mathematicians, who disproved Euler's conjecture on orthogonal Latin squares. Anil Kumar Bhattacharya is credited with introduction of the measures Bhattacharyya distance and Bhattacharya coefficient. | |

The topic of multilinear algebra is applied in some studies of multivariate calculus and manifolds where the Jacobian matrix comes into play. The infinitesimal differentials of single variable calculus become differential forms in multivariate calculus, and their manipulation is done with exterior algebra. After Grassmann, developments in multilinear algebra were made in 1872 by Victor Schlegel when he published the first part of his System der Raumlehre, and by Elwin Bruno Christoffel. A major advance in multilinear algebra came in the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita (see references). | |

Joseph Lee Rodgers III (born February 9, 1953) is an American psychologist and the Lois Autrey Betts Professor of Psychology and Human Development at Vanderbilt University. He is also the George Lynn Cross Research Professor Emeritus at the University of Oklahoma, where he taught from 1981 to 2012. He is a past president of the Society for the Study of Social Biology, the Society of Multivariate Experimental Psychology, and Divisions 5 and 34 of the American Psychological Association. From 2006 to 2011, he was editor-in-chief of Multivariate Behavioral Research. | |

Goldstein et al., in 2014, studied 439,457 US paediatric patients who underwent a range of surgical procedures. After multivariate adjustment and regression, patients undergoing a weekend procedure were more likely to die (OR = 1.63; 95% CI 1.21-2.20). | |

Johanna G. Nešlehová is a Czech mathematical statistician who works in Canada at McGill University as a professor in the department of mathematics and statistics. Her research interests include copulas, extreme value theory, multivariate statistics, and operational risk. | |

Testing hypothesis of convergence with multivariate data: Morphological and functional convergence among herbivorous lizards. Evolution 60:824–841. Additionally, herbivorous lizards often possess a fleshy tongue, which is used to manipulate food in the mouth.Throckmorton, G. S. 1976. | |

The vegetation of exposed sea cliffs at South Stack, Anglesey: I. The multivariate approach. Journal of Ecology 61(3): 787–818.Goldsmith, F. B. 1973b. The vegetation of exposed sea cliffs at South Stack, Anglesey: II. Experimental studies. | |

Radar charts are helpful for small-to-moderate-sized multivariate data sets. Their primary weakness is that their effectiveness is limited to data sets with less than a few hundred points. After that, they tend to be overwhelming. | |

Dynamic neural networks address nonlinear multivariate behaviour and include (learning of) time-dependent behaviour, such as transient phenomena and delay effects. Techniques to estimate a system process from observed data fall under the general category of system identification. | |

The multivariate generalization of the continuous Bernoulli is called the continuous categorical.Gordon-Rodriguez, E., Loaiza- Ganem, G., & Cunningham, J. P. (2020). The continuous categorical: a novel simplex-valued exponential family. In 36th International Conference on Machine Learning, ICML 2020. | |

Tanagra makes a good compromise between statistical approaches (e.g. parametric and nonparametric statistical tests), multivariate analysis methods (e.g. factor analysis, correspondence analysis, cluster analysis, regression) and machine learning techniques (e.g. neural network, support vector machine, decision trees, random forest). | |

"grounded theory is multivariate. It happens sequentially, subsequently, simultaneously, serendipitously, and scheduled" (Glaser, 1998). Open coding or substantive coding is conceptualizing on the first level of abstraction. Written data from field notes or transcripts are conceptualized line by line. | |

New York: ACM Search analytics includes search volume trends and analysis, reverse searching (entering websites to see their keywords), keyword monitoring, search result and advertisement history, advertisement spending statistics, website comparisons, affiliate marketing statistics, multivariate ad testing, et al. | |

Peter C. M. Molenaar (born 1946) is a Dutch developmental and mathematical psychologist who is Distinguished Professor of Human Development and Family Studies at Pennsylvania State University (Penn State). He is the editor-in- chief of Multivariate Behavioral Research. | |

Greig-Smith was among the first ecologists to understand that multivariate methods were destined to become important tools in quantitative plant ecology. His inclusion of multivariate classification and ordination in his book opened the door for the explosion of analytical approaches and methodological discussions that surged in the 1970s and 1980s. As with pattern analysis, Greig-Smith was ahead of his time: although the potential of multivariate methods—based on matrix algebra—was already recognized in the 1950s, mostly as a result of a growing need for more-rigorous methods in ecology, performing an analysis of a modestly sized floristic matrix of, say, fifty sites and one hundred species was a complicated and enormously time-consuming task. The generalized access to electronic computers, which began in the 1960s, opened the doors for the widespread use of these methods; Bangor’s first computer became available in 1964. | |

Firstly, if this was the case, it is unlikely that the effect would be seen in elective admissions. Secondly, in most of the papers described above, weekend vs weekday data has been studied by multivariate analysis (i.e. taking comorbidities into account). | |

"Using Benthic Macroinvertebrate and Fish Communities as Bioindicators of the Tanshui River Basin Around the Greater Taipei Area — Multivariate Analysis of Spatial Variation Related to Levels of Water Pollution". International Journal of Environmental Research and Public Health 11(7): 7116–7143. . | |

Research Goals # Determine crude outcome incidence rates among centres. # Examine variations in outcomes and practices among tertiary perinatal units, using staged multivariate logistic and linear regression analysis. # Associate obstetric practice differences with outcomes variation. # Compare crude measures of resource use. | |

Tapinoma schreiberi is a species of ant in the genus Tapinoma. Described by Hamm in 2010, the species is endemic to the United States.Hamm, C.A. 2010. Multivariate discrimination and description of a new species of Tapinoma from the western United States. | |

The Fourier amplitude sensitivity test (FAST) uses the Fourier series to represent a multivariate function (the model) in the frequency domain, using a single frequency variable. Therefore, the integrals required to calculate sensitivity indices become univariate, resulting in computational savings. | |

R package. Analysis of multivariate dichotomous and polytomous data using latent trait models under the Item Response Theory approach. It includes the Rasch, the Two-Parameter Logistic, the Birnbaum's Three-Parameter, the Graded Response, and the Generalized Partial Credit Models. | |

Although this method cannot elucidate the multivariate nature of background factors, it can gauge the effects they have on the time- series at a given point in time even without measuring them. This observation can be used to make a forecast. | |

Multiple Discriminant Analysis (MDA) is a multivariate dimensionality reduction technique. It has been used to predict signals as diverse as neural memory traces and corporate failure.Duda R, Hart P, Stork D (2001) Pattern Classification, Second Edition. New York, NY, Uand Sons. | |

Alexandra M. (Alex) Schmidt is a Brazilian biostatistician and epidemiologist who works as an associate professor of biostatistics at McGill University in Canada. She is known for her research on spatiotemporal and multivariate statistics and their applications in environmental statistics. | |

Overnight Bartlett worked out a proof using characteristic functions. Bartlett was Wishart's first post-graduate student and they wrote two papers together. This was the beginning of Bartlett's involvement with multivariate analysis. During his Queens years, he rowed for the college. | |

In univariate statistics, the Student's t-test makes use of Student's t-distribution. Hotelling's T-squared distribution is a distribution that arises in multivariate statistics. The matrix t-distribution is a distribution for random variables arranged in a matrix structure. | |

Univariate distribution is a dispersal type of a single random variable described either with a probability mass function (pmf) for discrete probability distribution, or probability density function (pdf) for continuous probability distribution. It is not to be confused with multivariate distribution. | |

To factorize the initial polynomial, it suffices to factorize each square-free factor. Square-free factorization is therefore the first step in most polynomial factorization algorithms. Yun's algorithm extends this to the multivariate case by considering a multivariate polynomial as a univariate polynomial over a polynomial ring. In the case of a polynomial over a finite field, Yun's algorithm applies only if the degree is smaller than the characteristic, because, otherwise, the derivative of a non-zero polynomial may be zero (over the field with p elements, the derivative of a polynomial in xp is always zero). | |

Multivariate analysis of covariance (MANCOVA) is an extension of analysis of covariance (ANCOVA) methods to cover cases where there is more than one dependent variable and where the control of concomitant continuous independent variables – covariates – is required. The most prominent benefit of the MANCOVA design over the simple MANOVA is the 'factoring out' of noise or error that has been introduced by the covariant. A commonly used multivariate version of the ANOVA F-statistic is Wilks' Lambda (Λ), which represents the ratio between the error variance (or covariance) and the effect variance (or covariance). Statsoft Textbook, ANOVA/MANOVA. | |

The extensive use of kernel smoothing and smoothing splines to ensure smoothness assumptions signify why functional data analysis is at its core a nonparametric statistical technique. Nevertheless, models for functional data and methods for their analysis may resemble those for conventional multivariate data, including linear and nonlinear regression models, principal components analysis among others; that is because functional data can be thought of as multivariate data with order on its dimensions. But the possibility of using derivative information greatly extends the power of these methods, and also leads to purely functional models such as those defined by differential equations, often called dynamical systems. | |

A recently growing way to analyze T-RFLP profiles is use multivariate statistical methods to interpret the T-RFLP data. Usually the methods applied are those commonly used in ecology and especially in the study of biodiversity. Among them ordinations and cluster analysis are the most widely used. In order to perform multivariate statistical analysis on T-RFLP data, the data must first be converted to table known as a “sample by species table“ which depicts the different samples (T-RFLP profiles) versus the species (T-RFS) with the height or area of the peaks as values. | |

Singular spectrum analysis applied to a time-series F, with reconstructed components grouped into trend, oscillations, and noise In time series analysis, singular spectrum analysis (SSA) is a nonparametric spectral estimation method. It combines elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. Its roots lie in the classical Karhunen (1946)–Loève (1945, 1978) spectral decomposition of time series and random fields and in the Mañé (1981)–Takens (1981) embedding theorem. SSA can be an aid in the decomposition of time series into a sum of components, each having a meaningful interpretation. | |

In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics. | |

He developed a multivariate statistical model based on ultrasound spectral parameters to differentiate metastatic carcinoma, and two subtypes of uveal malignant melanoma. Publication of these findings in 1983Coleman DJ, Lizzi FL, Silverman RH,, Rondeau MJ, Smith ME, Torpey JH. Acoustic biopsy as a means for characterization of intraocular tumors. American Academy of Ophthalmology, Acta: XXIV International Congress of Ophthalmology, edited by Paul Henkind, MD, J.B. Lippincott Company, Philadelphia, PA, 1983, pp. 115-118. represented one of the first reports in the literature of medical diagnosis based on multivariate statistical analysis and one of the earliest applications of ultrasound tissue characterization. | |

A machine-learning algorithm that involves a Gaussian process uses lazy learning and a measure of the similarity between points (the kernel function) to predict the value for an unseen point from training data. The prediction is not just an estimate for that point, but also has uncertainty information—it is a one-dimensional Gaussian distribution. For multi-output predictions, multivariate Gaussian processes are used, for which the multivariate Gaussian distribution is the marginal distribution at each point. For some kernel functions, matrix algebra can be used to calculate the predictions using the technique of kriging. | |

In the case where the sum of the mutual information is less than I(X,Y;Z), the multivariate mutual information will be negative. In this case, knowing both X and Y together provides more information about Z than the sum of the information yielded by knowing either one alone. That is to say, there is a "synergy" in the information about Z provided by the X and Y variables. The above explanation is intended to give an intuitive understanding of the multivariate mutual information, but it obscures the fact that it does not depend upon which variable is the subject (e.g. | |

After receiving his PhD, Millsap taught industrial/organizational psychology at Baruch College until 1997, where he eventually became a full professor. In 1997, he joined the quantative psychology faculty of Arizona State University, where he taught until his death in 2014. He served as editor-in-chief of Multivariate Behavioral Research from 1996 to 2006, and of Psychometrika from 2007 until his death. He also served as president of the Society for Multivariate Experimental Psychology in 2001–2002, of Division 5 of the American Psychological Association in 2004–2005, and of the Psychometric Society in 2006–2007. | |

Molenaar received two bachelor's degrees from the University of Utrecht: one in 1972 and one in 1976. He also received two master's degrees from the University of Utrecht in 1976, one in mathematical psychology and one in psychophysiology, before earning his Ph.D. in Social Sciences from the same university in 1981. He then served on the faculty of the University of Amsterdam, where he eventually became head of the Department of Methodology, before joining the faculty of Penn State in 2005. In 2013, he received the Sells Award for Distinguished Multivariate Research from the Society of Multivariate Experimental Psychology. | |

Johannes Müller, Martin Hinz and Markus Ullrich, "Bell Beakers – chronology, innovation and memory: a multivariate approach", chapter 6 in The Bell Beaker Transition in Europe: Mobility and local evolution during the 3rd millennium BC, eds. Maria Pilar Prieto Martinez and Laure Salanova (2015). | |

They are nocturnal, migratory and colonial. Hunting starts early in the morning where they leave their roost and begin preying on larger insects than other bats.Freeman, P. March 31, 1981. A Multivariate Study of the Family Molossidae (Mammalia, Chiroptera): Morphology, Ecology, Evolution. | |

The probability that all balls have the same color is easier to calculate. See the formula below under multivariate distribution. No exact formula for the mean is known (short of complete enumeration of all probabilities). The equation given above is reasonably accurate. | |

For empirical research, such patterns correspond to a multivariate combination of independent 'consciousness factors', which can be quantified via questionnaires. The 'phenomenological pattern' results from the factor structure of the applied psychometric assessment, i.e. the individual ratings, or factor scores, of a questionnaire. | |

Presented at the SPE Kuwait International Petroleum Conference and Exhibition, The technique has been applied to Raman spectroscopy, fluorescence spectroscopy,Priore, R.J. (2013). "OPTICS FOR BIOPHOTONICS: Multivariate optical elements beat bandpass filters in fluorescence analysis". Laser Focus World. 49 (6): 49–52. | |

References J. B. Carroll (1993), Human cognitive abilities: A survey of factor-analytic studies, Cambridge University Press, New York, NY, USA. Horn, J. L. (1988). Thinking about human abilities. In J. R. Nesselroade & R. B. Cattell (Eds.), Handbook of multivariate experimental psychology. | |

GGobi is a free statistical software tool for interactive data visualization. GGobi allows extensive exploration of the data with Interactive dynamic graphics. It is also a tool for looking at multivariate data. R can be used in sync with GGobi (through rggobi). | |

2002: StatServer 6. Student edition of S-PLUS now free. 2003: S-PLUS 6.2 New reporting, database integration, improved Graphlets, ported to AIX, libraries for correlated data, Bayesian methods, multivariate regressions. 2004: Insightful purchases the S language from Lucent Technologies for $2 million. | |

Ordination can be used on the analysis of any set of multivariate objects. It is frequently used in several environmental or ecological sciences, particularly plant community ecology. It is also used in genetics and systems biology for microarray data analysis and in psychometrics. | |

These estimators can be applied to fMRI data, if the required image sequences are available. Among estimators of connectivity, there are linear and non-linear, bivariate and multivariate measures. Certain estimators also indicate directionality. Different methods of connectivity estimation vary in their effectiveness. | |

Decision tree learning algorithms can be applied to learn to predict a dependent variable from data. Although the original Classification And Regression Tree (CART) formulation applied only to predicting univariate data, the framework can be used to predict multivariate data, including time series. | |

Dyar, M.D., Speicher, E.A., Gunter, M.E., Lanzirotti, A., Tucker, J.M., Carey, CJ, Peel, S.A., Brown, E.B., Oberti, R., and Delaney, J.S. (2016) Use of multivariate analysis for synchrotron micro-XANES analysis of iron valence state in amphiboles. American Mineralogist, 101, 1171-1189. | |

Modern phytosociologists try to include higher levels of complexity in the perception of vegetation, namely by describing whole successional units (vegetation series) or, in general, vegetation complexes. Other developments include the use of multivariate statistics for the definition of syntaxa and their interpretation. | |

These include, e.g., time-series analysis using multiple regression, Box–Jenkins analysis, and seasonality analysis. Analysis may be univariate (modeling one series) or multivariate (from several series). Econometricians, economic statisticians, and financial analysts formulate models, whether for past relationships or for economic forecasting. | |

Normaliz also computes enumerative data, such as multiplicities (volumes) and Hilbert series. The kernel of Normaliz is a templated C++ class library. For multivariate polynomial arithmetic it uses CoCoALib. Normaliz has interfaces to several general computer algebra systems: CoCoA, GAP, Macaulay2 and Singular. | |

In the field of multivariate statistics, kernel principal component analysis (kernel PCA) is an extension of principal component analysis (PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are performed in a reproducing kernel Hilbert space. | |

Like univariate analysis, bivariate analysis can be descriptive or inferential. It is the analysis of the relationship between the two variables. Bivariate analysis is a simple (two variable) special case of multivariate analysis (where multiple relations between multiple variables are examined simultaneously). | |

McArdle was president of the Society of Multivariate Experimental Psychology from 1992 to 1993, and of the Federation of Behavioral, Psychological & Cognitive Sciences from 1996 to 1999. In 2012, he was elected as a fellow of the American Association for the Advancement of Science. | |

They found, on multivariate analysis, mortality was associated with patient age (10 years: OR = 1.31, p < 0.01), severity of illness (extreme: OR = 34.68, p < 0.01), insurance status (Medicaid: OR = 1.24, p < 0.01; uninsured: OR = 1.40, p < 0.01), and weekend admission (OR = 1.09, p = 0.04). | |

Latent class analysis (LCA) is a subset of structural equation modeling, used to find groups or subtypes of cases in multivariate categorical data. These subtypes are called "latent classes".Lazarsfeld, P.F. and Henry, N.W. (1968) Latent structure analysis. Boston: Houghton MifflinFormann, A. K. (1984). | |

If the random draws are with simple replacement (no balls over and above the observed ball are added to the urn), then the distribution follows a multinomial distribution and if the random draws are made without replacement, the distribution follows a multivariate hypergeometric distribution. | |

Many writers have considered the growth curve analysis, among them Wishart (1938), Box (1950) and Rao (1958). Potthoff and Roy in 1964;R.F. Potthoff and S.N. Roy, “A generalized multivariate analysis of variance model useful especially for growth curve problems,” Biometrika, vol. 51, pp. | |

A multivariate analysis done by the United States Sentencing Commission found that women of all races get much lighter sentencing than white male offenders. Other papers have confirmed the hypothesis that women get significantly more lenient sentences than men in the criminal justice system. | |

Box's M test is susceptible to errors if the data does not meet model assumptions or if the sample size is too large or small. Box's M test is especially prone to error if the data does not meet the assumption of multivariate normality. | |

Greig-Smith’s later papers on multivariate methods, with Mike SwaineSwaine, Michael D., and Peter Greig-Smith. 1980. An application of principal components analysis to vegetation change in permanent plots. Journal of Ecology 68(1): 33–41. and Carlos MontañaMontaña, Carlos, and Peter Greig-Smith. 1990. | |

Gerard Saucier is an American psychologist. He is a professor in the department of Psychology at the University of Oregon. He has co-authored many academic articles on personality. He won the 1999 Cattell Early Career Research Award from the Society of Multivariate Experimental Psychology. | |

The authors demonstrate that variables like year of diagnosis, higher education, sexual exposure category, gender, the presence of specific pathogens all appeared to predict survival in a univariate analysis; however, in a multivariate analysis only antiretroviral treatment, diagnostic criteria, and transmission category remained significant. | |

Maurice Stevenson Bartlett FRS (18 June 1910 – 8 January 2002) was an English statistician who made particular contributions to the analysis of data with spatial and temporal patterns. He is also known for his work in the theory of statistical inference and in multivariate analysis. | |

Taught research methods, statistics/computer use, and graduate statistical analysis courses—including bivariate and multivariate statistics. Responsible for managing and conducting wide variety of research and for analysis of data resulting from research projects. Served as a consultant on statistics, SPSSX computer programs and more. | |

In statistics, precision is the reciprocal of the variance, and the precision matrix (also known as concentration matrix) is the matrix inverse of the covariance matrix. Thus, if we are considering a single random variable in isolation, its precision is the inverse of its variance: p=1/σ². Some particular statistical models define the term precision differently. One particular use of the precision matrix is in the context of Bayesian analysis of the multivariate normal distribution: for example, Bernardo & Smith prefer to parameterise the multivariate normal distribution in terms of the precision matrix, rather than the covariance matrix, because of certain simplifications that then arise. | |

Thus, it can generally produce the same accuracy as laboratory grade spectroscopic systems, but with the fast speed inherent with a pure, passive, optical computer. The multivariate optical computer makes use of optical computing to realize the performance of a full spectroscopic system using traditional multivariate analysis. A side benefit is that the throughput and efficiency of the system is higher than conventional spectrometers, which increases the speed of analysis by orders of magnitude. While each chemical problem presents its own unique challenges and opportunities, the design of a system for a specific analysis is complex and requires the assembly of several pieces of a spectroscopic puzzle. | |

Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). Ordination orders objects that are characterized by values on multiple variables (multivariate objects) so that similar objects are near each other and dissimilar objects are farther from each other. Such relationships between the objects, on each of several axes (one for each variable), are then characterized numerically and/or graphically. Many ordination techniques exist, including principal components analysis (PCA), non-metric multidimensional scaling (NMDS), correspondence analysis (CA) and its derivatives (detrended CA (DCA), canonical CA (CCA)), Bray-Curtis ordination, and redundancy analysis (RDA), among others. | |

In order to estimate the edges between nodes, correlation coefficient at zero time lag between all possible pairs of nodes were estimated. A pair of nodes was considered to be connected, if their correlation coefficient is above a threshold of 0.5. The team of Havlin introduced the weighted links method which considers (i) the time delay of the link, (ii) the maximum of the cross- correlation at the time delay and (iii) the level of noise in the cross- correlation function. Steinhaeuser and team introduced the novel technique of multivariate networks in climate by constructing networks from several climate variables separately and capture their interaction in multivariate predictive model. | |

Many datasets include multiple measurements like time, space, demographic, phenotypic and functional recording. For instance, the annual US Housing Price Index dataset includes dozens of variable including location (State and US region), year, unemployment rate, state population, percent subprime loans, etc. Motion charts provide a dynamic data visualization paradigm that facilitates the representation and understanding of large and multivariate data. Using the familiar 2D Bubble charts, motion Charts enable the display of large multivariate data with thousands of data points and allow for interactive visualization of the data using additional dimensions like time, the size of the blobs, and color) to show different characteristics of the data. | |

Macaulay's resultant, named after Francis Sowerby Macaulay, also called the multivariate resultant, or the multipolynomial resultant,, Chapter 3. Resultants is a generalization of the homogeneous resultant to homogeneous polynomials in indeterminates. Macaulay's resultant is a polynomial in the coefficients of these homogeneous polynomials that vanishes if and only if the polynomials have a common non-zero solution in an algebraically closed field containing the coefficients, or, equivalently, if the hyper surfaces defined by the polynomials have a common zero in the dimensional projective space. The multivariate resultant is, with Gröbner bases, one of the main tools of effective elimination theory (elimination theory on computers). | |

The information given by a correlation coefficient is not enough to define the dependence structure between random variables. The correlation coefficient completely defines the dependence structure only in very particular cases, for example when the distribution is a multivariate normal distribution. (See diagram above.) In the case of elliptical distributions it characterizes the (hyper-)ellipses of equal density; however, it does not completely characterize the dependence structure (for example, a multivariate t-distribution's degrees of freedom determine the level of tail dependence). Distance correlation was introduced to address the deficiency of Pearson's correlation that it can be zero for dependent random variables; zero distance correlation implies independence. | |

Univariate analysis is perhaps the simplest form of statistical analysis. Like other forms of statistics, it can be inferential or descriptive. The key fact is that only one variable is involved. Univariate analysis can yield misleading results in cases in which multivariate analysis is more appropriate. | |

More recently, multivariate methods have been proposed that derive ICP by combining the transit times with measured acoustic impedance, resonant frequency and ultrasound velocity,6\. Bridger et al. US5919144 (1999). or with a dispersion of the ultrasound wave on its way through the brain parenchyma.7\. | |

This method adapts the univariate algorithm to the multivariate case by substituting a hyperrectangle for the one-dimensional w region used in the original. The hyperrectangle H is initialized to a random position over the slice. H is then shrunk as points from it are rejected. | |

At the end of his career, Wold turned away from econometric modelling and developed multivariate techniques for what he called "soft" modelling. Some of these methods were developed through interactions with his student K. G. Jöreskog, although the latter's focus was primarily on maximum likelihood methods. | |

The term has been used in the books of Hair and his colleagues,Hair, J. F., Anderson, R. E., Tatham, R. L., & Black, W. C. (1995). Multivariate data analysis with readings (4th ed.). Prentice Hall.Hair, J. F., Babin, B. J., Anderson, R. E., & Black, W. C. (2018). | |

If X and Y are normally distributed and independent, this implies they are "jointly normally distributed", i.e., the pair (X,Y) must have multivariate normal distribution. However, a pair of jointly normally distributed variables need not be independent (would only be so if uncorrelated, \rho = 0 ). | |

In marketing, geodemographic segmentation is a multivariate statistical classification technique for discovering whether the individuals of a population fall into different groups by making quantitative comparisons of multiple characteristics with the assumption that the differences within any group should be less than the differences between groups. | |

A MAR model is indexed by the nodes of a tree, whereas a standard (discrete time) autoregressive model is indexed by integers. Note that the ARMA model is a univariate model. Extensions for the multivariate case are the vector autoregression (VAR) and Vector Autoregression Moving-Average (VARMA). | |

For example, if you have data about a group of people, you might want to arrange their ages into a smaller number of age intervals (for example, grouping every five years together). It can also be used in multivariate statistics, binning in several dimensions at once. | |

And the weight of the competing balls depends on the outcomes of all preceding draws. A multivariate version of Wallenius' distribution is used if there are more than two different colors. The distribution of the balls that are not drawn is a complementary Wallenius' noncentral hypergeometric distribution. | |

Ada Dietz (1882 - 1950) was an American weaver best known for her 1949 monograph Algebraic Expressions in Handwoven Textiles, which defines weaving patterns based on the expansion of multivariate polynomials. used the Rule 90 cellular automaton to design tapestries depicting both trees and abstract patterns of triangles. | |

If the default mode network is altered, this can change the way one perceives events and their social and moral reasoning, thus making a person more susceptible to major depressive-like symptoms. Multivariate analysis reveals genetic associations of the resting DMN in psychotic bipolar disorder and schizophrenia. | |

J. Econ. Entomol. 101: 759–768.Lapointe, Stephen L.; Stelinski, Lukasz L.; Evens, Terence J.; Niedz, Randall P.; Hall, David G.; Mafra-Neto, Agenor. 2009. Sensory Imbalance as Mechanism of Orientation Disruption in the Leafminer Phyllocnistis citrella: Elucidation by Multivariate Geometric Designs and Response Surface Models. | |

Sometimes, topics from heterodox economics are introduced. Econometrics extends the undergraduate domain to multiple linear regression and multivariate time series, and introduces simultaneous equation methods and generalized linear models. Game theory and computational economics are often included. Some (doctoral) programs include core work in economic history. | |

These constrained optimization values not only reflect the naturally selected preferred gait parameters that are observed by fixing a single parameter at different values, but also form part of a predictive map that allows for the identification of the cost of transport for a multivariate system. | |

There is some evidence linking experiences of sexual abuse in childhood or adolescence with patterns of victimization during adulthood.Acierno R et al. Risk factors for rape, physical assault, and post- traumatic stress disorder in women: examination of differential multivariate relationships. Journal of Anxiety Disorders, 1999, 13:541–563. | |

Also, box splines can be used to compute the volume of polytopes. In the context of multidimensional signal processing, box splines can provide multivariate interpolation kernels (reconstruction filters) tailored to non- Cartesian sampling lattices,Entezari, Alireza. Optimal sampling lattices and trivariate box splines. [Vancouver, BC.]: Simon Fraser University, 2007. . | |

They found through multivariate analysis, that weekend admission (OR = 1.53; 95% CI 1.26-1.86; p<0.001) was a significant predictor of inhospital mortality. In 2015, Tadisina et al. carried out the only study so far published on plastic surgery, investigating 50,346 US patients who had body contouring procedures. | |

Steinhaeuser et al. applied complex networks to explore the multivariate and multi-scale dependence in climate data. Findings of the group suggested a close similarity of observed dependence patterns in multiple variables over multiple time and spatial scales. Tsonis and Roeber investigated the coupling architecture of the climate network. | |

In July 2015, Royen supplemented his proof with a further paper in arXiv Some probability inequalities for multivariate gamma and normal distributions. On March 28, 2017 Natalie Wolchover of Quanta Magazine published a story about Royen's proof, after which he gained more academic and public recognition for his achievement. | |

Many of the methods of classical multivariate analysis turn out to be special cases of projection pursuit. Examples are principal component analysis and discriminant analysis, and the quartimax and oblimax methods in factor analysis. One serious drawback of projection pursuit methods is their high demand on computer time. | |

Once normalized, data between portions of the triad are able to be compared even when large differences in measurements or units exits (Chapman, 1990). From the combination of the results from each portion of the triad a multivariate figure is developed and used to determine the level of degradation. | |

A Linguistic Time-Capsule: The Newcastle Electronic Corpus of Tyneside English (NECTE), AHRB project code RE11776 Project Leaders: K. Corrigan, J. Beal, H. Moisl Postgraduate Supervision Natural language modelling and text processing. My specific areas of interest are natural language understanding systems and multivariate analysis of text corpora. | |

2008: & F Steele, I Moustaki, J Galbraith, Analysis of Multivariate Social Science Data, (second edition) Boca Raton, Florida: Chapman & Hall/CRC 25\. 2011: & M Knott, I Moustaki, Latent Variable Models and factor Analysis: A unified approach. Chichester: John Wiley and Sons Ltd 26\. 2013: Unobserved Variables: Models and Misunderstandings. | |

In cryptography, the unbalanced oil and vinegar (UOV) scheme is a modified version of the oil and vinegar scheme designed by J. Patarin. Both are digital signature protocols. They are forms of multivariate cryptography. The security of this signature scheme is based on an NP-hard mathematical problem. | |

The issue is particularly relevant in multivariate and regression problems. Thus, some care is needed to ensure that good starting points are chosen. Robust starting points, such as the median as an estimate of location and the median absolute deviation as a univariate estimate of scale, are common. | |

In his early work Joachim Engel specialized in nonparametric curve estimation and signal detection applying methods of Harmonic Analysis (Engel, 1994)Engel, J. (1994). A simple wavelet approach to nonparametric regression from recursive partitioning schemes. J. Multivariate Analysis, 49, 242 – 254. (Engel & Kneip 1996)Engel, J. & Kneip, A. (1996). | |

John Frederick MacGregor (born 1943 in Ontario, Canada) is a statistician whose work in the field of statistical process control has received significant recognition. His pioneering work was in the area of latent variable/multivariate analysis (MVA) methods (principal components analysis and partial least squares) applied to industrial processes. | |

Neighbourhood components analysis is a supervised learning method for classifying multivariate data into distinct classes according to a given distance metric over the data. Functionally, it serves the same purposes as the K-nearest neighbors algorithm, and makes direct use of a related concept termed stochastic nearest neighbours. | |

In statistical theory, the field of high-dimensional statistics studies data whose dimension is larger than dimensions considered in classical multivariate analysis. High-dimensional statistics relies on the theory of random vectors. In many applications, the dimension of the data vectors may be larger than the sample size. | |

The use of this marble began in antiquity and has continued ever since. It was among the first types of Cycladic "island marble" to be used. It is the largest-grained marble which was used in ancient times.Thomas Cramer: Multivariate Herkunftsanalyse von Marmor auf petrographischer und geochemischer Basis. | |

11, p. 172-182(1793) Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems: > When the long-known finite step algorithms were first put on computers, they > turned out to be highly inefficient. The fact that almost any uni- or > multivariate polynomial of degree up to 100 and with coefficients of a > moderate size (up to 100 bits) can be factored by modern algorithms in a few > minutes of computer time indicates how successfully this problem has been > attacked during the past fifteen years. | |

Greig-Smith saw in multivariate methods an opportunity to analyze and understand the complex floristic composition of tropical forests, which had fascinated him since his visit to Trinidad in 1948. In 1963 he applied for a grant to investigate the use of quantitative methods to establish whether the highly diverse tropical rainforests showed any organized pattern, particularly in relation to environment, an open question at the time. With these funds, in 1964 he appointed Michael Austin as a postdoctoral research assistant to the project. Three seminal papers were published as a result of their collaboration, highlighting the potential of multivariate methods to analyze floristic patterns in complex, multispecies communitiesGreig-Smith, Peter, Michael P. Austin, and Timothy C. Whitmore. 1967. | |

After two years he returned to India in 1990 and joined the Indian Statistical Institute, Kolkata, as a lecturer. He was promoted to full professorship in 1997. Some of the widely used statistical techniques and concepts that he has invented and developed include: local polynomial nonparametric quantile regression, a geometric notion of quantiles for multivariate data, adaptive transformation and re-transformation technique for the construction of affine invariant distribution-free tests and robust estimates from multivariate data and the scale-space approach in function estimation and smoothing. He was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology in 2005, the highest science award in India, in the mathematical sciences category. | |

The autoregressive fractionally integrated moving average (ARFIMA) model generalizes the former three. Extensions of these classes to deal with vector-valued data are available under the heading of multivariate time-series models and sometimes the preceding acronyms are extended by including an initial "V" for "vector", as in VAR for vector autoregression. An additional set of extensions of these models is available for use where the observed time-series is driven by some "forcing" time-series (which may not have a causal effect on the observed series): the distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter's control. For these models, the acronyms are extended with a final "X" for "exogenous". | |

This includes the use of multivariate t-distributions, and skew variants of multivariate t- and normal distributions. His works have found applications in numerous areas of practical research including biology, bioinformatics, cardiology, engineering, psychology, neuroimaging, among numerous other fields. McLachlan's research has been published in various well-regarded journals such as Biometrics; Biometrika; Journal of the Royal Statistical Society; Journal of the American Statistical Association; Proceedings of the National Academy of Sciences of the USA; Nature Methods; the Computer Journal; and the IEEE Transactions on Pattern Analysis and Machine Intelligence, Medical Imaging, and Neural Networks. He is a featured researcher in Journeys to Data Mining: Experiences from 15 Renowned Researchers, edited by Mohamed Medhat Gaber. | |

Here are described some facets of statistical literacy that underpin the ability to engage with statistics about social issues – i.e. Civic Statistics. Data that can be used to inform social policy are complex. Data are often multivariate; aggregated data and indicator systems are common; variables interact; data may be time critical. | |

Richard G. Lomax (born September 30, 1954) is a tenured professor of education at the School of Educational Policy and Leadership and the College of Education and Human Ecology at Ohio State University. His research interests include multivariate analysis, models of literacy acquisition, structural equation models, graphics, and statistics in sports. | |

11.4]Das, S., Dey, D. K. (2010) "On Bayesian inference for generalized multivariate gamma distribution". Statistics and Probability Letters, 80, 1492-1499.Karagiannidis, K., Sagias, N. C., Tsiftsis, T. A. (2006) "Closed-form statistics for the sum of squared Nakagami-m variates and its applications". Transactions on Communications, 54, 1353-1359. | |

Other land management activities can affect the effectiveness of post fire seeding. Grazing seeded burned areas exacerbates the problem of non-native annual grass invasions, even when conducted after a two-year hiatus.Eiswerth, M.E. and J.S. Shonkwiler. 2006. Examining post-wildfire reseeding on arid rangeland: a multivariate tobit modeling approach. | |

The EATCS--IPEC Nerode Prize is a theoretical computer science prize awarded for outstanding research in the area of multivariate algorithmics. It is awarded by the European Association for Theoretical Computer Science and the International Symposium on Parameterized and Exact Computation.. The prize was offered for the first time in 2013.. | |

Díaz, D. M. V. (2013). Multivariate analysis of morphological and anatomical characters of Calophyllum (Calophyllaceae) in South America. Botanical Journal of the Linnean Society 171(3), 587-626. The inflorescence is a cyme or a thyrse of flowers that grows from the leaf axils or at the ends of branches. | |

The generalized variance is a scalar value which generalizes variance for multivariate random variables. It was introduced by Samuel S. Wilks. The generalized variance is defined as the determinant of the covariance matrix, \det(\Sigma). It can be shown to be related to the multidimensional scatter of points around their mean. | |

In probability theory and in particular in information theory, total correlation (Watanabe 1960) is one of several generalizations of the mutual information. It is also known as the multivariate constraint (Garner 1962) or multiinformation (Studený & Vejnarová 1999). It quantifies the redundancy or dependency among a set of n random variables. | |

Alarcon, M., Plomin, R., Fulker, D.W., Corley, R. & DeFries, J.C. (1998). Multivariate path analysis of specific cognitive abilities: data at 12 years of age in the Colorado Adoption Project. Behavior Genetics 28:255-264. DeFries' articles have been cited over 29,000 times and he has an h-index of 87. | |

Hsieh et al., in Taiwan, analysed 46,007 ischaemic stroke admissions. They found, in multivariate analysis without adjustment for stroke severity, weekend admission was associated with increased 30-day mortality (OR = 1.20; 95% CI 1.08-1.34). But this association did not persist after adjustment for stroke severity (OR = 1.07; 95% CI 0.95-1.20). | |

Shadish was the founding secretary- treasurer of the Society for Research Synthesis Methodology, later serving as its president from 2013 to 2014. He was also elected president of the American Evaluation Association in 1996, and of the Society for Multivariate Experimental Psychology in 2014. He was a fellow of the American Psychological Association. | |

There are several common parametric empirical Bayes models, including the Poisson–gamma model (below), the Beta- binomial model, the Gaussian–Gaussian model, the Dirichlet-multinomial model, as well specific models for Bayesian linear regression (see below) and Bayesian multivariate linear regression. More advanced approaches include hierarchical Bayes models and Bayesian mixture models. | |

Publications TETRAD. Retrieved December 16, 2019. Using multivariate statistical data as input, TETRAD rapidly searches from among all possible causal relationship models and returns the most plausible causal models based on conditional dependence relationships between those variables. The algorithm is based on principles from statistics, graph theory, philosophy of science, and artificial intelligence. | |

Dempster received his B.A. in mathematics and physics (1952) and M.A. in mathematics (1953), both from the University of Toronto. He obtained his Ph.D. in mathematical statistics from Princeton University in 1956. His thesis, titled The two-sample multivariate problem in the degenerate case, was written under the supervision of John Tukey. | |

Multivariate demographic segmentation involves using at least two types of demographic variables in conjunction with each other to make the market more precise for the advertiser to target. This way of further segmentation is effective because it allows advertisers to filter out more consumers who won't match the demographics of the target market. | |

Autocorrelations should be near-zero for randomness; if the analyst does not check for randomness, then the validity of many of the statistical conclusions becomes suspect. The correlogram is an excellent way of checking for such randomness. Sometimes, corrgrams, color-mapped matrices of correlation strengths in multivariate analysis, are also called correlograms. | |

English translation by Michael Abramson. An analogous concept for multivariate power series was developed independently by Heisuke Hironaka in 1964, who named them standard bases. This term has been used by some authors to also denote Gröbner bases. The theory of Gröbner bases has been extended by many authors in various directions. | |

Peter M. Bentler is an American psychologist, statistician, and distinguished professor at the University of California, Los Angeles. In multivariate analysis and psychometrics, Bentler is the developer of the structural equation modeling software EQS.EQS. Bentler received a doctorate in clinical psychology from Stanford University in 1964. His publications have over 190,000 citations . | |

In mathematics and multivariate statistics, the centering matrixJohn I. Marden, Analyzing and Modeling Rank Data, Chapman & Hall, 1995, , page 59. is a symmetric and idempotent matrix, which when multiplied with a vector has the same effect as subtracting the mean of the components of the vector from every component of that vector. | |

Anuška Ferligoj is a Slovenian mathematician, born August 19, 1947 in Ljubljana, Slovenia, whose specialty is network analysis.. Her specific interests include multivariate analysis (constrained and multicriteria clustering), social networks (measurement quality and blockmodeling), and survey methodology (reliability and validity of measurement). She is Fellow of the European Academy of Sociology. She is Professor of Multivariate statistical methods at the University of Ljubljana and head of the graduate program on Statistics at the University of Ljubljana. She is also editor of the journal Advances in Methodology and Statistics (Metodoloski zvezki) since 2004 and is a member of the editorial boards of the Journal of Mathematical Sociology, Journal of Classification, Social Networks, Statistic in Transition, Methodology, Structure and Dynamics: eJournal of Anthropology and Related Sciences. | |

In the analysis of multivariate observations designed to assess subjects with respect to an attribute, a Guttman Scale (named after Louis Guttman) is a single (unidimensional) ordinal scale for the assessment of the attribute, from which the original observations may be reproduced. The discovery of a Guttman Scale in data depends on their multivariate distribution's conforming to a particular structure (see below). Hence, a Guttman Scale is a hypothesis about the structure of the data, formulated with respect to a specified attribute and a specified population and cannot be constructed for any given set of observations. Contrary to a widespread belief, a Guttman Scale is not limited to dichotomous variables and does not necessarily determine an order among the variables. | |

Venn diagram of information theoretic measures for three variables x, y, and z, represented by the lower left, lower right, and upper circles, respectively. The multivariate mutual information is represented by gray region. Since it may be negative, the areas on the diagram represent signed measures. In information theory there have been various attempts over the years to extend the definition of mutual information to more than two random variables. The expression and study of multivariate higher-degree mutual- information was achieved in two seemingly independent works: McGill (1954) who called these functions “interaction information”, and Hu Kuo Ting (1962) who also first proved the possible negativity of mutual-information for degrees higher than 2 and justified algebraically the intuitive correspondence to Venn diagrams . | |

He received numerous awards, including: Research Career Development Award, National Institutes of Health (1968–1972); Annual Prize for Distinguished Publications in Multivariate Psychology (SMEP) (1972); Lifetime Achievement Award, SMEP (1992). Horn also served as president of the National Association for the Advancement of Colored People and the American Civil Liberties Union. He died in 2006. | |

"El Niño/Southern Oscillation Behaviour since 1871 as Diagnosed in an Extended Multivariate ENSO Index (MEI.ext)." International Journal of Climatology 31.7 (2011): 1074-87. This was accomplished by using reconstructed data on sea level pressure and sea surface temperature, the two components thought to be most influential to determining MEI. Plots comparing MEI and MEI. | |

Lohr graduated from Calvin College in 1982. She completed her Ph.D. in statistics in 1987 at the University of Wisconsin–Madison. Her dissertation, Accurate Multivariate Estimation Using Double and Triple Sampling, was supervised by Mark Finster. After retiring from Arizona State, she served a five-year term as vice president and senior statistician at Westat. | |

These programs do not include or entail "Business mathematics" per se. Where mathematical economics is not a degree requirement, graduate economics programs often include "quantitative techniques", which covers (applied) linear algebra, multivariate calculus, and optimization, and may include the above topics; regardless, econometrics is usually a separate course, and is dealt with in depth. | |

Ingram Olkin (July 23, 1924 - April 28, 2016) was a professor emeritus and chair of statistics and education at Stanford University and the Stanford Graduate School of Education. He is known for developing statistical analysis for evaluating policies, particularly in education, and for his contributions to meta-analysis, statistics education, multivariate analysis, and majorization theory. | |

The exploratory factor analysis begins without a theory or with a very tentative theory. It is a dimension reduction technique. It is useful in psychometrics, multivariate analysis of data and data analytics. Typically a k-dimensional correlation matrix or covariance matrix of variables is reduced to k X r factor pattern matrix where r < k. | |

Because 'time' is treated as a qualitative factor in the ANOVA decomposition preceding ASCA, a nonlinear multivariate time trajectory can be modeled. An example of this is shown in Figure 10 of this reference.Smilde, A. K., Hoefsloot, H. C. and Westerhuis, J. A. (2008), "The geometry of ASCA". Journal of Chemometrics, 22, 464–471. | |

Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type of stochastic process model. VAR models generalize the single-variable (univariate) autoregressive model by allowing for multivariate time series. VAR models are often used in economics and the natural sciences. | |

Optimizely is an American company that makes progressive delivery and experimentation software for other companies. The Optimizely platform technology provides A/B testing and multivariate testing tools, website personalization, and feature toggle capabilities. The company's headquarters are in San Francisco, California with offices in Amsterdam, Netherlands, Cologne, Germany, London, United Kingdom and Sydney, Australia. | |

The first application of an experimental design for MVLPO was performed by Moskowitz Jacobs Inc. in 1998 as a simulation/demonstration project for LEGO. MVLPO did not become a mainstream approach until 2003 or 2004. Multivariate landing page optimization can be executed in a live (production) environment, or through simulations and market research surveys. | |

As such, multilevel models provide an alternative type of analysis for univariate or multivariate analysis of repeated measures. Individual differences in growth curves may be examined. Furthermore, multilevel models can be used as an alternative to ANCOVA, where scores on the dependent variable are adjusted for covariates (e.g. individual differences) before testing treatment differences. | |

The results of the preceding section remain valid if the ring of integers and the field of rationals are respectively replaced by any unique factorization domain and its field of fractions . This is typically used for factoring multivariate polynomials, and for proving that a polynomial ring over a unique factorization domain is also a unique factorization domain. | |

The above arithmetic can be generalized to calculate second order and higher derivatives of multivariate functions. However, the arithmetic rules quickly grow complicated: complexity is quadratic in the highest derivative degree. Instead, truncated Taylor polynomial algebra can be used. The resulting arithmetic, defined on generalized dual numbers, allows efficient computation using functions as if they were a data type. | |

Pattern based morphometry (PBM) is a method of brain morphometry first put forth in PBM. It builds upon DBM and VBM. PBM is based on the application of sparse dictionary learning to morphometry. As opposed to typical voxel based approaches which depend on univariate statistical tests at specific voxel locations, PBM extracts multivariate patterns directly from the entire image. | |

In 1962 Pillai was appointed Professor of Statistics and Mathematics at Purdue University. Pillai's research was in statistics, in particular in multivariate statistical analysis. Pillai was honoured by being elected a Fellow of the American Statistical Association and a Fellow of the Institute of Mathematical Statistics. He was an elected member of the International Statistical Institute. | |

Univariate normality is not needed for least squares estimates of the regression parameters to be meaningful (see Gauss–Markov theorem). However confidence intervals and hypothesis tests will have better statistical properties if the variables exhibit multivariate normality. Transformations that stabilize the variance of error terms (i.e. those that address heteroscedaticity) often also help make the error terms approximately normal. | |

It is when there is an elevated level of solutes within the soil that inhibit the growth and metabolic capabilities of crops. Salinity stress is a problem that affects A. desertorum in the more semiarid parts of North America.Golpalvar, A. R. (2011). Multivariate analysis of germination ability and tolerance to salinity in Agropyron desertorum genotypes in greenhouse condition. | |

In multivariate statistics, the congruence coefficient is an index of the similarity between factors that have been derived in a factor analysis. It was introduced in 1948 by Cyril Burt who referred to it as unadjusted correlation. It is also called Tucker's congruence coefficient after Ledyard Tucker who popularized the technique. Its values range between -1 and +1. | |

Chance is originally from San Diego, California. She graduated from Harvey Mudd College in 1990, majoring in mathematics with a minor in psychology. She completed a Ph.D. in operations research, concentrating in statistics, at Cornell University in 1994. Her dissertation, Behavior Characterization and Estimation for General Hierarchical Multivariate Linear Regression Models, was supervised by Martin Wells. | |

Mark Richard Jerrum (born 1955) is a British computer scientist and computational theorist. Jerrum received his Ph.D. in computer science 'On the complexity of evaluating multivariate polynomials' in 1981 from University of Edinburgh under the supervision of Leslie Valiant. He is professor of pure mathematics at Queen Mary, University of London.Personnel page, Queen Mary, University of London. | |

Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic.See and . | |

MEI is determined as the first principal component of six different parameters: sea level pressure, zonal and meridional components of the surface wind, sea surface temperature, surface air temperature and cloudiness using data from the International Comprehensive Ocean-Atmosphere Data Set (ICOADS ).Wolter, Klaus. "Multivariate ENSO Index (MEI)." NOAA Earth System Research Laboratory - Physical Sciences Division. | |

For example, a generalization of Gaussian elimination called Buchberger's algorithm has for its complexity an exponential function of the problem data (the degree of the polynomials and the number of variables of the multivariate polynomials). Because exponential functions eventually grow much faster than polynomial functions, an exponential complexity implies that an algorithm has slow performance on large problems. | |

Stein's example is an important result in decision theory which can be stated as : The ordinary decision rule for estimating the mean of a multivariate Gaussian distribution is inadmissible under mean squared error risk in dimension at least 3. The following is an outline of its proof. The reader is referred to the main article for more information. | |

Once an oscillation has been detected, the system can perform modal analysis using the multichannel matrix pencil technique. This analysis reveals the dominant oscillation modes and shows which parts of the power grid tend to oscillate together. Recent studies showed some time-frequency analysis methods are useful for multi-channel mode analysis, such as multivariate empirical mode decomposition methods. | |

Typically the generator is seeded with randomized input that is sampled from a predefined latent space (e.g. a multivariate normal distribution). Thereafter, candidates synthesized by the generator are evaluated by the discriminator. Independent backpropagation procedures are applied to both networks so that the generator produces better images, while the discriminator becomes more skilled at flagging synthetic images. | |

In mathematics, in abstract algebra, a multivariate polynomial over a field such that the Laplacian of is zero is termed a harmonic polynomial. The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In fact, they form a graded subspace. For the real field, the harmonic polynomials are important in mathematical physics. | |

While physico- chemical descriptors like molecular weight, (partial) charge, solubility, etc. can mostly be computed directly based on the molecule's structure, pharmacological descriptors can be derived only indirectly using involved multivariate statistics or experimental (screening, bioassay) results. All of those descriptors can for reasons of computational effort be stored along with the molecule's representation and usually are. | |

A signature scheme has a signing key, which is kept private, and a verification key, which is publicly revealed. For instance, in signature schemes based on RSA the keys are both exponents. In the UOV scheme, and in every other multivariate signature scheme the keys are more complex. The mathematical problem is to solve m equations with n variables. | |

Nearly all real-world regression models involve multiple predictors, and basic descriptions of linear regression are often phrased in terms of the multiple regression model. Note, however, that in these cases the response variable y is still a scalar. Another term, multivariate linear regression, refers to cases where y is a vector, i.e., the same as general linear regression. | |

Hofstra MS The M.Fin and MSc will often require more advanced topics such as multivariate calculus, linear algebra and differential equations; these may also require a greater background in Finance or Economics than the MSF.See Fordham's MSGF and MSQF Some programs may require work experience (sometimes at the managerial level), particularly if the candidate lacks a relevant undergraduate degree. | |

Manatunga graduated from the University of Colombo in Sri Lanka with first class honors in 1978. She has master's degrees in statistics from Purdue University (1984) and the University of Rochester (1986). She completed her Ph.D. at the University of Rochester in 1990. Her dissertation, Inference for Multivariate Survival Distributions Generated by Stable Frailties, was supervised by David Oakes. | |

For example, the degree of -7 is 0. The degree of a monomial is sometimes called order, mainly in the context of series. It is also called total degree when it is needed to distinguish it from the degree in one of the variables. Monomial degree is fundamental to the theory of univariate and multivariate polynomials. | |

Let n denote the number of observations. In all cases below, the data is assumed to consist of n points x_1,\ldots,x_n (which will be random vectors in the multivariate cases). If the likelihood function belongs to the exponential family, then a conjugate prior exists, often also in the exponential family; see Exponential family: Conjugate distributions. | |

In applied statistics, the Marshall–Olkin exponential distribution is any member of a certain family of continuous multivariate probability distributions with positive-valued components. It was introduced by Albert W. Marshall and Ingram Olkin. One of its main uses is in reliability theory, where the Marshall–Olkin copula models the dependence between random variables subjected to external shocks. | |

Mustonen, Seppo (2007): "On Survo cross sum puzzles". In J. Niemelä, S. Puntanen, and E. P. Liski (eds.) Abstracts of the Annual Conference of Finnish Statisticians 2007, "Multivariate Methods", pp. 23-26\. Dept. of Mathematics, Statistics and Philosophy, University of Tampere. . Certain properties of the Survo system like editorial computing and the COMB operation, making e.g. | |

Part III is concerned with the discussion of some important concepts in time series analysis, the discussion focuses on the techniques which can be readily applied in practice. Parts IV-VIII suggest different modeling methods and model structures. Part IX extends the concepts in chapter three to multivariate time series. Part X examines common aspects across time series. | |

EBSCOhost, doi:10.1111/mec.14803.Feeding from ants and mites on the ground is popular after sunrise and not later than 16:00, when people are either on the ground or elevated on fallen logs finding refugee under leaf litterPosso, Terranova, Andrés, and Jose Andrés. “Multivariate Species Boundaries and Conservation of Harlequin Poison Frogs.” Molecular Ecology, vol. | |

Piecewise constant interpolation, or nearest-neighbor interpolation. The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation, this could be a favourable choice for its speed and simplicity. | |

Simulation of 1000 observations drawn from a Dirichlet mixture model. Each observation within a cluster is drawn independently from the multivariate normal distribution N(\mu_k,1/4). The cluster means \mu_k are drawn from a distribution G which itself is drawn from a Dirichlet process with concentration parameter \alpha=0.5 and base distribution H=N(2,16). Each row is a new simulation. | |

Besides the above mentioned root finding techniques, there are also methods that approximate the multivariate inverse function directly. Often they are based on polynomials or rational functions. For the Bachelier ("normal", as opposed to "lognormal") model, Jaeckel published a fully analytic and comparatively simple two-stage formula that gives full attainable (standard 64 bit floating point) machine precision for all possible input values. | |

The marginal median is defined for vectors defined with respect to a fixed set of coordinates. A marginal median is defined to be the vector whose components are univariate medians. The marginal median is easy to compute, and its properties were studied by Puri and Sen.Puri, Madan L.; Sen, Pranab K.; Nonparametric Methods in Multivariate Analysis, John Wiley & Sons, New York, NY, 197l. | |

In statistics, a latent class model (LCM) relates a set of observed (usually discrete) multivariate variables to a set of latent variables. It is a type of latent variable model. It is called a latent class model because the latent variable is discrete. A class is characterized by a pattern of conditional probabilities that indicate the chance that variables take on certain values. | |

R. isosceles was also present in the Aguja Formation, roughly the same age. All other referred teeth most likely belong to different species, which have not been named due to the lack of body fossils for comparison.Larson DW, Currie PJ (2013) Multivariate Analyses of Small Theropod Dinosaur Teeth and Implications for Paleoecological Turnover through Time. PLoS ONE 8(1): e54329. doi:10.1371/journal.pone. | |

Intact samples can be imaged in transmittance or diffuse reflectance. The lineshapes for overtone and combination bands tend to be much broader and more overlapped than for the fundamental bands seen in the MIR. Often, multivariate methods are used to separate spectral signatures of sample components. NIR chemical imaging is particularly useful for performing rapid, reproducible and non- destructive analyses of known materials. | |

In 1980 Reiss and two of his colleagues published a research and theory paper on factors that predicted a person's attitudes towards extramarital sexuality.Reiss, I.L., Anderson, R.E., & Sponaugle, G.C. (1980). A multivariate model of the determinants of extramarital sexual permissiveness, Journal of Marriage and the Family, 42, 395–411.Saunders, J.M. & Edwards, J.N. (1984) Extramarital sexuality: A predictive model of permissive attitudes. | |

It may include video, animation, native components, and interactive elements. Test variables represent the parts of the ad creative that are varied in the multivariate testing framework. These commonly include graphical elements, ad copy, colors, and click-through actions. It is helpful to have digital assets managed in a digital asset management system, especially when digital rights need to be enforced. | |

Olkin was born in 1924 in Waterbury, Connecticut. He received a B.S. in mathematics at the City College of New York, an M.A. from Columbia University, and his Ph.D. from the University of North Carolina. Olkin also studied with Harold Hotelling. Olkin's advisor was S. N. Roy and his Ph.D. thesis was "On distribution problems in multivariate analysis" submitted in 1951. | |

Usage of a generalized linear model showed that 9 of the 11 traits demonstrated mimicry by the vine to its host tree. Gianoli et al. also sampled more individuals that were prostrated, that grew on leafless tree trunks, and more individuals that have climbed on the 8 most common host species. To analyze these samples, the researchers used multivariate analysis of variance (MANOVA). | |

In Gaussian process regression, also known as Kriging, a Gaussian prior is assumed for the regression curve. The errors are assumed to have a multivariate normal distribution and the regression curve is estimated by its posterior mode. The Gaussian prior may depend on unknown hyperparameters, which are usually estimated via empirical Bayes. The hyperparameters typically specify a prior covariance kernel. | |

Box's M test is a multivariate statistical test used to check the equality of multiple variance-covariance matrices. The test is commonly used to test the assumption of homogeneity of variances and covariances in MANOVA and linear discriminant analysis. It is named after George E. P. Box who first discussed the test in 1949. The test uses a chi-squared approximation. | |

Overall survival (OS) observed for tasquinimod-treated patients was longer than previously reported in this patient population. Median overall survival was 33.4 months for the tasquinimod group versus 30.4 months for the placebo group (p=0.49). Using a multivariate analysis, treatment with tasquinimod was associated with an OS advantage with a HR of 0.64 (95% CI 0.42, 0.97, p=0.034). | |

In Bayesian statistics, a credible interval is an interval within which an unobserved parameter value falls with a particular probability. It is an interval in the domain of a posterior probability distribution or a predictive distribution.Edwards, Ward, Lindman, Harold, Savage, Leonard J. (1963) "Bayesian statistical inference in psychological research". Psychological Review, 70, 193-242 The generalisation to multivariate problems is the credible region. | |

Journal of Ecology 60(2): 305–324.. The power of his approach, merging multivariate descriptive methods with experimental hypothesis testing, is especially noticeable in the work of some of his students like Mike Austin (cited previously), Exequiel EzcurraEzcurra, E. 1987. A comparison of Reciprocal Averaging and Non-Centred Principal Component Analysis. Vegetatio 71(1): 41–48., and F.B. GoldsmithGoldsmith, F. B. 1973a. | |

The orthogonal decomposition of a PSD matrix is used in multivariate analysis, where the sample covariance matrices are PSD. This orthogonal decomposition is called principal component analysis (PCA) in statistics. PCA studies linear relations among variables. PCA is performed on the covariance matrix or the correlation matrix (in which each variable is scaled to have its sample variance equal to one). | |

Males attract females to burrows through production of advertisement calls, wherein females will mount the males should they be interested - therefore, song production is a key determinant for fitness of the male.Bentsen C.L., Hunt J, Jennions M.D., Brooks R. 2006. Complex Multivariate Sexual Selection on Male Acoustic Signaling in a Wild Population of Teleogryllus commodus. The American Naturalist 167:102-116. | |

However, correction strategies should be used to reduce false discoveries when multiple comparisons are conducted. For multivariate analysis, models should always be validated to ensure the results can be generalized. Machine learning is also a powerful tool that can be used in metabolomics analysis. Recently, the authors of a paper published in Analytical Chemistry, developed a retention time prediction software called Retip. | |

The literature on forecasting daily electricity prices has concentrated largely on models that use only information at the aggregated (i.e., daily) level. On the other hand, the very rich body of literature on forecasting intra-day prices has used disaggregated data (i.e., hourly or half-hourly), but generally has not explored the complex dependence structure of the multivariate price series. | |

1996: The Statistical Approach to Social Measurement, San Diego: Academic Press. 19\. 1999: & M. Knott, Latent Variable Models and Factor Analysis, London Arnold, 2nd edition: 20\. 2002: & F. Steele, I Moustaki & J. I. Galbraith, The Analysis and Interpretation of Multivariate Data for Social Scientists, Boca Raton, Florida: Chapman & Hall/CRC. 21\. 2004: Measuring Intelligence: Facts and Fallacies, Cambridge: Cambridge University Press. 22\. | |

Implementations can be found in C, C++, Matlab and Python. Sampling from the multivariate truncated normal distribution is considerably more difficult. Exact or perfect simulation is only feasible in the case of truncation of the normal distribution to a polytope region. In more general cases, Damien and Walker (2001) introduce a general methodology for sampling truncated densities within a Gibbs sampling framework. | |

Multivariate behavioral research is becoming very popular in psychology. These methods include Multiple Regression and Prediction; Moderated and Mediated Regression Analysis; Logistics Regression; Canonical Correlations; Cluster analysis; Multi-level modeling; Survival-Failure analysis; Structural Equations Modeling; hierarchical linear modelling etc. are very useful for psychological statistics (Hayes, 2013;Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis. | |

There are many specialized journals that publish advances in statistical analysis for psychology. Psychometrika is at the forefront. Educational and Psychological Measurement, Assessment, American Journal of Evaluation, Applied Psychological Measurement, Behavior Research Methods, British Journal of Mathematical and Statistical Psychology, Journal of Educational and Behavioral Statistics, Journal of Mathematical Psychology, Multivariate Behavioral Research, Psychological Assessment, Structural Equation Modeling are other useful journals. | |

Little is a Fellow of the American Association for the Advancement of Science (AAAS) as well as the American Psychological Association (APA Divisions 5, 7, & 15) and the Association for Psychological Science (APS). In 2001, Little was elected to the Society for Multivariate Experimental Psychology. In 2009, he was elected President of the APS's Division 5 (then called Evaluation, Measurement, and Statistics). | |

Michael Louis Friendly (born 1945) is an American psychologist, Professor of Psychology at York University in Ontario, Canada, and director of its Statistical Consulting Service, especially known for his contributions to graphical methods for categorical and multivariate data, and on the history of data and information visualisation.Rosenberg, Daniel, and Anthony Grafton. Cartographies of time: A history of the timeline. Princeton Architectural Press, 2013. | |

Ludovic Lebart (born 1942) is a French statistician. He is a senior researcher at the Centre National de la Recherche Scientifique and a professor at the Ecole Nationale Supérieure des Télécommunications in Paris. His research interests are the exploratory analysis of qualitative and textual data. He has coauthored several books on descriptive multivariate statistics, survey methodology, and exploratory analysis of textual data. | |

Recently, copula functions have been successfully applied to the database formulation for the reliability analysis of highway bridges, and to various multivariate simulation studies in civil engineering, reliability of wind and earthquake engineering, and mechanical & offshore engineering. Researchers are also trying these functions in the field of transportation to understand the interaction between behaviors of individual drivers which, in totality, shapes traffic flow. | |

The Blank Slate. New York: Penguin. 2002. Critically, multivariate studies show that the distinct faculties of the mind, such as memory and reason, fractionate along genetic boundaries. Cultural universals such as emotion and the relative resilience of psychological adaptation to accidental biological changes (for instance the David Reimer case of gender reassignment following an accident) also support basic biological mechanisms in the mind. | |

Wayne DeSarbo is the Mary Jean and Frank P. Smeal Distinguished Professor of Marketing at the Smeal College of Business at Pennsylvania State University at University Park and executive director the Center for Sports Business and Research. He is known for his work on multidimensional scaling, and multivariate statistics in relation to marketing research. He is a fellow of the American Statistical Association. | |

Yvonne Millicent Mahala Bishop (died May 26, 2015) was an English-born statistician who spent her working life in America. She wrote a "classic" book on multivariate statistics, and made important studies of the health effects of anesthetics and air pollution. Later in her career, she became the Director of the Office of Statistical Standards in the Energy Information Administration. | |

Multivariate landing page optimization (MVLPO) is a specific form of landing page optimization where multiple variations of visual elements (e.g., graphics, text) on a webpage are evaluated. For example, a given page may have k choices for the title, m choices for the featured image or graphic, and n choices for the company logo. This example yields k×m×n landing page configurations. | |

Multivariate analysis, design of experiments, and survey sampling By Subir Ghosh, Jagdish Narain Srivastava, CRC Press, 1999 Srivastava code was invented by him. Gödel's incompleteness theorems inspired him to recognize the limitations of science. He slowly turned toward spirituality and studied all the major religions of the world. This led him to obtain his 1991 joint appointment in the philosophy department of CSU. | |

First, data is collected regarding geographical location of a target mosquito species. Next, a multivariate regression model establishes the conditions under which the target species can survive. Finally, the model determines the likelihood of the mosquito species to become established in a new location based on similar living conditions. The model can further predict future distributions based on environmental emissions data. | |

It is proved that the t-cherry junction trees provide a better or at least as good approximation for a discrete multivariate probability distribution as the Chow–Liu tree gives. For the third order t-cherry junction tree see , for the kth-order t-cherry junction tree see . The second order t-cherry junction tree is in fact the Chow–Liu tree. | |

Bock, R. D. (1975). Multivariate statistical methods. New York: McGraw-Hill. This set of variables can then be expressed by the following regression equation: Y = β0 \+ β1M + β2G + β3MG where β0 corresponds to the intercept or the probability of a response when M and G are equal to 0 with remaining βs corresponding to weight coefficients for each independent variable. | |

Gábor J. Székely (; born February 4, 1947 in Budapest) is a Hungarian-American statistician/mathematician best known for introducing the Energy of dataE- Statistics: The energy of statistical samples (2002), G.J.Szekely, PDF [see E-statistics or Package energy in R (programming language)], e.g. the distance correlationSzékely and Rizzo (2009). which is a bona fide dependence measure, equals zero exactly when the variables are independent, the distance skewness which equals zero exactly when the probability distribution is diagonally symmetric, the E-statistic for normality testSzékely, G. J. and Rizzo, M. L. (2005) A new test for multivariate normality, Journal of Multivariate Analysis 93, 58-80. and the E-statistic for clustering. Other important discoveries include the Hungarian semigroups,Raja, C.R.E. (1999) On a class of Hungarian semigroups and the factorization theorem of Khinchin, J. Theoretical Probability 12/2, 561-569. | |

Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing. Traditionally, algebraic statistics has been associated with the design of experiments and multivariate analysis (especially time series). In recent years, the term "algebraic statistics" has been sometimes restricted, sometimes being used to label the use of algebraic geometry and commutative algebra in statistics. | |

The Hong Kong Baptist University honored Professor Fang with the President's Award for Outstanding Performance in Scholarly Work in 2001. Fang and Zhang's book Generalized multivariate analysis was honored as a "most excellent book in China"Loie describes the award as only "most excellent book", without the indefinite article "a", which here is inserted (in accordance with standard written English). by the Government Information and Publication Administration. | |

Springer, 1999. are useful for sampling multivariate functions in 2-D, 3-D and higher dimensions. In the 2-D setting the three- direction box spline is used for interpolation of hexagonally sampled images. In the 3-D setting, four-direction and six-direction box splines are used for interpolation of data sampled on the (optimal) body-centered cubic and face- centered cubic lattices respectively. | |

BirdMorpho Traditional morphometrics is the study of morphological variations between or within groups using multivariate statistical tools. Shape is defined by collecting and analyzing length measurements, counts, ratios, and angles. The statistical tools are able to quantify the covariation within and between samples. Some of the typical statistical tools used for traditional morphometrics are: principal components, factor analysis, canonical variate, and discriminant function analysis. | |

In applied statistics, canonical correspondence analysis (CCA) is a multivariate constrained ordination technique that extracts major gradients among combinations of explanatory variables in a dataset. The requirements of a CCA are that the samples are random and independent. Also, the data are categorical and that the independent variables are consistent within the sample site and error-free.McGarigal, K., S. Cushman, and S. Stafford (2000). | |

Previously, this article discussed the univariate median, when the sample or population had one-dimension. When the dimension is two or higher, there are multiple concepts that extend the definition of the univariate median; each such multivariate median agrees with the univariate median when the dimension is exactly one.Small, Christopher G. "A survey of multidimensional medians." International Statistical Review/Revue Internationale de Statistique (1990): 263–277. | |

If a low involvement consumer continues to use variety-seeking behavior, brand loyalty is unlikely to be established. Factors influencing brand loyalty Loyalty includes some degree of predisposition toward a brand. It is determined by several distinct psychological processes, and it entails multivariate measurements. Customer perceived value, brand trust, customer satisfaction, repeat purchase behavior, and commitment are found to be the key influencing factors of brand loyalty. | |

One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry. Wiles' proof of Fermat's Last Theorem uses advanced methods of algebraic geometry for solving a long-standing problem of number theory. In general, algebraic geometry studies geometry through the use of concepts in commutative algebra such as multivariate polynomials. It has applications in many areas, including cryptography and string theory. | |

It is possible to use multivariate statistics to determine the main trends in phenotypic variability in a range of organisms, which for various major animal groups (most prominently vertebrates), has been shown to have three main endpoints consistent with UAST. UAST is a key part of the twin-filter model describing how species with similar overall strategies but divergent sets of minor traits coexist in ecological communities. | |

L1-PCA compared with PCA. Nominal data (blue points); outlier (red point); PC (black line); L1-PC (red line); nominal maximum-variance line (dotted line). L1-norm principal component analysis (L1-PCA) is a general method for multivariate data analysis. L1-PCA is often preferred over standard L2-norm principal component analysis (PCA) when the analyzed data may contain outliers (faulty values or corruptions). | |

The package was initially developed by the Spatial Analysis Laboratory of the University of Illinois at Urbana-Champaign under the direction of Luc Anselin. From 2016 development continues at the Center for Spatial Data Science (CSDS) at the University of Chicago. GeoDa has powerful capabilities to perform spatial analysis, multivariate exploratory data analysis, and global and local spatial autocorrelation. It also performs basic linear regression. | |

In mathematics, polynomial identity testing (PIT) is the problem of efficiently determining whether two multivariate polynomials are identical. More formally, a PIT algorithm is given an arithmetic circuit that computes a polynomial p in a field, and decides whether p is the zero polynomial. Determining the computational complexity required for polynomial identity testing is one of the most important open problems in algebraic computing complexity. | |

Waiting time between eruptions and the duration of the eruption for the Old Faithful Geyser in Yellowstone National Park, Wyoming, USA. This chart suggests there are generally two types of eruptions: short-wait-short- duration, and long-wait-long-duration. A 3D scatter plot allows the visualization of multivariate data. This scatter plot takes multiple scalar variables and uses them for different axes in phase space. | |

In finance, the shape is widely called a "hockey stick", due to the shape being similar to an ice hockey stick. hinge functions with a knot at x=3.1 In statistics, hinge functions of multivariate adaptive regression splines (MARS) are ramps, and are used to build regression models. In machine learning, it is commonly known as the rectifier used in rectified linear units (ReLUs). | |

In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. A toric ideal is an ideal generated by differences of monomials (provided the ideal is a prime ideal). An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or projective toric variety, possibly non-normal. | |

Data Desk was developed in 1985 by Paul F. Velleman, a statistics professor at Cornell University who had studied exploratory data analysis with John Tukey. Data Desk was released in 1986 for the Macintosh. It provided most standard statistical methods accessed through its own desktop interface. In 1997, Data Desk was released for Windows, and included a General Linear Model (GLM), multivariate statistics, and nonlinear curve fitting. | |

Starting in 1997, Mondrian was first developed with a focus on visualization techniques for categorical data and enhanced selection techniques. Over the years, a complete suite of visualizations for univariate and multivariate data measured on any scale were added. The link to R offers well tested statistical procedures, which integrate seamlessly into the interactive graphics. Today, even geographical data is supported with highly interactive maps. | |

There are two methods of doing this: balloon and pointwise estimation. In a balloon estimator, the kernel width is varied depending on the location of the test point. In a pointwise estimator, the kernel width is varied depending on the location of the sample. For multivariate estimators, the parameter, h, can be generalized to vary not just the size, but also the shape of the kernel. | |

Lanczos resampling is typically used to increase the sampling rate of a digital signal, or to shift it by a fraction of the sampling interval. It is often used also for multivariate interpolation, for example to resize or rotate a digital image. It has been considered the "best compromise" among several simple filters for this purpose. The filter is named after its inventor, Cornelius Lanczos (). | |

However, on the other hand, Cliff also suggested that there are viable and robust ordinal alternatives to mean comparisons. He introduced a measure of proportional difference (or dominance) between two sets of data often referred to as Cliff's delta. He has been president of the Psychometric Society and of the Society for Multivariate Experimental Psychology. Now an Emeritus Professor, he lives in New Mexico. | |

Lawrence C. Rafsky (Larry Rafsky), is an American data scientist, inventor, and entrepreneur. Rafsky created search algorithms and methodologies for the financial and news information industries. He invented the Friedman-Rafsky Test commonly used to test goodness-of-fit for the multivariate normal distribution. Rafsky founded and became chief scientist for Acquire Media, a news and information syndication company, now a subsidiary of Newscycle Solutions. | |

Theodore Wilbur Anderson (June 5, 1918 - September 17, 2016) was an American mathematician and statistician who has specialized in the analysis of multivariate data. He was born in Minneapolis, Minnesota. He was on the faculty of Columbia University from 1946 until moving to Stanford University in 1967, becoming Emeritus Professor in 1988. He served as Editor of Annals of Mathematical Statistics from 1950 to 1952. | |

By 1964 the site had an IBM 1620 computer. In 1965 the site formed an Operational Research Section at Port Sunlight, and their computers used PL/I and Fortran IV. In 1967 statisticians used control charts, timeseries analysis, multivariate analysis and stochastic processes. From early 1969 the consoles at the site were IBM 2780 with the MFT2 and HASPII operating systems. By 1969, new laboratories were built. | |

Then, already available and optimized solution methods can be used. The TFC theory has been developed for multivariate rectangular domains subject to absolute, integral, relative, and linear combinations of constraints . Numerically efficient applications of TFC have already been implemented in optimization problems, especially in solving differential equations . In this area, TFC has uniﬁed initial, boundary, and multi-value problems by providing fast solutions at machine-error accuracy. | |

From this, he constructed phylogenetic trees that showed genetic distances diagrammatically. His team also performed principal component analyses, which is good at analysing multivariate data with minimal loss of information. The information that is lost can be partly restored by generating a second principal component, and so on. In turn, the information from each individual principal component (PC) can be presented graphically in synthetic maps. | |

As a result of this method, care must be taken in the interpretation of Ferguson's three factors, as factor analysis will output an abstract factor whether an objectively real factor exists or not.SAS(R) 3.11 Users Guide, Multivariate Analysis: Factor Analysis Although replication of the nationalism factor was inconsistent, the finding of religionism and humanitarianism had a number of replications by Ferguson and others. | |

Regression analysis has become so sophisticated that some gamblers actually perform it as a full-time job. For example, Advanced Football Analytics ran a multivariate linear regression on the outcomes of American football games. The results determined that the most important aspect to winning the game was passing efficiency. One of the problems that results from using linear regression is determining causation vs. correlation. | |

The lattice condition for μ is also called multivariate total positivity, and sometimes the strong FKG condition; the term (multiplicative) FKG condition is also used in older literature. The property of μ that increasing functions are positively correlated is also called having positive associations, or the weak FKG condition. Thus, the FKG theorem can be rephrased as "the strong FKG condition implies the weak FKG condition". | |

It has been used to estimate the differences between the members of two populations. It has been generalized from univariate populations to multivariate populations, which produce samples of vectors. It is based on the Wilcoxon signed-rank statistic. In statistical theory, it was an early example of a rank-based estimator, an important class of estimators both in nonparametric statistics and in robust statistics. | |

The deviance information criterion (DIC) is a hierarchical modeling generalization of the Akaike information criterion (AIC). It is particularly useful in Bayesian model selection problems where the posterior distributions of the models have been obtained by Markov chain Monte Carlo (MCMC) simulation. DIC is an asymptotic approximation as the sample size becomes large, like AIC. It is only valid when the posterior distribution is approximately multivariate normal. | |

John Clinton Loehlin (January 13, 1926 - August 9, 2020) was an American behavior geneticist, computer scientist, and psychologist. Loehlin served as president of the Behavior Genetics Association and of the Society for Multivariate Experimental Psychology. He was an ISIR lifetime achievement awardee. He received an A.B. in English from Harvard in 1947, and a Ph.D. in Psychology from the University of California, Berkeley in 1957. | |

Validity in prescription-driven research is approached in different ways than descriptive research. The first difference deals with what some researchers call 'messy situations' (Brown 1992; Collins, Joseph, and Bielaczuc 2004). A messy situation is a real-life, a highly multivariate one is where independent variables cannot be minimized nor completely accounted for. In explanatory science, experiments are in controlled laboratories, where variables can be minimalized. | |

This is a nonexhaustive list of schools that offer degrees in quantitative psychology or related fields such as psychometrics or research methodology. Programs are typically offered in departments of psychology, educational psychology, or human development. Various organizations, including the American Psychological Association's Division 5, the Canadian Psychological Association, the National Council for Measurement in Education, and the Society of Multivariate Experimental Psychology have compiled lists of programs. | |

Originally, algebraic geometry was the study of common zeros of sets of multivariate polynomials. These common zeros, called algebraic varieties belong to an affine space. It appeared soon, that in the case of real coefficients, one must consider all the complex zeros for having accurate results. For example, the fundamental theorem of algebra asserts that a univariate square-free polynomial of degree has exactly complex roots. | |

The larva is an obligate egg-feeder and will starve without this form of nutrition.Feeding from ants and mites on the ground is popular after sunrise and not later than 16:00, when people are either on the ground or elevated on fallen logs finding refugee under leaf litter.Posso, Terranova, Andrés, and Jose Andrés. “Multivariate Species Boundaries and Conservation of Harlequin Poison Frogs.” Molecular Ecology, vol. | |

This implicit equation defines f as a function of x only if -1 \leq x \leq 1 and one considers only non- negative (or non-positive) values for the values of the function. The implicit function theorem provides conditions under which some kinds of relations define an implicit function, namely relations defined as the indicator function of the zero set of some continuously differentiable multivariate function. | |

Insana MF, Wagner RF, Garra BS, Momenan R, Shawker TH. Pattern recognition methods for optimizing multivariate tissue signatures in diagnostic ultrasound. Ultrason Imaging. 1986 Jul;8(3):165-80Bohs LN, Friemel BH, Trahey GE. Experimental velocity profiles and volumetric flow via two-dimensional speckle tracking. Ultrasound Med Biol. 1995;21(7):885-98 The method thus tracks a kernels motion from one frame to the next. | |

Having laid the foundations of the discipline of multivariate methods, as a committed naturalist and field biologist Greig-Smith never took much interest in abstruse theoretical discussions, always expressing his concern about the risk of becoming too engaged with theoretical refinements of methodology and repeatedly stating his belief that numerical methods are worth developing only if they are to be used on real data in attempts to answer real questions with relevance in the field. The ultimate evidence, for him, was not in the computer output but in the field. In a synthesis paper published in 1980, he insisted that multivariate methods are simply hypothesis-generating procedures, and counseled students to do experimental tests addressing the insights from the classification and ordination analyses rather than taking the results of their analyses as an unchallengeable truthGreig-Smith, Peter. 1980. The development of numerical classification and ordination. | |

Matiyasevich proved that there is no algorithm that, given a multivariate polynomial p(x1, x2,...,xk) with integer coefficients, determines whether there is an integer solution to the equation p = 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language of arithmetic, if a multivariate integer polynomial equation p = 0 does have a solution in the integers then any sufficiently strong system of arithmetic T will prove this. Moreover, if the system T is ω-consistent, then it will never prove that a particular polynomial equation has a solution when in fact there is no solution in the integers. Thus, if T were complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation has a solution by merely enumerating proofs of T until either "p has a solution" or "p has no solution" is found, in contradiction to Matiyasevich's theorem. | |

StatPlus is a software product developed by AnalystSoft for basic univariate and multivariate statistical analysis (MANOVA, GLM, Latin squares), as well as time series analysis, nonparametric statistics, survival analysis and statistical charts including control charts. It was originally developed for use in biomedical sciences and known as BioStat. It is nowadays mostly used in biomedicine and natural sciences. The software has a version for the Mac OS X known as StatPlus:mac. | |

The word problem for groups was proved algorithmically unsolvable by Pyotr Novikov in 1955 and independently by W. Boone in 1959. The busy beaver problem, developed by Tibor Radó in 1962, is another well-known example. Hilbert's tenth problem asked for an algorithm to determine whether a multivariate polynomial equation with integer coefficients has a solution in the integers. Partial progress was made by Julia Robinson, Martin Davis and Hilary Putnam. | |

Most of his paintings have vibrant colours, and the backgrounds are moving is dominant. Initially, multivariate fragility noted in his paintings that often reminds the artworks of the Fauvist artists. Paul Klee's painting are often remembered when analyzing the space and intention of Tareque's painting "Antarer Anusandhan" and "Naree Banam Naree". Tareque has often used the Arabic alphabet and the use of words in the art of painting. | |

In 1940, Raymond Cattell retained the adjectives, and eliminated synonyms to reduce the total to 171. He constructed a self-report instrument for the clusters of personality traits he found from the adjectives, which he called the Sixteen Personality Factor Questionnaire. In 1949, the first systematic multivariate research of personality was conducted by Joy P. Guilford. Guilford analyzed ten factors of personality, which he measured by the Guilford-Zimmerman Temperament Survey. | |

K C Sreedharan Pillai (1920–1985) was an Indian statistician who was known for his works on multivariate analysis and probability distributions. Pillai studied at the University of Travancore in Trivandrum. He graduated in 1941 and obtained his master's degree in 1945. He was appointed a lecturer at the University of Kerala in 1945 and worked there for six years until he went to the United States in 1951. | |

For univariate distributions that are symmetric about one median, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population median; for non-symmetric distributions, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population pseudo-median, which is the median of a symmetrized distribution and which is close to the population median. The Hodges–Lehmann estimator has been generalized to multivariate distributions. | |

Here each of the k included studies in turn is omitted and compared with the summary estimate derived from aggregating the remaining k- 1 studies. A general validation statistic, Vn based on IOCV has been developed to measure the statistical validity of meta-analysis results. For test accuracy and prediction, particularly when there are multivariate effects, other approaches which seek to estimate the prediction error have also been proposed. | |

Echinocactus horizonthalonius is a species of cactus known by several common names, including devilshead, turk's head cactus, blue barrel cactus, eagle's claw,Baker, M. (2007). A multivariate study of morphological characters for Echinocactus horizonthalonius and E. texensis (Cactaceae). USFWS. horse maimer, horse crippler, and visnaga meloncillo. It is native to the southwestern United States and northern Mexico, where it occurs in Chihuahuan Desert and Sonoran Desert habitats, particularly on limestone substrates. | |

Nešlehová was named as an Elected Member of the International Statistical Institute in 2011. In 2020 she was named a Fellow of the Institute of Mathematical Statistics. She was the 2019 winner of the CRM-SSC Prize in Statistics "for fundamental contributions to multivariate statistics, and in particular stochastic dependence modeling and extreme-value theory, and for her efforts to promote the sound application of statistics in risk management". | |

Goldberg, L. R (Ed), (2004) Personality Topics in Honor of Jerry S. Wiggins Multivariate Behavioral Research, Vol. 39, 2. As well as including references to Dr Wiggins' circumplex models it made particular reference to his contributions to the Minnesota Multiphasic Personality Inventory (MMPI) His work was also mentioned following his death in the newsletter for the professional society he helped found, the Society for Interpersonal Theory and Research. | |

The data necessary for a successful design are spectral characteristics of light sources, detectors and a variety of optics to be used in the final assemblage, dispersion characteristics of the materials used in the wavelength range of interest, and a set of calibrated sample spectra for pattern-recognition-based analysis. With these pieces assembled, suitable application specific multivariate optical computer designs can be generated and the performance accurately modeled and predicted. | |

Quasi-experimental approaches can remove bias arising from selection on observables and, where panel data are available, time invariant unobservables. Quasi-experimental methods include matching, differencing, instrumental variables and the pipeline approach; they are usually carried out by multivariate regression analysis. If selection characteristics are known and observed, they can be controlled for to remove the bias. Matching involves comparing program participants with non-participants based on observed selection characteristics. | |

Soil sampling and analysis is one of the most popular mineral exploration tools Martins-Ferreira, M. A. C., Campos, J. E. G., & Pires, A. C. B. (2017). Near-mine exploration via soil geochemistry multivariate analysis at the Almas gold province, Central Brazil: A study case. Journal of Geochemical Exploration, 173, 52-63. Mann, A. W., Birrell, R. D., Fedikow, M. A. F., & De Souza, H. A. F. (2005). | |

An algebraic hypersurface is an algebraic variety that may be defined by a single implicit equation of the form :p(x_1, \ldots, x_n)=0, where is a multivariate polynomial. Generally the polynomial is supposed to be irreducible. When this is not the case, the hypersurface is not an algebraic variety, but only an algebraic set. It may depend on the authors or the context whether a reducible polynomial defines a hypersurface. | |

Despite the variations, measurements are routinely taken in the process of bird ringing and for other studies. Several of the measurements are considered quite constant and well defined, at least in the vast majority of birds. Although field measurements are usually univariate, laboratory techniques can often make use of multivariate measurements derived from an analysis of variation and correlations of these univariate measures. These can often indicate variations more reliably. | |

Her 1980 dissertation, Approximation of Functions of Several Variables concerned function approximation for multivariate functions, and was supervised by David Sprecher. In 1977, before completing her doctorate, Diefenderfer took a faculty position at Hollins College, where she would remain for the rest of her career. She also served as chief reader for the AP Calculus exam for 2004–2007, and as president of the National Numeracy Network for 2011–2013. | |

Dynamic creative optimization (DCO), is a form of programmatic advertising that allows advertisers to optimize the performance of their creative using real-time technology. While the actual optimization approaches may vary, they almost always involve the use of multivariate testing. The DCO process consists of creative development, identification of test variables, definition of the optimization objective, and method of optimization. Creative development is done using creative studio tools like Adobe Photoshop. | |

In statistics, multivariate adaptive regression splines (MARS) is a form of regression analysis introduced by Jerome H. Friedman in 1991. It is a non- parametric regression technique and can be seen as an extension of linear models that automatically models nonlinearities and interactions between variables. The term "MARS" is trademarked and licensed to Salford Systems. In order to avoid trademark infringements, many open-source implementations of MARS are called "Earth". | |

Binford and her then-husband, Lewis Binford, co-founded the movement, however Binford was often denied credit for her involvement. Sally and Lewis co-edited New Perspectives in Archaeology (1968), deriving from a symposium held in 1965 in Denver at the annual American Anthropological Association Conference. Its success has been attributed to Sally's editing skills. A 1966 article on Mousterian Levallois lithics was an early application of multivariate statistics in archaeology. | |

For the within-subject effects, it is important to ensure normality and homogeneity of variance are not being violated. If the assumptions are violated, a possible solution is to use the Greenhouse–Geisser correctionGeisser, S. and Greenhouse, S.W. (1958). An extension of Box's result on the use of the F distribution in multivariate analysis. Annals of Mathematical Statistics, 29, 885–891 or the Huynh & FeldtHyunh, H. and Feldt, L.S. (1970). | |

Temporal map animation shows the ongoing gradual changes over time. Temporal maps can also be termed as animated timeline maps and can be a useful reference to examine the changes ongoing on each step and analyze the progression occurring gradually as time passes. There are many purposes which temporal animation might serve to depict: displaying and analyzing geographic patterns, meteorological events, climate, natural disasters, and other multivariate data. | |

Norman Cliff (born September 1, 1930) is an American psychologist. He received his Ph.D. from Princeton in psychometrics in 1957. After research positions in the US Public Health Service and at Educational Testing Service he joined the University of Southern California in 1962. He has had a number of research interests, including quantification of cognitive processes, scaling and measurement theory, computer-interactive psychological measurement, multivariate statistics, and ordinal methods. | |

His earlier publications were on h-statistics, l-statistics, k-statistics and related finite sampling methods. Later his research focused on order statistics and statistical inference based on optimal spacing and goodness-of- fit procedures. He has also published in diverse areas such as ranking and selection, multivariate distributions, characterization, and mixtures of distributions. Most recently he has published in the areas of Bayesian inference and reliability theory. | |

Drawing upon the background of his thesis, Wilks worked with the Educational Testing Service in developing the standardized tests like the SAT that have had a profound effect on American education. He also worked with Walter Shewhart on statistical applications in quality control in manufacturing. Wilks's lambda distribution is a probability distribution related to two independent Wishart distributed variables. It is important in multivariate statistics and likelihood-ratio tests. | |

In signal processing, independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents. This is done by assuming that the subcomponents are non-Gaussian signals and that they are statistically independent from each other. ICA is a special case of blind source separation. A common example application is the "cocktail party problem" of listening in on one person's speech in a noisy room. | |

This species is conspicuous in North America, where it may locally be known as the Halloween ladybeetle. It earns this name as it often invades homes during October to overwinter. When the species first arrived in the UK, it was labelled in jest as the "many- named ladybird" due to the great quantity of vernacular names. Among those already listed other names include multivariate, southern, Japanese, and pumpkin ladybird. | |

1.Traditionally, using pedigree data in humans, plants, and livestock species to estimate additive genetic variance. 2\. Using a single- nucleotide polymorphisms (SNP) regression method to quantify the contribution of additive, dominance, and imprinting variance to the total genetic variance 3\. Genetic variance–covariance (G ) matrices conveniently summarize the genetic relationships among a suite of traits and are a central parameter in the determination of the multivariate response to selection. | |

In the United States, 1973 was the most active tornado year up to that time, with over 1,100 confirmed tornadoes. Tornadoes killed 89 people nationwide, which exceeded the annual average of about 60, and there were almost 2,400 injuries. Greg Carbin of the Storm Prediction Center (SPC), upon examining data maintained, concluded that strong El Niño events—as measured by the multivariate ENSO index—may foster better conditions for more tornadoes. | |

Vector field reconstruction has several applications, and many different approaches. Some mathematicians have not only used radial basis functions and polynomials to reconstruct a vector field, but they have used Lyapunov exponents and singular value decomposition. Gouesbet and Letellier used a multivariate polynomial approximation and least squares to reconstruct their vector field. This method was applied to the Rössler system, and the Lorenz system, as well as thermal lens oscillations. | |

2008 Sep;48(9):1733-46. To investigate the AD of a training set of chemicals one can directly analyse properties of the multivariate descriptor space of the training compounds or more indirectly via distance (or similarity) metrics. When using distance metrics care should be taken to use an orthogonal and significant vector space. This can be achieved by different means of feature selection and successive principal components analysis. | |

Demonstrating that PLS regression is of a similar kind to the well-known canonical correlation in multivariate statistics. In 1996 he published "Prediction Methods in Science and Technology", a review of latent structure regression, including PLS regression. The following year Höskuldsson received the Herman Wold gold medal from the Swedish Chemical Society at the Scandinavian Symposium on Chemometrics as recognition of his contributions and pioneering work in the field of chemometrics. | |

Yuen began his career as an assistant professor at Chung-Ang University, Seoul, South Korea. He spent two and a half years at the university teaching maritime transport, transport logistics and economics, innovation logistics, principle of economics, statistics, and multivariate data analysis in English. Subsequently, he moved to Nanyang Technological University in Jan 2020, teaching quality management in shipping, shipping and the environment, maritime economics and maritime strategy. | |

A combination of HOSVD and SVD also has been applied for real-time event detection from complex data streams (multivariate data with space and time dimensions) in disease surveillance. It is also used in tensor product model transformation-based controller design. In multilinear subspace learning,Haiping Lu, K.N. Plataniotis and A.N. Venetsanopoulos, "A Survey of Multilinear Subspace Learning for Tensor Data", Pattern Recognition, Vol. 44, No. 7, pp. | |

When a parameterised kernel is used, optimisation software is typically used to fit a Gaussian process model. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. | |

In statistics and econometrics, the multinomial probit model is a generalization of the probit model used when there are several possible categories that the dependent variable can fall into. As such, it is an alternative to the multinomial logit model as one method of multiclass classification. It is not to be confused with the multivariate probit model, which is used to model correlated binary outcomes for more than one independent variable. | |

For instance, on an e-commerce website the purchase funnel is typically a good candidate for A/B testing, as even marginal improvements in drop-off rates can represent a significant gain in sales. Significant improvements can be seen through testing elements like copy text, layouts, images and colors. Multivariate testing or bucket testing is similar to A/B testing but tests more than two versions at the same time. | |

The difference between an agglomeration of an urban area and the daily urban system is that an agglomeration is a multivariate means of combining townships, counties, and other defined areas. It looks at shared economic relationships and other factors. Daily urban system, on the other hand, only attempts to show how far away people who commute into a city are living. It shows how much sprawl has occurred. | |

The Journal of Grey System, 2, pp. 111–123Liu, SF., Forrest, J., and Yang, Y. (2012), A brief introduction to grey systems theory, Grey Systems: Theory and Application, 2(2), pp. 89–104 The works of Professor Lotfi A. Zadeh significantly inspired Deng and the uncertainty of the Deng Xiaoping era inspired him to work on uncertain systems. In 1965, he proposed the theory of multivariate system control. | |

This is augmented by the continuous development of computer programs that can perform complex multivariate and multilevel analyses of data and models. Nonetheless, flaws inherent in social research must be considered, identified and moderated, and this is particularly relevant in Jury research. Jury research can be done with 'real jurors' or 'mock jurors'. Each of these methods have their downfalls and each provides a different slant on the juror experience. | |

Prof. Pleszczyńska is known for her criticism of the classic statistical approach. Classic parametric methods, like Pearson correlation coefficient, or least squares method produce comparable results only for comparable distribution types (in practice multivariate normal distribution is being assumed). Parametric statistical tests are derived from distribution assumptions. Classic methods fail if the input data contain strong outliers, and interpretation of their results should be different for different distribution types. | |

It uses multiple scatter plots to represent a pairwise relation among variables. Another statistical distribution approach to visualize multivariate data is parallel coordinates. Rather than graphing every pair of variables in two dimensions, the data is repeatedly plotted on a parallel axis and corresponding points are then connected with a line. The advantage of parallel coordinates is that they are relatively compact, allowing many variables to be shown simultaneously. | |

A study by Patronek et al. (1996) found in a univariate analysis that declawed cats were only 63% as likely to be relinquished as non-declawed cats. A multivariate analysis conducted in the same study shows odds of being relinquished to a shelter were 89% higher for declawed cats. The authors concluded that the conflicting results of the two analyses made it difficult to interpret the effects of declawing. | |

Nature Reviews Genetics 7:510-523. formalized its definition as “The study of the genetic interactions that occur between species and their abiotic environment in complex communities.” The field aims to bridge the gaps in the study of evolution and ecology, within the multivariate community context that ecological and evolutionary phenomena are embedded within. The documentary movie A Thousand Invisible Cords provides an introduction to the field and its implications. | |

Steven Holland's research interests combine sequence stratigraphy and paleobiology. He uses a mixture of computer simulation, field work, and multivariate data analysis to understand how the processes of sediment accumulation control the expression of the fossil record, and how to use this understanding to interpret the fossil record. This approach suggests that most ancient mass extinction events took place over hundreds of thousands of years, rather than as brief events. | |

Tidy data is an alternative name for the common statistical form called a model matrix or data matrix. A data matrix is defined in Krzanowski, W. J., F. H. C. Marriott, Multivariate Analysis Part 1, Edward Arnold, 1994 as follows: > A standard method of displaying a multivariate set of data is in the form of > a data matrix in which rows correspond to sample individuals and columns to > variables, so that the entry in the ith row and jth column gives the value > of the jth variate as measured or observed on the ith individual. Hadley Wickham later defined "Tidy Data" as data sets that are arranged such that each variable is a column and each observation (or case) is a row. (Originally with additional per-table conditions that made the definition equivalent to the Boyce–Codd 3rd normal form.) Data arrangement is an important consideration in data processing, but should not be confused with the also important task of data cleansing. | |

The amount of information about Z which is yielded by knowing both X and Y together is the information that is mutual to Z and the X,Y pair, written I(X,Y;Z) (yellow, gray and cyan in the Venn diagram above) and it may be greater than, equal to, or less than the sum of the two mutual information, this difference being the multivariate mutual information: I(X;Y;Z)=I(Y;Z)+I(X;Z)-I(X,Y;Z). In the case where the sum of the two mutual information is greater than I(X,Y;Z), the multivariate mutual information will be positive. In this case, some of the information about Z provided by knowing X is also provided by knowing Y, causing their sum to be greater than the information about Z from knowing both together. That is to say, there is a "redundancy" in the information about Z provided by the X and Y variables. | |

From this point of view, we can define the pseudo-determinant for a singular matrix to be the product of its nonzero eigenvalues (the density of multivariate normal distribution will need this quantity). In many applications, such as PageRank, one is interested in the dominant eigenvalue, i.e. that which is largest in absolute value. In other applications, the smallest eigenvalue is important, but in general, the whole spectrum provides valuable information about a matrix. | |

In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis. This theorem is also particularly important in particle physics, where it is known as Wick's theorem after the work of . Other applications include the analysis of portfolio returns, quantum field theory and generation of colored noise. | |

An elliptical distribution with a zero mean and variance in the form \alpha I where I is the identity-matrix is called a spherical distribution. For spherical distributions, classical results on parameter-estimation and hypothesis-testing hold have been extended. Similar results hold for linear models, and indeed also for complicated models ( especially for the growth curve model). The analysis of multivariate models uses multilinear algebra (particularly Kronecker products and vectorization) and matrix calculus. | |

Although the concept of using a single optical element for analyte regression and detection was suggested in 1986, the first full MOC concept device was published in 1997 from the Myrick group at the University of South Carolina, with a subsequent demonstration in 2001. The technique has received much recognition in the optics industry as a new method to perform optical analysis with advantages for harsh environment sensing.Myrick, M.L. (2002). "Multivariate optical elements simplify spectroscopy". | |

A multivariate polynomial is SOS-convex (or sum of squares convex) if its Hessian matrix H can be factored as H(x) = ST(x)S(x) where S is a matrix (possibly rectangular) which entries are polynomials in x. In other words, the Hessian matrix is a SOS matrix polynomial. An equivalent definition is that the form defined as g(x,y) = yTH(x)y is a sum of squares of forms. | |

More recent studies have continued the research into a correlation between height and intelligence, but again were often not directly related to height and intelligence. Some of the earlier large studies cited for height and intelligence are the Scottish Mental Surveys in 1932 and 1947. However, the studies were largely meant to analyze the genetic and environmental contributions to cognitive ability differences. Height (and weight) were added to provide a multivariate analysis. | |

Compared to modern day property tax evaluations, land valuations involve fewer variables and have smoother gradients than valuations that include improvements. This is due to variation of building style, quality and size between lots. Modern statistical techniques have eased the process; in the 1960s and 1970s, multivariate analysis was introduced as an assessment means. Usually, such a valuation process commences with a measurement of the most and least valuable land within the taxation area. | |

Multivariate analysis techniques are frequently used to examine landscape level vegetation patterns. Studies use statistical techniques, such as cluster analysis, canonical correspondence analysis (CCA), or detrended correspondence analysis (DCA), for classifying vegetation. Gradient analysis is another way to determine the vegetation structure across a landscape or to help delineate critical wetland habitat for conservation or mitigation purposes (Choesin and Boerner 2002). Climate change is another major component in structuring current research in landscape ecology. | |

A large number of tests were developed in the latter half of the 20th century (e.g., all the multivariate tests). Popular techniques (such as Hierarchical Linear Model, Arnold, 1992, Structural Equation Modeling, Byrne, 1996 and Independent Component Analysis, Hyvarinën, Karhunen and Oja, 2001) are relatively recent. In 1946, psychologist Stanley Smith Stevens organized levels of measurement into four scales, Nominal, Ordinal, Ratio, and Interval, in a paper that is still often cited. | |

Since closing down his fund in 1997, he began trading for his own account again in 1998, after mortgaging his house and selling his antique silver collection. This original fund is called Wimbledon Fund, the name reflecting his love of tennis. He began managing money for offshore clients in February 2002, with the Matador Fund. Niederhoffer employs proprietary computer programs that purports to predict short-term moves using multivariate time series analysis. | |

Before data mining algorithms can be used, a target data set must be assembled. As data mining can only uncover patterns actually present in the data, the target data set must be large enough to contain these patterns while remaining concise enough to be mined within an acceptable time limit. A common source for data is a data mart or data warehouse. Pre-processing is essential to analyze the multivariate data sets before data mining. | |

In cases where the tests are not independent, the null distribution of X2 is more complicated. A common strategy is to approximate the null distribution with a scaled random variable. Different approaches may be used depending on whether or not the covariance between the different p-values is known. Brown's method can be used to combine dependent p-values whose underlying test statistics have a multivariate normal distribution with a known covariance matrix. | |

Susan P. Holmes is a statistician and professor at Stanford University. She is noted for her work in applying nonparametric multivariate statistics, bootstrapping methods, and data visualization to biology. She received her PhD in 1985 from Université Montpellier II. She served as a tenured research scientist at INRA for ten years. She then taught at MIT, Harvard and was an associate professor of biometry at Cornell before moving to Stanford in 1998. | |

Canonical analysis is a multivariate technique which is concerned with determining the relationships between groups of variables in a data set. The data set is split into two groups X and Y, based on some common characteristics. The purpose of canonical analysis is then to find the relationship between X and Y, i.e. can some form of X represent Y. It works by finding the linear combination of X variables, i.e. X1, X2 etc. | |

Regression techniques can be used to determine if a specific case within a sample population is an outlier via the combination of two or more variable scores. Even for normal distributions, a point can be a multivariate outlier even if it is not a univariate outlier for any variable (consider a probability density concentrated along the line x_1 = x_2, for example), making Mahalanobis distance a more sensitive measure than checking dimensions individually. | |

Inverse distance weighting (IDW) is a type of deterministic method for multivariate interpolation with a known scattered set of points. The assigned values to unknown points are calculated with a weighted average of the values available at the known points. The name given to this type of methods was motivated by the weighted average applied, since it resorts to the inverse of the distance to each known point ("amount of proximity") when assigning weights. | |

Nearest neighbor interpolation (blue lines) in one dimension on a (uniform) dataset (red points). Nearest neighbor interpolation on a uniform 2D grid (black points). Each coloured cell indicates the area in which all the points have the black point in the cell as their nearest black point. Nearest- neighbor interpolation (also known as proximal interpolation or, in some contexts, point sampling) is a simple method of multivariate interpolation in one or more dimensions. | |

The most exciting feature of projection pursuit is that it is one of the very few multivariate methods able to bypass the "curse of dimensionality" caused by the fact that high- dimensional space is mostly empty. In addition, projection pursuit are able to ignore irrelevant (i.e. noisy and information-poor) variables. This is a distinct advantage over methods based on interpoint distances like minimal spanning trees, multidimensional scaling and most clustering techniques. | |

The human visual system perceives visual information as a pattern on the retina, which is 2-dimensional. Thus walking around the sculpture to understand it better creates a temporal sequence of 2-dimensional images in the brain. The multivariate data that is the original input for any grand tour visualization is a (finite) set of points in some high-dimensional Euclidean space. This kind of set arises naturally when data is collected. | |

Most simply, one may use a simple line graph, particularly for time series. For graphical qualitative comparison of 2-dimensional tabular data in several variables, a common alternative are Harvey balls, which are used extensively by Consumer Reports. Comparison in Harvey balls (and radar charts) may be significantly aided by ordering the variables algorithmically to add order. An excellent way for visualising structures within multivariate data is offered by principal component analysis (PCA). | |

It is rarely the case that there is a single treatment and control group. Often the "treatment" can be a variety of simple variations of a message or a multi-stage contact strategy that is classed as a single treatment. In the case of A/B or multivariate testing, uplift modelling can help in understanding whether the variations in tests provide any significant uplift compared to other targeting criteria such as behavioural or demographic indicators. | |

In general, random variables may be uncorrelated but statistically dependent. But if a random vector has a multivariate normal distribution then any two or more of its components that are uncorrelated are independent. This implies that any two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. | |

These add on a quadratic in the latent variable to the RR-VGLM class. The result is a bell- shaped curve can be fitted to each response, as a function of the latent variable. For R = 2, one has bell-shaped surfaces as a function of the 2 latent variables---somewhat similar to a bivariate normal distribution. Particular applications of QRR-VGLMs can be found in ecology, in a field of multivariate analysis called ordination. | |

The central object of a motion chart is a blob (or bubble), which is a solid object homeomorphic to a disc. Blobs have 3 important characteristics – size, position and appearance. Using variable mapping, motion charts allow control over the appearance of the blobs at different time points. This mechanism enhances the dynamic appearance of the data in the motion chart and facilitates the visual inspection of associations, patterns and trends in multivariate datasets. | |

Multivariate modeling can give answers to questions about the genetic relationship between variables that appear independent. For instance: do IQ and long-term memory share genes? Do they share environmental causes? Additional benefits include the ability to deal with interval, threshold, and continuous data, retaining full information from data with missing values, integrating the latent modeling with measured variables, be they measured environments, or, now, measured molecular genetic markers such as SNPs. | |

This analysis was described by his first scientific paper in 1922. During the course of these studies he found a way of comparing and grouping populations using a multivariate distance measure. This measure, denoted "D2" and now eponymously named Mahalanobis distance, is independent of measurement scale. Mahalanobis also took an interest in physical anthropology and in the accurate measurement of skull measurements for which he developed an instrument that he called the "profiloscope". | |

3 projects the multivariate signal down to an M−1 dimensional space where M is the number of categories. MDA is useful because most classifiers are strongly affected by the curse of dimensionality. In other words, when signals are represented in very-high-dimensional spaces, the classifier's performance is catastrophically impaired by the overfitting problem. This problem is reduced by compressing the signal down to a lower- dimensional space as MDA does. | |

Statistical testing relies on design of experiments. Several methods in use for multivariate testing include: #Full factorial the most straightforward method whereby all possible combinations of content variants are served with equal probability. #Discrete choice and what has mutated to become choice modeling is the complex technique that won Daniel McFadden the Nobel Prize in Economics in 2000. Choice modeling models how people make tradeoffs in the context of a purchase decision. | |

Analysis of variance – simultaneous component analysis (ASCA or ANOVA–SCA) is a method that partitions variation and enables interpretation of these partitions by SCA, a method that is similar to principal components analysis (PCA). This method is a multivariate or even megavariate extension of analysis of variance (ANOVA). The variation partitioning is similar to ANOVA. Each partition matches all variation induced by an effect or factor, usually a treatment regime or experimental condition. | |

In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval [0, 1]. Copulas are used to describe the dependence between random variables. Their name comes from the Latin for "link" or "tie", similar but unrelated to grammatical copulas in linguistics. Copulas have been used widely in quantitative finance to model and minimize tail risk and portfolio-optimization applications. | |

75 or nothing is known about sphericity, the Greenhouse–Geisser correction should be applied. Another alternative procedure is using the multivariate test statistics (MANOVA) since they do not require the assumption of sphericity. However, this procedure can be less powerful than using a repeated measures ANOVA, especially when sphericity violation is not large or sample sizes are small. O’Brien and Kaiser suggested that when you have a large violation of sphericity (i.e. | |

Sociologists of religion have stated that religious behaviour may have a concrete impact on a person's life. These consequences of religiosity are thought to include emotional and physical health, spiritual well-being, personal, marital, and family happiness. Although a simple correlation between religiosity and well-being is repeatedly reported in the research literature, recent multivariate research (which controls for other predictors of well- being) suggests religiosity's contribution to happiness is minuscule and sometimes negative. | |

While the online tools were primitive at the time it was deemed to be valuable in collecting consumer insights. This service was first brought to market by www.userlytics.com, and initially focused on the website usability and user experience field. However, its uses have since expanded to hosted prototype testing, ad and campaign optimization prior to multivariate testing, understanding analytics results, desktop and enterprise user interface (UI) testing, and software as a service (SaaS) testing. | |

The typical use of discriminants in algebraic geometry is for studying algebraic curve and, more generally algebraic hypersurfaces. Let be such a curve or hypersurface; is defined as the zero set of a multivariate polynomial. This polynomial may be considered as a univariate polynomial in one of the indeterminates, with polynomials in the other indeterminates as coefficients. The discriminant with respect to the selected indeterminate defines a hypersurface in the space of the other indeterminates. | |

An example of Gaussian Process Regression (prediction) compared with other regression models.The documentation for scikit-learn also has similar examples. A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference. Given any set of N points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian. | |

When bivariate Gaussian copulas are assigned to edges of a vine, then the resulting multivariate density is the Gaussian density parametrized by a partial correlation vine rather than by a correlation matrix. The vine pair-copula construction, based on the sequential mixing of conditional distributions has been adapted to discrete variables and mixed discrete/continuous response . Also factor copulas, where latent variables have been added to the vine, have been proposed (e.g., ). | |

It is also used for various applications of finite fields, such as coding theory (cyclic redundancy codes and BCH codes), cryptography (public key cryptography by the means of elliptic curves), and computational number theory. As the reduction of the factorization of multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with one variable are considered in this article. | |

NeuroSolutions is a neural network development environment developed by NeuroDimension. It combines a modular, icon-based (component-based) network design interface with an implementation of advanced learning procedures, such as conjugate gradients, Levenberg-Marquardt and backpropagation through time. The software is used to design, train and deploy neural network (supervised learning and unsupervised learning) models to perform a wide variety of tasks such as data mining, classification, function approximation, multivariate regression and time-series prediction. | |

The multivariate mutual-information functions generalize the pairwise independence case that states that X_1,X_2 if and only if I(X_1;X_2)=0, to arbitrary numerous variable. n variables are mutually independent if and only if the 2^n-n-1 mutual information functions vanish I(X_1;...;X_k)=0 with n \ge k \ge 2 (theorem 2 ). In this sense, the I(X_1;...;X_k)=0 can be used as a refined statistical independence criterion. | |

The multivariate mutual-information functions generalize the pairwise independence case that states that X_1,X_2 if and only if I(X_1;X_2)=0, to arbitrary numerous variable. n variables are mutually independent if and only if the 2^n-n-1 mutual information functions vanish I(X_1;...;X_k)=0 with n \ge k \ge 2 (theorem 2 ). In this sense, the I(X_1;...;X_k)=0 can be used as a refined statistical independence criterion. | |

A complication is that this multivariate mutual information (as well as the interaction information) can be positive, negative, or zero, which makes this quantity difficult to interpret intuitively. In fact, for n random variables, there are 2^n-1 degrees of freedom for how they might be correlated in an information- theoretic sense, corresponding to each non-empty subset of these variables. These degrees of freedom are bounded by the various inequalities in information theory. | |

Multivariate statistical methods can be used to test statistical hypotheses about factors that affect shape and to visualize their effects. To visualize the patterns of variation in the data, the data need to be reduced to a comprehensible (low- dimensional) form. Principal component analysis (PCA) is a commonly employed tool to do summarize the variation. Simply put, the technique projects as much of the overall variation as possible into a few dimensions. | |

Multivariate optimization is one of the most common methods for product optimization. In this method, multiple product attributes are specified and then tested with consumers. Due to complex interaction effects between different attributes (for example, consumers frequently associate certain flavors with packaging colors), it is problematic to use mathematical methods, such as Conjoint Analysis, typically used in industrial process optimization. More recently companies started to adopt Evolutionary Optimization techniques for Product optimization. | |

Monti is the daughter of Katherine (Kit) Buckley Nuckolls, the former chair of pediatric nursing at Yale University. She graduated from Oberlin College in 1971, married sociologist Daniel J. Monti Jr., and completed a Ph.D. in biostatistics in 1975 at the University of North Carolina at Chapel Hill. Her dissertation, The Locally Optimal Combination of Certain Multivariate Test Statistics, was supervised by Pranab K. Sen. She became a faculty member at the University of Missouri–St. | |

The Bott Hypothesis is a thesis first advanced in Elizabeth Bott's Family and Social Networks (1957), one of the most influential works published in the sociology of the family. Elizabeth Bott's hypothesis holds that the connectedness or the density of a husband's and wife's separate social networks is positively associated with marital role segregation.Michael Gordon and Helen Downing. 1978. A Multivariate Test of the Bott Hypothesis in an Urban Irish Setting, Journal of Marriage and Family, Vol. | |

This algorithm is often confused with the k-medoids algorithm. However, a medoid has to be an actual instance from the dataset, while for the multivariate Manhattan- distance median this only holds for single attribute values. The actual median can thus be a combination of multiple instances. For example, given the vectors (0,1), (1,0) and (2,2), the Manhattan-distance median is (1,1), which does not exist in the original data, and thus cannot be a medoid. | |

His Ph.D. dissertation was entitled Robust Estimation for Multivariate Location and Scale in the Presence of Asymmetry and was supervised by John R. Collins.. After receiving his Ph.D. in 1982, Wiens took a faculty position at Dalhousie University, and moved in 1987 to Alberta. Wiens was editor-in-chief of The Canadian Journal of Statistics from 2004 to 2006CJS editorial board, retrieved 2010-01-07. and program chair of the 2003 annual meeting of the Statistical Society of Canada. | |

McNicholas started his faculty career at the University of Guelph in 2007, and, in 2014, he moved to McMaster University. He has authored more than 100 scientific works and has been cited over 4000 times. The majority of his work is in the area of model-based clustering, specifically in developing novel finite mixture models for clustering and classification of multivariate data. He has published works on clustering high-dimensional data and the use of non-Gaussian mixtures. | |

She uses complementary image analysis technique (including multivariate, univariate and connectivity analysis) to answer her research questions. Most recently, Rajah has begun to study memory decline in healthy middle-aged adults over the age of 40. She designed an experiment that showed participants a series of faces, and subsequently asked them to identify where and when the image of a particular face appeared on a screen. During the study Rajah monitored the participant's brains using MRI. | |

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gateaux derivative. | |

Multivariate pattern analysis using EEG has suggested that an evidence-based perceptual decision model may be applicable to free-will decisions. It was found that decisions could be predicted by neural activity immediately after stimulus perception. Furthermore, when the participant was unable to determine the nature of the stimulus, the recent decision history predicted the neural activity (decision). The starting point of evidence accumulation was in effect shifted towards a previous choice (suggesting a priming bias). | |

Training for quantitative psychology can begin informally at the undergraduate level. Many graduate schools recommend that students have some coursework in psychology and complete the full college sequence of calculus (including multivariate calculus) and a course in linear algebra. Quantitative coursework in other fields such as economics and research methods and statistics courses for psychology majors are also helpful. Historically, however, students without all these courses have been accepted if other aspects of their application show promise. | |

When we have two or three sources acting simultaneously, which is a common situation, we shall get dense and disorganized structure of connections, similar to random structure (at best some "small world" structure may be identified). This kind of pattern is usually obtained in case of application of bivariate measures. In fact, effective connectivity patterns yielded by EEG or LFP measurements are far from randomness, when proper multivariate measures are applied, as we shall demonstrate below. | |

For an example of the application of this formula, see the article on the Lyapunov equation. This formula also comes in handy in showing that the matrix normal distribution is a special case of the multivariate normal distribution. This formula is also useful for representing 2D image processing operations in matrix-vector form. Another example is when a matrix can be factored as a Hadamard product, then matrix multiplication can be performed faster by using the above formula. | |

Webtrends acquired Seattle based Widemile, a provider of multivariate testing and targeting on July 30, 2009. The product was rebranded and relaunched as Webtrends Optimize the day the acquisition was made public. Webtrends acquired San Francisco based Transpond, a maker of social microsites and applications that can be distributed over the web or Facebook, on August 10, 2010. The product was rebranded as Webtrends Apps at the time of the announcement and was later rebranded as Webtrends Social. | |

Peter Hans Schönemann (July 15, 1929 – April 7, 2010) was a German born psychometrician and statistical expert. He was professor emeritus in the Department of Psychological Sciences at Purdue University. His research interests included multivariate statistics, multidimensional scaling and measurement, quantitative behavior genetics, test theory and mathematical tools for social scientists. He published around 90 papers dealing mainly with the subjects of psychometrics and mathematical scaling. Schönemann’s influences included Louis Guttman, Lee Cronbach, Oscar Kempthorne and Henry Kaiser. | |

Dyar, M.D., Fassett, C.I., Giguere, S., Lepore, K., Byrne, S., Boucher, T., Carey, CJ, and Mahadevan, S. (2016) Comparison of univariate and multivariate models for prediction of major and minor elements from laser-induced breakdown spectra with and without masking. Spectrochimica Acta Part B, 123, 93-104. Dyar, M.D., Giguere, S., Carey, CJ, and Boucher, T. (2016) Comparison of baseline removal methods for laser-induced breakdown spectroscopy of geological samples. Spectrochimica Acta Part B, 126, 53-64. | |

The parametric Goldfeld–Quandt test offers a simple and intuitive diagnostic for heteroskedastic errors in a univariate or multivariate regression model. However some disadvantages arise under certain specifications or in comparison to other diagnostics, namely the Breusch–Pagan test, as the Goldfeld–Quandt test is somewhat of an ad hoc test. Primarily, the Goldfeld–Quandt test requires that data be ordered along a known explanatory variable. The parametric test orders along this explanatory variable from lowest to highest. | |

Mahalanobis distance is preserved under full-rank linear transformations of the space spanned by the data. This means that if the data has a nontrivial nullspace, Mahalanobis distance can be computed after projecting the data (non- degenerately) down onto any space of the appropriate dimension for the data. We can find useful decompositions of the squared Mahalanobis distance that help to explain some reasons for the outlyingness of multivariate observations and also provide a graphical tool for identifying outliers. | |

PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. | |

Meng Chen, Gregory Philip Wilson, A multivariate approach to infer locomotor modes in Mesozoic mammals, Article in Paleobiology 41(02) · February 2015 DOI: 10.1017/pab.2014.14 The evolution of Didelphodon and other large stagodontids (as well as large deltatheroideans like Nanocuris) occurs after the local extinction of eutriconodont mammals, suggesting passive or direct ecological replacement.G. W. Rougier, B. M. Davis, and M. J. Novacek. 2015. A deltatheroidan mammal from the Upper Cretaceous Baynshiree Formation, eastern Mongolia. | |

Christian Genest (born January 11, 1957, in Chicoutimi, Quebec) is a professor in the Department of Mathematics and Statistics at McGill University (Montréal, Canada), where he holds a Canada Research Chair. He is the author of numerous research papers in multivariate analysis, nonparametric statistics, extreme-value theory, and multiple-criteria decision analysis. He is a recipient of the Statistical Society of Canada's Gold Medal for Research and was elected a Fellow of the Royal Society of Canada in 2015. | |

Ada K. Dietz (June 16, 1882 – May 13, 1950) was an American weaver best known for her 1949 monograph Algebraic Expressions in Handwoven Textiles, which defines a novel method for generating weaving patterns based on algebraic patterns. Her method employs the expansion of multivariate polynomials to devise a weaving scheme. Dietz' work is still well-regarded today, by both weavers and mathematicians. Along with the references listed below, Griswold (2001) cites several additional articles on her work. | |

The Master of Science in QM and Analytics is a three to four semesters Master’s Degree that is designed to prepare students to use data-driven methods to contribute to organizational effectiveness and to help guide decisions. Key topics may include decision making, optimization and simulation methods, predictive modeling, and multivariate statistics. Teamwork, written and oral communication, presentation and other skills that are crucial for students to prepare themselves for a professional career are also addressed throughout the curriculum. | |

To create and validate signatures, a minimal quadratic equation system must be solved. Solving equations with variables is NP-hard. While the problem is easy if is either much much larger or much much smaller than , importantly for cryptographic purposes, the problem is thought to be difficult in the average case when and are nearly equal, even when using a quantum computer. Multiple signature schemes have been devised based on multivariate equations with the goal of achieving quantum resistance. | |

In statistics, the rational quadratic covariance function is used in spatial statistics, geostatistics, machine learning, image analysis, and other fields where multivariate statistical analysis is conducted on metric spaces. It is commonly used to define the statistical covariance between measurements made at two points that are d units distant from each other. Since the covariance only depends on distances between points, it is stationary. If the distance is Euclidean distance, the rational quadratic covariance function is also isotropic. | |

In statistics, the Matérn covariance, also called the Matérn kernel, is a covariance function used in spatial statistics, geostatistics, machine learning, image analysis, and other applications of multivariate statistical analysis on metric spaces. It is named after the Swedish forestry statistician Bertil Matérn. It is commonly used to define the statistical covariance between measurements made at two points that are d units distant from each other. Since the covariance only depends on distances between points, it is stationary. | |

Extreme value theory in more than one variable introduces additional issues that have to be addressed. One problem that arises is that one must specify what constitutes an extreme event. Although this is straightforward in the univariate case, there is no unambiguous way to do this in the multivariate case. The fundamental problem is that although it is possible to order a set of real-valued numbers, there is no natural way to order a set of vectors. | |

The calculated effect partitions are called effect estimates. Because even the effect estimates are multivariate, interpretation of these effects estimates is not intuitive. By applying SCA on the effect estimates one gets a simple interpretable result.Smilde, Age K.; Jansen, Jeroen J.; Hoefsloot, Huub C. J.; Lamers, Robert-Jan A. N.; van der Greef, Jan; Timmerman, Marieke E. (2005) "ANOVA- simultaneous component analysis (ASCA): a new tool for analyzing designed metabolomics data", Bioinformatics, 21 (13), 3043-3048. | |

In engineering, mathematics and the physical, and biological sciences, common terms for the points around which the system gravitates include: attractors, stable states, eigenstates/eigenfunctions, equilibrium points, and setpoints. In control theory, negative refers to the sign of the multiplier in mathematical models for feedback. In delta notation, −Δoutput is added to or mixed into the input. In multivariate systems, vectors help to illustrate how several influences can both partially complement and partially oppose each other. | |

An example of two connected graphs In multivariate statistics and the clustering of data, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is provided as an input and consists of a quantitative assessment of the relative similarity of each pair of points in the dataset. In application to image segmentation, spectral clustering is known as segmentation-based object categorization. | |

Ms Gowen joined UCD in 2007 as a postdoctoral researcher, working on Hyperspectral Imaging (HSI).Her research involves the application and growth of multivariate analysis and image processing techniques for non-destructive assessment of biological products using HSI. In 2010 Ms Gowen spent two years in Kobe University, Japan, studying and researchingHyperspectral imaging. This research gave rise to her current endeavour, which is probing what happens to water at interfaces where it interacts with various materials. | |

Hardin is a Pomona graduate, earning a bachelor's degree there in mathematics in 1995. She initially planned to do actuarial science, but was led to statistics by a faculty mentor, Donald Bentley. She went to the University of California, Davis for her graduate studies, earning a master's degree in 1997 and a Ph.D. in 2000. Her dissertation, supervised by David Rocke, was Multivariate Outlier Detection and Robust Clustering with Minimum Covariance Determinant Estimation and S-Estimation. | |

For electrical circuits feeding outdoor lighting, the number of hours of darkness can be employed. For a borehole pump, the quantity of water delivered would be used; and so on. What these examples all have in common is that on a weekly basis (say) numerical values can be recorded for each factor and one would expect particular streams of energy consumption to correlate with them either singly or in a multivariate model. Correlation is arguably more important than causality. | |

He obtained a PhD from the University of Paris in 1968 for a thesis entitled Autour de la platitude. His doctoral supervisor was Pierre Samuel.. Lazard began his academic career by working in commutative algebra, especially on flat modules. Around 1970, he began to work in computer algebra, which, soon after, became his main research area. In this field, he is specially interested in multivariate polynomials and more generally in computational algebraic geometry, with emphasis on polynomial system solving. | |

J. N. Srivastava received a Ph.D. in 1962 Srivastava's Ph.D. from the University of North Carolina at Chapel Hill. Prof. R. C. Bose was Srivastava's advisor. He joined Colorado State University in 1966.Some Prehistory of the Department of Statistics and Statistical Laboratory at Colorado State University He was known for his contributions in design of experiments as well as in multivariate analysis, survey sampling, reliability, coding theory, combinatorial theory, and other areas of statistics and mathematics. | |

Royen published this proof in an article with the title A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions on arXiv Supplemented by and subsequently in the Far East Journal of Theoretical Statistics,Thomas Royen: A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions, in: Far East Journal of Theoretical Statistics, Part 48 Nr. 2, Pushpa Publishing House, Allahabad 2014, p.139–145 a relatively unknown periodical based in Allahabad, India, for which Royen was at the time voluntarily working as a referee himself. Due to this, his proof went at first largely unnoticed by the scientific community,In the Quanta magazine article for instance Tilmann Gneiting, a statistician at the Heidelberg Institute for Theoretical Studies, just 65 miles from Bingen, said he was shocked to learn in July 2016, two years after the fact, that the GCI had been proved. until in late 2015 two Polish mathematicians, Rafał Latała and Dariusz Matlak, wrote a paper in which they reorganized Royen's proof in a way that was intended to be easier to follow. | |

Nowadays, this methodology is being adapted from traditional multi-variate techniques to carry out analysis on financial data sets such as stock market indices, generation of implied volatility graphs and so on.Functional Data Analysis with Applications in Finance by Michal Benko A very nice example of the advantages of the functional approach is the Smoothed FPCA (SPCA), proposed by Silverman [1996] and studied by Pezzulli and Silverman [1993] that enables direct combination of the FPCA analysis together with a general smoothing approach that makes the use of the information stored in some linear diﬀerential operators possible. An important application of the FPCA already known from multivariate PCA, is motivated by the Karhunen-Loève decomposition of a random function to the set of functional parameters – factor functions and corresponding factor loadings (scalar random variables). This application is much more important than in the standard multivariate PCA since the distribution of the random function is in general too complex to be directly analyzed and the Karhunen-Loève decomposition reduces the analysis to the interpretation of the factor functions and the distribution of scalar random variables. | |

1–17 Other techniques supported include principal component analysis (PCA),S. De Vries, Cajo J.F. Ter Braak (1995) Prediction error in partial least squares regression: a critique on the deviation used in The Unscrambler Chemometrics and Intelligent Laboratory Systems 30:239-245 PDF 3-way PLS, multivariate curve resolution, design of experiments, supervised classification, unsupervised classification and cluster analysis.Kristian Helland (1991) UNSCRAMBLER 11, version 3.10: A program for multivariate analysis with PLS and PCA/PCR Journal of Chemometrics 5(4):413-415 The software is used in spectroscopy (IR, NIR, Raman, etc.), chromatography, and process applications in research and non-destructive quality control systems in pharmaceutical manufacturing,M.R. Maleki, A.M. Mouazen, H. Ramon and J. De Baerdemaeker (2006) Multiplicative Scatter Correction during On-line Measurement with Near Infrared Spectroscopy Biosystems Engineering 96(3):427-433 Tatavarti AS, Fahmy R, Wu H, Hussain AS, Marnane W, Bensley D, Hollenbeck G, Hoag SW. Assessment of NIR Spectroscopy for Nondestructive Analysis of Physical and Chemical Attributes of Sulfamethazine Bolus Dosage Forms AAPS PharmSciTech. 2005; 06(01): E91-E99. | |

When the mutation step is drawn from a multivariate normal distribution using an evolving covariance matrix, it has been hypothesized that this adapted matrix approximates the inverse Hessian of the search landscape. This hypothesis has been proven for a static model relying on a quadratic approximation. The (environmental) selection in evolution strategies is deterministic and only based on the fitness rankings, not on the actual fitness values. The resulting algorithm is therefore invariant with respect to monotonic transformations of the objective function. | |

Another imputation technique involves replacing any missing value with the mean of that variable for all other cases, which has the benefit of not changing the sample mean for that variable. However, mean imputation attenuates any correlations involving the variable(s) that are imputed. This is because, in cases with imputation, there is guaranteed to be no relationship between the imputed variable and any other measured variables. Thus, mean imputation has some attractive properties for univariate analysis but becomes problematic for multivariate analysis. | |

In 2013, Busi et al., investigated 14,093 US patients with acute non-traumatic subdural haemorrhage. In multivariate analysis, weekend admission (OR = 1.19; 95% CI 1.02-1.38) was an independent predictor of in-hospital mortality. Similarly, in 2017, Rumalia et al., in an American study of 404,212 patients with traumatic SDH, showed that weekend admission was associated with an increased likelihood of in-hospital complication (OR = 1.06-1.12), prolonged length of stay (OR = 1.08-1.17), and in-hospital mortality (OR: 1.04-1.11). | |

Hotelling is known to statisticians because of Hotelling's T-squared distribution which is a generalization of the Student's t-distribution in multivariate setting, and its use in statistical hypothesis testing and confidence regions. He also introduced canonical correlation analysis. At the beginning of his statistical career Hotelling came under the influence of R.A. Fisher, whose Statistical Methods for Research Workers had "revolutionary importance", according to Hotelling's review. Hotelling was able to maintain professional relations with Fisher, despite the latter's temper tantrums and polemics. | |

This proposal does restrict each trial to two interventions, but also introduces a workaround for multiple arm trials: a different fixed control node can be selected in different runs. It also utilizes robust meta-analysis methods so that many of the problems highlighted above are avoided. Further research around this framework is required to determine if this is indeed superior to the Bayesian or multivariate frequentist frameworks. Researchers willing to try this out have access to this framework through a free software. | |

To compare two such signatures with the EMD, one must define a distance between features, which is interpreted as the cost of turning a unit mass of one feature into a unit mass of the other. The EMD between two signatures is then the minimum cost of turning one of them into the other. EMD analysis has been used for quantitating multivariate changes in biomarkers measured by flow cytometry, with potential applications to other technologies that report distributions of measurements. | |

Dagstuhl Seminar 12241 Data Reduction and Problem Kernels June 10 – 15, 2012 was the occasion to honour Michael R. Fellows on the Occasion of His 60th Birthday. He was presented with a Springer festschrift: The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated Michael R. Fellows on the Occasion of His 60th Birthday. Editors: Hans L. Bodlaender and Rod Downey and Fedor V. Fomin and Daniel Marx. Springer LNCS 7370, DOI 10.1007/978-3- 642-30891-8_8), 2012. 1) Academy Europaea (MAE) 2018. | |

Arguedas' novel reveals his proposed solution to indigenous problems: Andean culture must not be destroyed, as part of some or other form of modernization that assimilates. Harmonious thinking with nature is accepted, in order to develop a revolutionary mindset that projects a future of well-being and freedom. The national ideal is that of multivariate Peru, with ecological, multicultural and multilingual diversity.as witnessed in the Peruvian Constitution of 1993, which aims to preserve diversity of customs, alternative medicine, languages etc. | |

This method can be applied to several multivariate signals but it seems that most works on it concern electroencephalographic signals. Particularly, the method is mostly used on brain–computer interface in order to retrieve the component signals which best transduce the cerebral activity for a specific task (e.g. hand movement).G. Pfurtscheller, C. Guger and H. Ramoser "EEG-based brain-computer interface using subject-specific spatial filters", Engineering applications of bio- inspired artificial neural networks, Lecture Notes in Computer Science, 1999, Vol. | |

Undertaken up to the present day provides only modest support to the proposal, that workforce diversity per se brings business benefits with it. Therefore, this idea remains open to debate and further research. In short, whether diversity pays off or not depends on environmental factors, internal or external to the firm. Dwyer, Richard & Chadwyck (2003) found that the effects of gender diversity at the management level are conditional on the firm's strategic orientation, the organizational culture and the multivariate interaction among these variables. | |

Ki-67 is an excellent marker to determine the growth fraction of a given cell population. The fraction of Ki-67-positive tumor cells (the Ki-67 labeling index) is often correlated with the clinical course of cancer. The best-studied examples in this context are prostate, brain and breast carcinomas, as well as nephroblastoma and neuroendocrine tumors. For these types of tumors, the prognostic value for survival and tumor recurrence have repeatedly been proven in uni- and multivariate analysis. | |

Butterfly way station sign Butterfly gardening provides a recreational activity to view butterflies interacting with the environment. Besides anthropocentric values of butterfly gardening, creating habitat reduces the impacts of habitat fragmentation and degradation. Habitat degradation is a multivariate issue; development, increased use of pesticides and herbicides, woody encroachment, and non-native plants are contributing factors to the decline in butterfly and pollinator habitat. Pollination is one ecological service butterflies provide; about 90% of flowering plants and 35% of crops rely on animal pollination. | |

Segmented regression, also known as piecewise regression or broken-stick regression, is a method in regression analysis in which the independent variable is partitioned into intervals and a separate line segment is fit to each interval. Segmented regression analysis can also be performed on multivariate data by partitioning the various independent variables. Segmented regression is useful when the independent variables, clustered into different groups, exhibit different relationships between the variables in these regions. The boundaries between the segments are breakpoints. | |

The testable definition of causality was introduced by Granger. Granger causality principle states that if some series Y(t) contains information in past terms that helps in the prediction of series X(t), then Y(t) is said to cause X(t). Granger causality principle can be expressed in terms of two-channel multivariate autoregressive model (MVAR). Granger in his later work pointed out that the determination of causality is not possible when the system of considered channels is not complete. | |

Grigoriev,Dima, Karpinski,Marek, and Singer,Michael F., "Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields", SIAM J. Comput., Vol 19, No.6, pp. 1059-1063, December 1990 A low degree PIT has an upper bound on the degree of the polynomial. Any low degree PIT problem can be reduced in subexponential time of the size of the circuit to a PIT problem for depth-four circuits; therefore, PIT for circuits of depth-four (and below) is intensely studied. | |

We need outside information about what the name means. Using a data base (such as the internet) and a means to search the database (such as a search engine like Google) provides this information. Every search engine on a data base that provides aggregate page counts can be used in the normalized Google distance (NGD). A python package for computing all information distances and volumes, multivariate mutual information, conditional mutual information, joint entropies, total correlations, in a dataset of n variables is available .. | |

The case of univariate polynomials over a field is especially important for several reasons. Firstly, it is the most elementary case and therefore appears in most first courses in algebra. Secondly, it is very similar to the case of the integers, and this analogy is the source of the notion of Euclidean domain. A third reason is that the theory and the algorithms for the multivariate case and for coefficients in a unique factorization domain are strongly based on this particular case. | |

Analyses have been conducted to test the relative fit of categorical and dimensional modals to evaluate whether single diagnostic categories are suited to either status. These types of analysis can include a range of data, including endophenotypes or other genetic or biological markers which increases their utility. Multivariate genetic analysis helps establish how well the current phenotypically developed structure of personality disorder diagnosis fits with the genetic structure underlying personality disorders. Results from these types of analysis support dimensional over categorical approaches. | |

" Researchers who conducted a study in Iowa reported that "In this population-based case-control investigation, we report further evidence that alcohol consumption decreases the risk of RCC among women but not among men. Our ability to show that the association remains after multivariate adjustment for several new confounding factors (i.e., diet, physical activity, and family history) strengthens support for a true association. Another study found no relationship between alcohol consumption and risk of kidney cancer among either men or women. | |

ICA on four randomly mixed videos Independent component analysis attempts to decompose a multivariate signal into independent non- Gaussian signals. As an example, sound is usually a signal that is composed of the numerical addition, at each time t, of signals from several sources. The question then is whether it is possible to separate these contributing sources from the observed total signal. When the statistical independence assumption is correct, blind ICA separation of a mixed signal gives very good results. | |

Near-infrared spectroscopy is, therefore, not a particularly sensitive technique, but it can be very useful in probing bulk material with little or no sample preparation. The molecular overtone and combination bands seen in the near-IR are typically very broad, leading to complex spectra; it can be difficult to assign specific features to specific chemical components. Multivariate (multiple variables) calibration techniques (e.g., principal components analysis, partial least squares, or artificial neural networks) are often employed to extract the desired chemical information. | |

In 1971 Robert Chinnock published a new species name, D. blackii, for some New Zealand material, and five years later he transferred M. clavellatum to Disphyma. In the early 1980s, Hugh Francis Glen determined, on the basic of a multivariate analysis, that Disphyma was monotypic. All other names were therefore given synonymy with D. crassifolium. This situation remained until 1986, when it was decided that the South African populations differed sufficiently from the Australian and New Zealand populations to merit distinct subspecies. | |

The most widely cited of these (Wilk and Gnanadesikan, 1968) describes the Q-q and P-p plots, which are used to compare different statistical distributions. This work arose in part out of work in speaker recognition, as it was found that the statistical distribution of energy across frequency bands was different for different speakers.Kettenring, p.298 His highly-cited 1977 monograph on multivariate data analysis has been translated into Japanese and Russian and a second edition was published in 1997. | |

The case where ordering doesn't matter, however, is comparable to describing a single multinomial distribution of N draws from an X-fold category, where only the number seen of each category matters. The case where ordering doesn't matter and sampling is without replacement is comparable to a single multivariate hypergeometric distribution, and the fourth possibility does not seem to have a correspondence. Note that in all the "injective" cases (i.e. sampling without replacement), the number of sets of choices is zero unless . | |

In addition, models avoid constraint problems in the crude correlation method: all parameters will lie, as they should, between 0–1 (standardized). Multivariate, and multiple-time wave studies, with measured environment and repeated measures of potentially causal behaviours are now the norm. Examples of these models include extended twin designs, simplex models, and growth- curve models. SEM programs such as OpenMx and other applications suited to constraints and multiple groups have made the new techniques accessible to reasonably skilled users. | |

Gross has been a visiting professor at the University of California, Irvine, the University of Utah, the Academia Sinica in Taiwan, Drexel University, Macquarie University, and Australia's University of Newcastle. He has twice served as a divisional program director for the National Science Foundation. He was the director of the Vermont Mathematics Initiative.Vermont Mathematics Initiative He did research on harmonic analysis, group representation theory, analysis on Lie groups and homogeneous spaces, special functions, Fourier analysis, and mathematical applications to physics and multivariate statistics. | |

The mixture model- based clustering is also predominantly used in identifying the state of the machine in predictive maintenance. Density plots are used to analyze the density of high dimensional features. If multi-model densities are observed, then it is assumed that a finite set of densities are formed by a finite set of normal mixtures. A multivariate Gaussian mixture model is used to cluster the feature data into k number of groups where k represents each state of the machine. | |

The developers of the BIS monitor collected many (around 1000) EEG records from healthy adult volunteers at specific clinically important end points and hypnotic drug concentrations. They then fitted bispectral and power spectral variables in a multivariate statistical model to produce the BIS index. As with other types of EEG analysis, the calculation algorithm that the BIS monitor uses is proprietary. Therefore, although the principles of BIS and other monitors are well known, the exact method used in each case is not. | |

Vernon M. Chinchilli (born March 12, 1952) is an American biostatistician and Distinguished Professor of Public Health Sciences at the Penn State College of Medicine, where he is also Chair of the Department of Public Health Sciences. He is also a professor of Statistics at Penn State University. Chinchilli earned his Ph.D. in 1979 at the University of North Carolina at Chapel Hill. His dissertation, Rank Tests for Restricted Alternative Problems in Multivariate Analysis, was supervised by Pranab K. Sen. | |

Structural Equation Modeling is a peer-reviewed scientific journal publishing methodological and applied papers on structural equation modeling, a blend of multivariate statistical methods from factor analysis to systems of regression equations, with applications across a broad spectrum of social sciences as well as biology. One of the founders and the current editor-in-chief of the journal is George Marcoulides (University of California, Riverside). According to the Journal Citation Reports, the journal has a 2017 impact factor of 3.531. | |

Together with Bernard Jeune, he demonstrated mathematically a continuum of plant forms that spans not only organ categories such as root, stem, and leaf, but also different hierarchical levels of organ systems, organs, and tissues.Sattler, R. and B. Jeune. 1992. Multivariate analysis confirms the continuum view of plant form. Annals of Botany 69: 249-262 Rutishauser and Isler regard him as one of the major contemporary proponents of continuum morphology (or Fuzzy Arberian Morphology: FAM).Rutishauser, R. and Isler, B. 2001. | |

Besides respecting multiplication, monomial orders are often required to be well- orders, since this ensures the multivariate division procedure will terminate. There are however practical applications also for multiplication-respecting order relations on the set of monomials that are not well-orders. In the case of finitely many variables, well-ordering of a monomial order is equivalent to the conjunction of the following two conditions: # The order is a total order. # If u is any monomial then 1 \leq u. | |

Post- retirement, he continues his association with the institute as an honorary emeritus professor. Sahu is known as a pioneer of mathematical geology and is credited with introducing mathematical and quantitative approaches to the science. He used multivariate and time series procedures to interpret the statistical and mathematical models of sediments and ore deposits for which he designed computer-aided techniques. His work is detailed in one book, Statistical Models in Earth Sciences, and chapters contributed to books edited by others. | |

Version 18, offered in both 32-bit and 64-bit editions, is available in five languages: English, French, Spanish, German and Italian. The 64-bit edition is capable of computing very large sized data sets bringing it into the realm of "big data" analytics. The current version is Statgraphics a Windows Desktop application with extensive capabilities for regression analysis, ANOVA, multivariate statistics, Design of Experiments, statistical process control, life data analysis, data visualization and beyond. It features 260 plus procedures. | |

Everything from summary statistics to advanced statistical models in an exceptionally easy to use format. It contains more than 260 data analysis procedures, including descriptive statistics, hypothesis testing, regression analysis, analysis of variance, survival analysis, time series analysis and forecasting, sample size determination, multivariate methods and Monte Carlo techniques. The SPC menu includes many procedures for quality assessment, capability analysis, control charts, measurement systems analysis, and acceptance sampling. The program also features a DOE Wizard that creates and analyzes statistically designed experiments. | |

The probability mass function is often the primary means of defining a discrete probability distribution, and such functions exist for either scalar or multivariate random variables whose domain is discrete. A probability mass function differs from a probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables. A PDF must be integrated over an interval to yield a probability. The value of the random variable having the largest probability mass is called the mode. | |

Livezey's research work dealt with controversial areas of bird phylogenetics and taxonomy. While Livezey's colleagues often used DNA analysis to support their research, Livezey demonstrated a more traditional approach, based on exhaustive studies of bone shape and other characteristics. His general interests included phylogenetic relationships of avian families, phylogenetic relationships of waterfowl, evolution of avian flightlessness, comparative osteology of birds, multivariate morphometrics, and avian paleontology. He was generally considered to be the world authority on the osteology—the study of skeletons—of birds. | |

Sklar's theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables. Copulas are popular in high-dimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginals and copulae separately. There are many parametric copula families available, which usually have parameters that control the strength of dependence. Some popular parametric copula models are outlined below. | |

Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt. It is fully independent of the Gröbner basis method, introduced by Bruno Buchberger (1965), even if Gröbner bases may be used to compute characteristic sets.P. Aubry, D. Lazard, M. Moreno Maza (1999). On the theories of triangular sets. | |

Bayesian neural network with two hidden layers, transforming a 3-dimensional input (bottom) into a two-dimensional output (y_1, y_2) (top). Right: output probability density function p(y_1, y_2) induced by the random weights of the network. Video: as the width of the network increases, the output distribution simplifies, ultimately converging to a multivariate normal in the infinite width limit. Bayesian networks are a modeling tool for assigning probabilities to events, and thereby characterizing the uncertainty in a model's predictions. | |

In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. | |

CranID is a free software program that utilizes multivariate linear discriminant analysis and nearest neighbor discriminant analysis in conjunction with 29 cranial measurements to assess the geographic origin, which can be used to infer the ancestry of an unknown cranium. CranID compares an unknown cranium with 74 geographic samples, from 3,163 crania from 39 different populations.Hughes, S., Wright, R., Barry, M. "Virtual Reconstruction and Morphological Analysis of the Cranium of an Ancient Egyptian Mummy." Australasian Physical & Engineering Sciences in Medicine. | |

Quantitative research mainly deals with application of bi-variate and multivariate statistics to marketing research. IBM SPSS is mostly widely used tool for data analysis though some companies would prefer to stay with excel and WinCross. Industrial application of statistics is limited to cross tabulations using Chi square estimation, but some companies do use linear and logistic regression analysis. More recently, Conjoint analysis is one the new techniques that has become popular for its application on product bundling and pricing. | |

A multivariate ordination algorithm is applied to derive a first-pass, hypothesized sequence of first and last appearances. The minimal constraint on this sequence is that if there is an observed, real- world C. edwardii before C. armbrusteri statement for any pair of taxa, the hypothesized event sequence must replicate it. Then, the program shuffles the events using a maximum likelihood criterion. The criterion basically seeks to pull apart as many hypothesized age range overlaps as possible, especially if they involve common taxa. | |

Also referred to as "pumping losses", the pressure drops shown in Figure 3 are caused by viscous flow through the heat exchangers. The red line represents the heater, green is the regenerator, and blue is the cooler. To properly design the heat exchangers, multivariate optimization is required to obtain sufficient heat transfer with acceptable flow losses. The flow losses shown here are relatively low, and they are barely visible in the following image, which will show the overall pressure variations in the cycle. | |

Florida Entomologist 89:274-276. It was determined that a 3:1 triene:diene blend of the synthetic pheromone was optimal for attracting males to an adhesive trap in the field in Florida. This is the same 3:1 blend that was first isolated from the female pheromone glands. However, the question of what blend (the "natural" 3:1 blend or some other "unnatural" ratio) was best for mating disruption in general was addressed for this species using geometric multivariate experiment designs combined with response surface modeling. | |

Factor Analysis and Principal Component Analysis are multivariate statistical procedures used to identify relationships between hydrologic variables,. Convolution is a mathematical operation on two different functions to produce a third function. With respect to hydrologic modeling, convolution can be used to analyze stream discharge's relationship to precipitation. Convolution is used to predict discharge downstream after a precipitation event. This type of model would be considered a “lag convolution”, because of the predicting of the “lag time” as water moves through the watershed using this method of modeling. | |

Invariant measures on locally compact groups have long been used in statistical theory, particularly in multivariate analysis. Beurling's factorization theorem and much of the work on (abstract) harmonic analysis sought better understanding of the Wold decomposition of stationary stochastic processes, which is important in time series statistics. Encompassing previous results on probability theory on algebraic structures, Ulf Grenander developed a theory of "abstract inference". Grenander's abstract inference and his theory of patterns are useful for spatial statistics and image analysis; these theories rely on lattice theory. | |

Yanoconodon was a small mammal, barely 5 inches (13 centimetres) long. It had a sprawling posture, and although previously inferred to be semi-aquatic,Meng Chen, Gregory Philip Wilson, A multivariate approach to infer locomotor modes in Mesozoic mammals, Article in Paleobiology 41(02) · February 2015 DOI: 10.1017/pab.2014.14 direct study of its postcrania indicates that Yanoconodon was likely a terrestrial mammal, and that it has features in common with digging, arboreal, and semiaquatic mammals. Yanocodon had lumbar ribs, a feature not seen in modern mammals. | |

In the 2-dimensional case, if the density exists, each iso-density locus (the set of x1,x2 pairs all giving a particular value of f(x)) is an ellipse or a union of ellipses (hence the name elliptical distribution). More generally, for arbitrary n, the iso-density loci are unions of ellipsoids. All these ellipsoids or ellipses have the common center μ and are scaled copies (homothets) of each other. The multivariate normal distribution is the special case in which g(z)=e^{-z/2}. | |

Djadochtatherioidea is a group of extinct mammals known from the upper Cretaceous of Central Asia. They were members of an also extinct order called Multituberculata. These were generally somewhat rodent-like creatures, who scurried around during the "age of the dinosaurs", though nonetheless very ecologically diverse; several were jerboa-like hoppers,Meng Chen, Gregory Philip Wilson, A multivariate approach to infer locomotor modes in Mesozoic mammals, Article in Paleobiology 41(02) · February 2015 DOI: 10.1017/pab.2014.14 while others like Mangasbaatar were large sized and fossorial. | |

The taxon Djadochtatheriidae was named by Z. Kielan-Jaworowska and J. H. Hurum in 1997. Multituberculates are a rather diverse group in terms of locomotion and diet. Forms like Kryptobaatar and Catopsbaatar were hopping, gerboa-like omnivores (and this is probably the ancestral condition for the group, given that Nemegtbaatar also had this lifestyle),Meng Chen, Gregory Philip Wilson, A multivariate approach to infer locomotor modes in Mesozoic mammals, Article in Paleobiology 41(02) · February 2015 DOI: 10.1017/pab.2014.14 while Mangasbaatar was a robust, digging herbivore. | |

In Chapter 10, a simple form for ranking tasks is presented that only involves the product of univariate normal distribution functions and includes rank-induced dependency parameters. A theorem is proven that shows that the particular form of the dependency parameters provides the only way that this simplification is possible. Chapter 6 links discrimination, identification and preferential choice through a common multivariate model in the form of weighted sums of central F distribution functions and allows a general variance-covariance matrix for the items. | |

At the centre of Werner Wittmann's work is an attempt to unite the sometimes contradictory methods and goals of experimental and non-experimental research approaches in the social and behavioural sciences. His work is based on works by Lee Cronbach, Donald T. Campbell, Thomas D. Cook, R.F. Boruch, Egon Brunswik, L. Sechrest, Gene V. Glass, Raymond Bernard Cattell and Kenneth Hammond. The central focus of Werner Wittmann's research and scientific theory is on considerations of a multivariate reliabilitys and validity theory (esp. Wittmann, 1985, 1988). | |

Information-based complexity (IBC) studies optimal algorithms and computational complexity for the continuous problems which arise in physical science, economics, engineering, and mathematical finance. IBC has studied such continuous problems as path integration, partial differential equations, systems of ordinary differential equations, nonlinear equations, integral equations, fixed points, and very-high-dimensional integration. All these problems involve functions (typically multivariate) of a real or complex variable. Since one can never obtain a closed-form solution of the problems of interest one has to settle for a numerical solution. | |

Finally the information is often contaminated by noise. The goal of information-based complexity is to create a theory of computational complexity and optimal algorithms for problems with partial, contaminated and priced information, and to apply the results to answering questions in various disciplines. Examples of such disciplines include physics, economics, mathematical finance, computer vision, control theory, geophysics, medical imaging, weather forecasting and climate prediction, and statistics. The theory is developed over abstract spaces, typically Hilbert or Banach spaces, while the applications are usually for multivariate problems. | |

Goldberg has served on the Personality and Cognition Research Review Committee and the Cognition, Emotion, and Personality Research Review Committee of the National Institute of Mental Health and on the Graduate Record Examination Board Research Committee. Goldberg has previously served as the president of both the Society of Multivariate Experimental Psychology (1974-1975) and the Association for Research in Personality (2004-2006). He is a fellow of the American Psychological Association, the Association for Psychological Science, and the Society for Personality and Social Psychology. | |

Early analysis of the large morphological diversity in Dikelocephalus resulted in splitting up the genus into many "species" during the first half of the 20th century. After applying modern analysis methods like multivariate analysis, including principal component analysis and nonmetric multidimensional scaling at the end of the 20th century, it turned out the variation was continuous, and all specimens belonged to the same morphospecies. This results in a large number of synonyms for D. minnesotensis (see box). The only other putative species may be D. freeburgensis. | |

Recent studies show that it led a possibly scansorial lifestyle, possessing long hindlimbs and a large plantar area on the foot, both optimal for climbing.Meng Chen, Gregory Philip Wilson, A multivariate approach to infer locomotor modes in Mesozoic mammals, Article in Paleobiology 41(02) · February 2015 The specimen GMV 2124 of the feathered dinosaur Sinosauropteryx? sp. contained two jaws of Zhangheotherium in its stomach region (Hurum et al. 2006). Thus, it seems to have preyed on this primitive mammal, possibly on a regular basis. | |

The Laplacian is the sum of second partials with respect to all the variables, and is an invariant differential operator under the action of the orthogonal group viz the group of rotations. The standard separation of variables theorem states that every multivariate polynomial over a field can be decomposed as a finite sum of products of a radical polynomial and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radical polynomialsCf. Corollary 1.8 of . | |

As an alternative, many methods have been suggested to improve the estimation of the covariance matrix. All of these approaches rely on the concept of shrinkage. This is implicit in Bayesian methods and in penalized maximum likelihood methods and explicit in the Stein-type shrinkage approach. A simple version of a shrinkage estimator of the covariance matrix is represented by the Ledoit- Wolf shrinkage estimatorO. Ledoit and M. Wolf (2004a) "A well-conditioned estimator for large-dimensional covariance matrices " Journal of Multivariate Analysis 88 (2): 365—411. | |

The multivariate probit model is a standard method of estimating a joint relationship between several binary dependent variables and some independent variables. For categorical variables with more than two values there is the multinomial logit. For ordinal variables with more than two values, there are the ordered logit and ordered probit models. Censored regression models may be used when the dependent variable is only sometimes observed, and Heckman correction type models may be used when the sample is not randomly selected from the population of interest. | |

The tarsier lineage is known to have split from other primate lineages around 58 mya, but it could very well be much earlier. Scientists have discovered Tarsiid fossils from Asia dating from the Eocene to the Miocene. Multivariate analyses have shown the T. lariang is significantly distinct from other species of Sulawesi tarsiers. The isolation of the Sulawesi islands archipelago (before they became one big island), due to a period of plate tectonic activity about 20 mya resulted in speciation of the species of Sulawesi tarsiers. | |

Presentation of sex work in two Costa Rican newspapers: a multivariate analysis of the roles of patriarchal prejudice and reporter gender. Research Journal of the Costa Rican Distance Education University 5 (2) Later research also deals with human intimate behavior,Monge-Nájera, J. & Vega Corrales, K. 2013. Sexual videos in Internet: a test of 11 hypotheses about intimate practices and gender interactions in Latin America. Research Journal of the Costa Rican Distance Education University 5 (2) and self-presentation of their personalities by glamour models. | |

Sparse principal component analysis (sparse PCA) is a specialised technique used in statistical analysis and, in particular, in the analysis of multivariate data sets. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by introducing sparsity structures to the input variables. A particular disadvantage of ordinary PCA is that the principal components are usually linear combinations of all input variables. Sparse PCA overcomes this disadvantage by finding linear combinations that contain just a few input variables. | |

Webinar On Reverse Bidding Strategy The quantum of discount would depend on project site location (i.e. solar radiation), technology used, simulated energy generation, capital cost and interest cost. Multivariate analysis was carried out using key variables like capital cost, interest and the capacity utilization factor (i.e. CUF which is actual generation of the plant and depends on the location (radiation) and technology used) to calculate the levelized tariff for a target equity IRR based on which the discount to be offered can be determined. | |

Intuitively, individuals scoring high on neuroticism are likely to score low on self-report EI measures. Studies have examined the multivariate effects of personality and intelligence on EI and also attempted to correct estimates for measurement error. For example, a study by Schulte, Ree, Carretta (2004), showed that general intelligence (measured with the Wonderlic Personnel Test), agreeableness (measured by the NEO-PI), as well as gender could reliably be used to predict the measure of EI ability. They gave a multiple correlation (R) of . | |

In multivariate statistics, random matrices were introduced by John Wishart for statistical analysis of large samples; see estimation of covariance matrices. Significant results have been shown that extend the classical scalar Chernoff, Bernstein, and Hoeffding inequalities to the largest eigenvalues of finite sums of random Hermitian matrices. Corollary results are derived for the maximum singular values of rectangular matrices. In numerical analysis, random matrices have been used since the work of John von Neumann and Herman Goldstine to describe computation errors in operations such as matrix multiplication. | |

Intuitively, this is evident during a financial crisis where all industry sectors experience a significant increase in correlations, as opposed to an upward trending market. This phenomenon is also known as asymmetric correlations or asymmetric dependence. Rather than using the historical simulation, Monte-Carlo simulations with well-specified multivariate models are an excellent alternative. For example, to improve the estimation of the variance-covariance matrix, one can generate a forecast of asset distributions via Monte-Carlo simulation based upon the Gaussian copula and well-specified marginals. | |

Byman, Waxman, and Larson (1999) They concluded that, "Although the United States and the USAF have scored some notable successes, the record is mixed."Byman, Waxman, and Larson (1999, p. iii, 5/195) Horowitz and Reiter applied "multivariate probit analysis [to] all instances of air power coercion from 1917 to 1999". Their quantitative analyses essentially matched Pape's qualitative assessment that attacking military targets has improved the chances of success, but "higher levels of civilian vulnerability have no effect on the chances of coercion success". | |

For a long time, the key ranking factor for Yandex was the number of third-party links to a particular site. Each page on the Internet was assigned a unique citation index, similar to the index for authors of scientific articles: the more links, the better. A similar mechanism was implemented in the Yandex and in the Google’s PageRank. In order to prevent cheating, Yandex uses multivariate analysis, in which only 70 of the 800 factors are affected by the number of third-party links. | |

In complex analysis, a discipline in mathematics, and in statistical physics, the Asano contraction or Asano–Ruelle contraction is a transformation on a separately affine multivariate polynomial. It was first presented in 1970 by Taro Asano to prove the Lee–Yang theorem in the Heisenberg spin model case. This also yielded a simple proof of the Lee–Yang theorem in the Ising model. David Ruelle proved a general theorem relating the location of the roots of a contracted polynomial to that of the original. | |

He then became interested in different problems of statistical signal processing. In particular, it contributes to the development of subspaces methods for the identification of multivariate linear systemsE Moulines, P Duhamel, JF Cardoso, S Mayrargue, « Subspace methods for the blind identification of multichannel FIR filters », IEEE Transactions on signal processing,, 1995, pp. 516–525 and source separationBelouchrani, Adel and Abed-Meraim, Karim and Cardoso, J-F and Moulines, Eric, « A blind source separation technique using second-order statistics », IEEE Transactions on signal processing, 1997, pp. | |

Scott A. Schwenter is Professor of Hispanic Linguistics at The Ohio State University in Columbus, Ohio, where he has taught since 1999. He is a variationist morphosyntactician and pragmaticist, whose research addresses grammatical issues in both Spanish and Portuguese. His work has included both experimental and corpus-based approaches, making use of multivariate statistical analysis to examine broad-scale patterns across different varieties of Spanish and Portuguese. Schwenter's research interests center around the contextual conditioning of linguistic variables and speakers' choice between variants to express meanings. | |

Moderated mediation, also known as conditional indirect effects,Preacher, K. J., Rucker, D. D., & Hayes, A. F. (2007) Addressing moderated mediation hypotheses: Theory, Methods, and Prescriptions. Multivariate Behavioral Research, 42, 185–227. occurs when the treatment effect of an independent variable A on an outcome variable C via a mediator variable B differs depending on levels of a moderator variable D. Specifically, either the effect of A on the B, and/or the effect of B on C depends on the level of D. | |

The Randomized Dependence CoefficientLopez-Paz D. and Hennig P. and Schölkopf B. (2013). "The Randomized Dependence Coefficient", "Conference on Neural Information Processing Systems" Reprint is a computationally efficient, copula-based measure of dependence between multivariate random variables. RDC is invariant with respect to non- linear scalings of random variables, is capable of discovering a wide range of functional association patterns and takes value zero at independence. For two binary variables, the odds ratio measures their dependence, and takes range non-negative numbers, possibly infinity: . | |

He was founding editor and editor-in-chief of the Journal of Statistical Software (1997–2015) and editor-in-chief for the Journal of Multivariate Analysis (1997–2015). De Leeuw was elected trustee at Psychometric Society in 1985–1986 and president in 1987–1988. He was elected fellow of the Royal Statistical Society in 1984; at the International Statistical Institute in 1986; at the Royal Netherlands Academy of Arts and Sciences in 1989;; at the Institute of Mathematical Statistics and at the American Statistical Association in 2001. | |

Michael Hechter is an American sociologist and Foundation Professor of Political Science at Arizona State University. He is also Emeritus Professor of Sociology at the University of Washington. Hechter first became known for his research in comparative-historical analysis. His book Internal Colonialism: The Celtic Fringe in British National Development, 1536-1966 (1975; 1998) presented a social structural analysis of nationalism – in contrast to then-popular cultural explanations of the phenomenon—and was one of the first such studies to employ multivariate statistical analysis. | |

SAS (previously "Statistical Analysis System") is a statistical software suite developed by SAS Institute for data management, advanced analytics, multivariate analysis, business intelligence, criminal investigation,SAS empowers crime fighters to crack complex cases and predictive analytics. SAS was developed at North Carolina State University from 1966 until 1976, when SAS Institute was incorporated. SAS was further in the 1980s and 1990s with the addition of new statistical procedures, additional components and the introduction of JMP. A point-and-click interface was added in version 9 in 2004. | |

In a classification setting, assigning outcome probabilities to observations can be achieved through the use of a logistic model (also called a logic model), which transforms information about the binary dependent variable into an unbounded continuous variable and estimates a regular multivariate model. The Wald and likelihood-ratio test are used to test the statistical significance of each coefficient b in the model (analogous to the t tests used in OLS regression; see above). A test assessing the goodness-of-fit of a classification model is the "percentage correctly predicted". | |

Flanders is known for advancing an approach to multivariate calculus that is independent of coordinates through treatment of differential forms. According to Shiing- Shen Chern, "an affine connection on a differentiable manifold gives rise to covariant differentiations of tensor fields. The classical approach makes use of the natural frames relative to local coordinates and works with components of tensor fields, thus giving the impression that this branch of differential geometry is a venture through a maze of indices. The author [Flanders] gives a mechanism which shows that this is not necessarily so."H. | |

Zaman has contributed extensively to various fields of social sciences including Econometrics, Economics and Islamic Economics. His early writing were mainly in Econometrics and was published to the top journals including Annals of Statistics, Journal of Econometrics and Journal of Multivariate Analysis. Gradually, his interests turned toward Islamic Economics and his writings are published in Journal of King Abdul Aziz University, Journal of Islamic Economics, Banking and Finance and Islamic Studies. He has written a book titled 'Statistical Foundations of Econometric Techniques' which has earned high repute in academic circles. | |

In 1957 he studied mathematics at Peking University, after which he entered the graduate program at the Institute of Mathematics of the Chinese Academy of Sciences, in Beijing. His doctoral supervisor was Pao-Lu Hsu, who suggested that Fang provide a multivariate generalization and correction of a univariate result, which had been given an incomplete proof in a Russian paper. With two weeks' work, Fang's submitted his extensions, which were declared by Hsu to suffice for his dissertation. Unfortunately, this paper remained unpublished for 19 years because the Cultural Revolution destroyed academic publishing in China. | |

Mertens had the idea to use linear competitive economies as an order book (trading) to model limit orders and generalize double auctions to a multivariate set up. Acceptable relative prices of players are conveyed by their linear preferences, money can be one of the goods and it is ok for agents to have positive marginal utility for money in this case (after all agents are really just orders!). In fact this is the case for most order in practice. More than one order (and corresponding order-agent) can come from same actual agent. | |

As of 2018, the company employs around 46. Upguard's Cyber Resilience platform determines a company's cyber-security risk factors by scanning both internal and external computer systems. The platform automatically scans every server, application, network and mobile devices in IT environments to create a living model of their configuration state, thereafter continually assessing this system of record for security vulnerabilities, configuration drift and procedural changes. From this model, the platform dynamically derives a unified cyber- security risk score, CSTAR, that determines the cyber risk posture of IT assets against multivariate factors. | |

This led to the first branch of multivariate morphometrics, which emphasized matrix manipulations involving variables. In the late 1970s and early 1980s, Fred Bookstein (currently a professor of Anthropology at the University of Vienna) began using Cartesian transformations and David George Kendall (statistician, 1918-2007) showed that figures that hold the same shape can be treated as separate points in a geometric space. Finally, in 1996, Leslie Marcus (paleontologist, 1930-2002) convinced colleagues to use morphometrics on the famous Ötzi skeleton, which helped expose the importance of the applications of these methods. | |

A model to detect fake news on social media by classifying propagation paths of news was proposed. The propagation path of each news story is modeled as a multivariate time series – Each tuple indicates the characteristics of the user who participates in the propagation of the news. A time series classifier is built with recurrent and convolutional networks to predict the veracity of the news story. Recurrent and convolutional networks are able to learn global and local variations of under characteristics which will in turn help characterize clues for the detection of fake news. | |

A crucial problem of multivariate statistics is finding the (direct-)dependence structure underlying the variables contained in high-dimensional contingency tables. If some of the conditional independences are revealed, then even the storage of the data can be done in a smarter way (see Lauritzen (2002)). In order to do this one can use information theory concepts, which gain the information only from the distribution of probability, which can be expressed easily from the contingency table by the relative frequencies. A pivot table is a way to create contingency tables using spreadsheet software. | |

Between 1967 and 1970, he was an assistant professor and taught courses on personality and factor analysis in the Department of Psychology at UNC. In 1970 he took a position as an associate professor in the School of Psychology at the Georgia Institute of Technology and rose to full professor in 1981. He taught courses in introductory statistics, psychometric theory, factor analysis, multivariate statistics, structural equation modelling, personality theory, and introduction to psychology during his career. In 1972 he published the well- received advanced text The Foundations of Factor Analysis. | |

An approach that has been tried since the late 1990s is the implementation of the multiple three-treatment closed-loop analysis. This has not been popular because the process rapidly becomes overwhelming as network complexity increases. Development in this area was then abandoned in favor of the Bayesian and multivariate frequentist methods which emerged as alternatives. Very recently, automation of the three-treatment closed loop method has been developed for complex networks by some researchers as a way to make this methodology available to the mainstream research community. | |

In characteristic zero, a better algorithm is known, Yun's algorithm, which is described below. Its computational complexity is, at most, twice that of the GCD computation of the input polynomial and its derivative. More precisely, if T_{n} is the time needed to compute the GCD of two polynomials of degree n and the quotient of these polynomial by the GCD, then 2T_{n} is an upper bound for the time needed to compute the square free decomposition. There are also known algorithms for the computation of the square-free decomposition of multivariate polynomials. | |

Univariate techniques are used for analyzing data when there is a single measurement of each element or unit in the sample, or, if there are several measurements of each element, each RCH variable is analyzed in isolation. On the other hand, multivariate techniques are used for analyzing data when there are two or more measurements on each element and the variables are analyzed simultaneously. The last stage is the report preparation and presentation. The entire project should be documented in a written report and the results and major findings must be presented. | |

The papers establishing the mathematical foundations of Kalman type filters were published between 1959 and 1961. The Kalman filter is the optimal linear estimator for linear system models with additive independent white noise in both the transition and the measurement systems. Unfortunately, in engineering, most systems are nonlinear, so attempts were made to apply this filtering method to nonlinear systems; Most of this work was done at NASA Ames. The EKF adapted techniques from calculus, namely multivariate Taylor series expansions, to linearize a model about a working point. | |

The multivariate aspect of the MANCOVA allows the characterisation of differences in group means in regards to a linear combination of multiple dependent variables, while simultaneously controlling for covariates. Example situation where MANCOVA is appropriate: Suppose a scientist is interested in testing two new drugs for their effects on depression and anxiety scores. Also suppose that the scientist has information pertaining to the overall responsivity to drugs for each patient; accounting for this covariate will grant the test higher sensitivity in determining the effects of each drug on both dependent variables. | |

Born in Bennekom, Bodlaender was educated at Utrecht University, earning a doctorate in 1986 under the supervision of Jan van Leeuwen with the thesis Distributed Computing – Structure and Complexity.. After postdoctoral research at the Massachusetts Institute of Technology in 1987, he returned to Utrecht as a faculty member. In 1987 he was appointed Assistant Professor and in 2003 Associate Professor. In 2014 he was awarded the Nerode Prize for an outstanding paper in the area of multivariate algorithmics. Bodlaender has written extensively about chess variants and founded the website The Chess Variant Pages in 1995. | |

Cumulative distribution function for the exponential distribution Cumulative distribution function for the normal distribution In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. | |

Another possibility is to present survey results by means of statistical models in the form of a multivariate distribution mixture. The statistical information in the form of conditional distributions (histograms) can be derived interactively from the estimated mixture model without any further access to the original database. As the final product does not contain any protected microdata, the model-based interactive software can be distributed without any confidentiality concerns. Another method is simply to release no data at all, except very large scale data directly to the central government. | |

When analyzing the response of materials to alternating electric fields (dielectric spectroscopy), in applications such as electrical impedance tomography,Otto H. Schmitt, University of Minnesota Mutual Impedivity Spectrometry and the Feasibility of its Incorporation into Tissue- Diagnostic Anatomical Reconstruction and Multivariate Time-Coherent Physiological Measurements. otto-schmitt.org. Retrieved on 2011-12-17. it is convenient to replace resistivity with a complex quantity called impedivity (in analogy to electrical impedance). Impedivity is the sum of a real component, the resistivity, and an imaginary component, the reactivity (in analogy to reactance). | |

He became Professor at Cooperative Graduate School, the University of Tsukuba in 1992 (to 2010), and Professor at the Graduate School of Information Science and Technology, the University of Tokyo in 2001 (to 2007). He retired from National Institute of Advanced Industrial Science and Technology in 2012 and awarded the title of Emeritus Researcher. He has engaged in mathematical fundamental research and its application concerning pattern recognition, image processing, multivariate analysis, artificial intelligence, and neurocomputing. Otsu's method, an image binarization technique, is still a standard technique widely used both in Japan and abroad. | |

For practical regression and prediction needs, Student's t-processes were introduced, that are generalisations of the Student t-distributions for functions. A Student's t-process is constructed from the Student t-distributions like a Gaussian process is constructed from the Gaussian distributions. For a Gaussian process, all sets of values have a multidimensional Gaussian distribution. Analogiusly, X(t) is a Student t-process on an interval I=[a,b] if the correspondent values of the process X(t_1),...,X(t_n) (t_i \in I) have a joint multivariate Student t-distribution. | |

This means that when the user has multiple views or windows in a project, selecting an object in one of them will highlight the same object in all other windows. GeoDa also is capable of producing histograms, box plots, Scatter plots to conduct simple exploratory analyses of the data. The most important thing, however, is the capability of mapping and linking those statistical devices with the spatial distribution of the phenomenon that the users are studying. Multivariate ESDA: multiple views linked to explore the relations in various characteristics of Colombian municipalities. | |

Comparison of performance of bivariate and multivariate estimators of connectivity may be found in, where it was demonstrated that in case of interrelated system of channels, greater than two, bivariate methods supply misleading information, even reversal of true propagation may be found. Consider the very common situation that the activity from a given source is measured at electrodes positioned at different distances, hence different delays between the recorded signals. When a bivariate measure is applied, propagation is always obtained when there is a delay between channels., which results in a lot of spurious flows. | |

Monte-Carlo simulations employ quantile functions to produce non-uniform random or pseudorandom numbers for use in diverse types of simulation calculations. A sample from a given distribution may be obtained in principle by applying its quantile function to a sample from a uniform distribution. The demands, for example, of simulation methods in modern computational finance are focusing increasing attention on methods based on quantile functions, as they work well with multivariate techniques based on either copula or quasi- Monte-Carlo methods and Monte Carlo methods in finance. | |

The program also comes with 50 multivariate data sets. Using TinkerPlots, students can make a large variety of graphs, including those specified for middle school in Common Core State Standards for Mathematics But rather than making these graphs directly using commands, students construct them by progressively organizing cases using basic operations including “stack,” “order,” and “separate.” Responding to these operations, case icons animate into different screen positions. The interface was based on observations of people organizing “data cards” on a table to make graphs to answer specific questions Harradine, A., & Konold, C. (2006). | |

Then, given a test sample, one computes the Mahalanobis distance to each class, and classifies the test point as belonging to that class for which the Mahalanobis distance is minimal. Mahalanobis distance and leverage are often used to detect outliers, especially in the development of linear regression models. A point that has a greater Mahalanobis distance from the rest of the sample population of points is said to have higher leverage since it has a greater influence on the slope or coefficients of the regression equation. Mahalanobis distance is also used to determine multivariate outliers. | |

The DES-IV is a version of the DES where it has 49 items. This version of mood-state inventory is a multidimensional instrument, and is used to look over and examine the frequency of multiple fundamental human emotions. The 49 items of the DES-IV help measure 12 basic emotions (interest, joy, surprise, sadness, anger, disgust, contempt, hostility, fear, shame, shyness and guilt). It was also suggested by Boyle (1985) that DES-IV and the Eight State Questionnaire are one of the more promising self-report multivariate mood-state instruments. | |

In mathematics, the Schwartz–Zippel lemma (also called the DeMillo-Lipton- Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing, i.e. in the problem of determining whether a given multivariate polynomial is the 0-polynomial (or identically equal to 0). It was discovered independently by Jack Schwartz, Richard Zippel, and Richard DeMillo and Richard J. Lipton, although DeMillo and Lipton's version was shown a year prior to Schwartz and Zippel's result. The finite field version of this bound was proved by Øystein Ore in 1922. | |

A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components of a multivariate random variable with a known distribution. Several types of correlation coefficient exist, each with their own definition and own range of usability and characteristics. They all assume values in the range from −1 to +1, where ±1 indicates the strongest possible agreement and 0 the strongest possible disagreement. | |

The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain. There exist algorithms to compute them as soon as one has a GCD algorithm in the ring of coefficients. These algorithms proceed by a recursion on the number of variables to reduce the problem to a variant of the Euclidean algorithm. They are a fundamental tool in computer algebra, because computer algebra systems use them systematically to simplify fractions. | |

In multivariate statistics, principal response curves (PRC) are used for analysis of treatment effects in experiments with a repeated measures design.ter Braak, Cajo J.F. & Šmilauer, Petr (2012). Canoco reference manual and user´s guide: software for ordination (version 5.0), p 292\. Microcomputer Power, Ithaca, NY. First developed as a special form of redundancy analysis, PRC allow temporal trends in control treatments to be corrected for, which allows the user to estimate the effects of the treatment levels without them being hidden by the overall changes in the system. | |

The Scientific approach relies heavily on the multivariate analysis of behaviors and any other information from the crime scene that could lead to the offender's characteristics or psychological processes. According to this approach, elements of the profile are developed by comparing the results of the analysis to those of previously caught offenders. Wilson, Lincon and Kocsis list three main paradigms of profiling: diagnostic evaluation, crime scene analysis, and investigative psychology. Ainsworth identified four: clinical profiling (synonymous with diagnostic evaluation), typological profiling (synonymous with crime scene analysis), investigative psychology, and geographical profiling. | |

Thus, the smoother a boundary is, the shorter coding length it attains. # Texture is encoded by lossy compression in a way similar to minimum description length (MDL) principle, but here the length of the data given the model is approximated by the number of samples times the entropy of the model. The texture in each region is modeled by a multivariate normal distribution whose entropy has a closed form expression. An interesting property of this model is that the estimated entropy bounds the true entropy of the data from above. | |

In 1977, Royen started working as a statistician for the pharmaceutical company Hoechst AG. From 1979 to 1985, he worked at the company's own educational facility teaching mathematics and statistics. Starting in 1985 until becoming an emeritus in 2010, he taught statistics and mathematics at the University of Applied Sciences Bingen in Rhineland-Palatinate. Royen worked mainly on probability distributions, in particular multivariate chi-squares and gamma distributions, to improve some frequently used statistical test procedures. Nearly half of his circa 30 publications were written when he was aged over sixty. | |

REML estimation is available in a number of general-purpose statistical software packages, including Genstat (the REML directive), SAS (the MIXED procedure), SPSS (the MIXED command), Stata (the mixed command), JMP (statistical software), and R (especially the lme4 and older nlme packages), as well as in more specialist packages such as MLwiN, HLM, ASReml, (ai)remlf90, wombat, Statistical Parametric Mapping and CropStat. REML estimation is implemented in Surfstat a Matlab toolbox for the statistical analysis of univariate and multivariate surface and volumetric neuroimaging data using linear mixed effects models and random field theory. | |

He was elected President of the Institute of Mathematical Statistics in 1962. Anderson's 1958 textbook, An Introduction to Multivariate Analysis, educated a generation of theorists and applied statisticians; it was "the classic" in the area until the book by Mardia, Kent and Bibby . Anderson's book emphasizes hypothesis testing via likelihood ratio tests and the properties of power functions: Admissibility, unbiasedness and monotonicity.(Pages 560–561) Anderson is also known for Anderson–Darling test of whether there is evidence that a given sample of data did not arise from a given probability distribution. | |

On the other hand, multivariate statistics are thriving methods for high-dimensional correlated metabolomics data, of which the most popular one is Projection to Latent Structures (PLS) regression and its classification version PLS-DA. Other data mining methods, such as random forest, support-vector machines, etc. are received increasing attention for untargeted metabolomics data analysis. In the case of univariate methods, variables are analyzed one by one using classical statistics tools (such as Student's t-test, ANOVA or mixed models) and only these with sufficient small p-values are considered relevant. | |

If we want to explore the structure of intra-day electricity prices, we need to use dimension reduction methods; for instance, factor models with factors estimated as principal components (PC). Empirical evidence indicates that there are forecast improvements from incorporating disaggregated (i.e., hourly or zonal) data for predicting daily system prices, especially when the forecast horizon exceeds one week. With the increase of computational power, the real-time calibration of these complex models will become feasible and we may expect to see more EPF applications of the multivariate framework in the coming years. | |

Example 5. Sort Big Data Objects by Revealed Preference When ranking big data observations, diverse consumers reveal heterogeneous preferences; but any revealed preference is a ranking between two observations, derived from a consumer’s rational consideration of many factors. Previous researchers have applied exogenous weighting and multivariate regression approaches, and spatial, network, or multidimensional analyses to sort complicated objects, ignoring the variety and variability of the objects. By recognizing the diversity and heterogeneity among both the observations and the consumers, Hu (2000) instead applies endogenous weighting to these contradictory revealed preferences. | |

Jöreskog proposed a reliable numerical method for computing maximum-likelihood estimates in factor analysis; similarly reliable methods were also proposed by Gerhard Derflinger, Robert Jennrich, and Stephen M. Robinson at roughly the same time. Jöreskog's Fortran codes helped to popularize factor analysis around the world. While working at the Educational Testing Service and giving lectures at Princeton University, Jöreskog proposed a linear model for the analysis of covariance structures, a fundamental contribution to structural equation modeling (SEM). His other research interests include multivariate analysis, item response theory, statistical computing, and factor-analysis in geology. | |

The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see list of probability distributions, which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.) All of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a mixture distribution. | |

Correspondence analysis (CA) or reciprocal averaging is a multivariate statistical technique proposedDodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP by Herman Otto Hartley (Hirschfeld)Hirschfeld, H.O. (1935) "A connection between correlation and contingency", Proc. Cambridge Philosophical Society, 31, 520-524 and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data. In a similar manner to principal component analysis, it provides a means of displaying or summarising a set of data in two-dimensional graphical form. | |

After graduation, Katz held several research positions. He worked for the Brain Research Laboratories (New York University) developing neurometric systems based on the multivariate statistical analysis of electroencephalographic signals (EEG). He later worked for HeartMap, a biomedical company, where he headed the design, hardware prototyping, software development and testing of a 64-channel cardiac monitor with special analytic capabilities, including neural network pattern recognition and the ability to generate 3-D images of the electrical potentials across the surface of the heart. He also worked for the American Society for Psychical Research. | |

He has been professor at Technical University of Madrid, and visiting professor at University of Wisconsin–Madison and University of Chicago. He is now professor at Charles III University of Madrid. He has been director of the Journal Revista Estadística Española and President of Sociedad española de Estadística e Investigación Operativa, Vicepresident of the Interamerican Statistical Institute and President of European Courses in Advanced Statistics. He has published fourteen books and more than 200 research articles in time series analysis, multivariate methods, Bayesian Statistics and Econometrics that have received more than 8,000 references . | |

In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the properties of finitely generated submodules. He proved an important theorem known as Hilbert's basis theorem which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated. However, the property is named after Emmy Noether who was the first one to discover the true importance of the property. | |

It is a basic tool of computer algebra, and is a built-in function of most computer algebra systems. It is used, among others, for cylindrical algebraic decomposition, integration of rational functions and drawing of curves defined by a bivariate polynomial equation. The resultant of n homogeneous polynomials in n variables (also called multivariate resultant, or Macaulay's resultant for distinguishing it from the usual resultant) is a generalization, introduced by Macaulay, of the usual resultant. It is, with Gröbner bases, one of the main tools of effective elimination theory (elimination theory on computers). | |

After the amplicons are sequenced, molecular phylogenetic methods are used to infer the composition of the microbial community. This is done by clustering the amplicons into operational taxonomic units (OTUs) and inferring phylogenetic relationships between the sequences. Due to the complexity of the data, distance measures such as UniFrac distances are usually defined between microbiome samples, and downstream multivariate methods are carried out on the distance matrices. An important point is that the scale of data is extensive, and further approaches must be taken to identify patterns from the available information. | |

Regression analysis is a type of statistical technique used to determine the important factors that affect the outcome of the event. In the case of sports betting this is usually done with multivariate linear regression. Because sports events are very complicated and there are many factors it is extremely difficult, if not impossible, to be able to accurately identify each variable that affects the outcome of the game. Also, regression analysis assigns a "weight" to each factors that identifies how much it affects the outcome of the event. | |

The goal of these investigations is to better understand how different groups respond to various messages and visual prompts, thereby providing an assessment of how well the advertisement meets its communications goals. Post-testing employs many of the same techniques as pre-testing, usually with a focus on understanding the change in awareness or attitude attributable to the advertisement. With the emergence of digital advertising technologies, many firms have begun to continuously post-test ads using real-time data. This may take the form of A/B split-testing or multivariate testing. | |

Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid. It approximates the value of a function at an intermediate point (x, y, z) within the local axial rectangular prism linearly, using function data on the lattice points. For an arbitrary, unstructured mesh (as used in finite element analysis), other methods of interpolation must be used; if all the mesh elements are tetrahedra (3D simplices), then barycentric coordinates provide a straightforward procedure. Trilinear interpolation is frequently used in numerical analysis, data analysis, and computer graphics. | |

Self-learning onsite behavioral targeting systems will monitor visitor response to site content and learn what is most likely to generate a desired conversion event. Some good content for each behavioral trait or pattern is often established using numerous simultaneous multivariate tests. Onsite behavioral targeting requires a relatively high level of traffic before statistical confidence levels can be reached regarding the probability of a particular offer generating a conversion from a user with a set behavioral profile. Some providers have been able to do so by leveraging its large user base, such as Yahoo!. | |

The notion of social structure may mask systematic biases, as it involves many identifiable sub-variables (e.g. gender). Some argue that men and women who have otherwise equal qualifications receive different treatment in the workplace because of their gender, which would be termed a "social structural" bias, but other variables (such as time on the job or hours worked) might be masked. Modern social structural analysis takes this into account through multivariate analysis and other techniques, but the analytic problem of how to combine various aspects of social life into a whole remains.Aberration, et al. | |

Moreover, for each consistent effectively generated system T, it is possible to effectively generate a multivariate polynomial p over the integers such that the equation p = 0 has no solutions over the integers, but the lack of solutions cannot be proved in T (Davis 2006:416, Jones 1980). Smorynski (1977, p. 842) shows how the existence of recursively inseparable sets can be used to prove the first incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are essentially undecidable (see Kleene 1967, p. 274). | |

Let R be the multivariate polynomial ring k[x1, ..., xn] over a field k. The variables are ordered linearly according to their subscript: x1 < ... < xn. For a non-constant polynomial p in R, the greatest variable effectively presenting in p, called main variable or class, plays a particular role: p can be naturally regarded as a univariate polynomial in its main variable xk with coefficients in k[x1, ..., xk−1]. The degree of p as a univariate polynomial in its main variable is also called its main degree. | |

If sphericity is violated, then the variance calculations may be distorted, which would result in an F-ratio that is inflated. Sphericity can be evaluated when there are three or more levels of a repeated measure factor and, with each additional repeated measures factor, the risk for violating sphericity increases. If sphericity is violated, a decision must be made as to whether a univariate or multivariate analysis is selected. If a univariate method is selected, the repeated- measures ANOVA must be appropriately corrected depending on the degree to which sphericity has been violated. | |

Epidemiological analysis: Using commands similar to those in EpiInfo, CIETanalysis produces basic frequencies (like the proportion with a given disease) through to multivariate models of gains (like the proportion that can be "saved" by a given intervention). Users can generate descriptive stats (mean, standard deviation, standard error); odds ratios, risk difference, gains and confidence intervals. An interface with R gives access to most statistical capabilities available in that language; some of these are available through customised drop-down menus. CIETmap can import data in other formats as SPSS, dBase or Excel. | |

The MCSF-test is a behaviour model used to study risk assessment, risk taking, anxiety and security seeking behaviour. It has a completely different design compared to the t-maze, but instead of using a battery of different behaviour models this test can be used to measure a variety of dependent and independent variables. In this context "multivariate" is defined as that the subject has a free choice of different environments contained in the same apparatus and session. The MCSF consists of different areas associated with risk-taking and shelter-seeking. | |

The alternative estimators have been characterized into two general type: (1) robust and (2) limited information estimator. When ML is implemented with data that deviates away from the assumptions of normal theory, CFA models may produce biased parameter estimates and misleading conclusions. Robust estimation typically attempts to correct the problem by adjusting the normal theory model χ2 and standard errors. For example, Satorra and Bentler (1994) recommended using ML estimation in the usual way and subsequently dividing the model χ2 by a measure of the degree of multivariate kurtosis. | |

Sediment or rock samples are collected from either cores or outcrops, and the microfossils they contain are extracted by a variety of physical and chemical laboratory techniques, including sieving, density separation by centrifuge or in heavy liquids, and chemical digestion of the unwanted fraction. The resulting concentrated sample of microfossils is then mounted on a slide for analysis, usually by light microscope. Taxa are then identified and counted. The enormous numbers of microfossils that a small sediment sample can often yield allows the collection of statistically robust datasets which can be subjected to multivariate analysis. | |

Most of the cases, where such a generalized discriminant is defined, are instances of the following. Let be a homogeneous polynomial in indeterminates over a field of characteristic 0, or of a prime characteristic that does not divides the degree of the polynomial. The polynomial defines a projective hypersurface, which has singular points if and only the partial derivatives of have a nontrivial common zero. This is the case if and only if the multivariate resultant of these partial derivatives is zero, and this resultant may be considered as the discriminant of . | |

In statistical theory, one long-established approach to higher-order statistics, for univariate and multivariate distributions is through the use of cumulants and joint cumulants.Kendall, MG., Stuart, A. (1969) The Advanced Theory of Statistics, Volume 1: Distribution Theory, 3rd Edition, Griffin. (Chapter 3) In time series analysis, the extension of these is to higher order spectra, for example the bispectrum and trispectrum. An alternative to the use of HOS and higher moments is to instead use L-moments, which are linear statistics (linear combinations of order statistics), and thus more robust than HOS. | |

Heike Hofmann (born 16 April 1972) is a statistician. She earned an MSc in Mathematics, with a minor in Computer Science, and a PhD in Statistics, from the University of Augsburg, Augsburg, Germany in 1998 and 2000, respectively. She is currently Professor in the Department of Statistics at Iowa State University, and faculty member of the Bioinformatics and Computational Biology and Human Computer Interaction programs. In her research on interactive data visualization she has provided new approaches for plotting multivariate categorical data using mosaic plots, and making interactions with these plots, and linking between plots. | |

Mutual information is also used in the area of signal processing as a measure of similarity between two signals. For example, FMI metric is an image fusion performance measure that makes use of mutual information in order to measure the amount of information that the fused image contains about the source images. The Matlab code for this metric can be found at.. A python package for computing all multivariate mutual informations, conditional mutual information, joint entropies, total correlations, information distance in a dataset of n variables is available . | |

Pragmatic validity in research looks to a different paradigms from more traditional, (post)positivistic research approaches. It tries to ameliorate problems associated with the rigour-relevance debate, and is applicable in all kinds of research streams. Simply put, pragmatic validity looks at research from a prescriptive-driven perspective. Solutions to problems that actually occur in the complex and highly multivariate field of practice are developed in a way that, while valid for a specific situation, need to be adjusted according to the context in which they are to be applied. | |

These artifacts are validated by the adoption rate of the practitioners within the community of practice associated with the field.Brown 1992; Hodkinson 2004; Zaritsky et al. 2003 Nowotny (2000) calls knowledge that has been validated by the multidisciplinary community of practice 'socially robust', meaning that it has been developed in (and for) contexts outside the laboratory and can be used by practitioners. In the following statement, Cook (1983) refers to the well- known educational researcher Cronbach about multivariate causal interdependency and validity, and the need for understanding the complexity of the situation being researched. | |

Design solutions are formulated in the following way; " If you want to achieve Y in situation Z, then you perform something like X" (van Aken & Romme, 2005; p. 6). In short, the resulting artifacts of pragmatic research can also be causal relationships, just typically not as specific or reductionist as those resulting from positivist research. The words 'something like' in the statement implicitly refer to the complexity in which the causal relationship is enacted. The causal agent (X, in the statement above) can also be seen as complex and multivariate (Cook, 1983). | |

Inge Koch is an Australian statistician, author, and advocate for gender diversity in mathematics. Koch is the author of Analysis of Multivariate and High-Dimensional Data (1993), and is a Professor in Statistics at the University of Western Australia. Previously, she has worked as an associate professor at University of Adelaide and taught statistics at the University of New South Wales. From 2015 to 2019, she was the Executive Director of Australian Mathematical Sciences Institute (AMSI)’s Choose Maths Program, encouraging girls and young women to participate in mathematics. | |

The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small number of parameters (for example, using an autoregressive or moving average model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure. Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate. | |

The disease is caused by a defect in a single gene on chromosome 12 that codes for enzyme phenylalanine hydroxylase, that affects multiple systems, such as the nervous and integumentary system. Pleiotropy not only affects humans, but also animals, such as chickens and laboratory house mice, where the mice have the "mini- muscle" allele. Pleiotropic gene action can limit the rate of multivariate evolution when natural selection, sexual selection or artificial selection on one trait favors one allele, while selection on other traits favors a different allele. Some gene evolution is harmful to an organism. | |

In mathematics, nonlinear modelling is empirical or semi-empirical modelling which takes at least some nonlinearities into account. Nonlinear modelling in practice therefore means modelling of phenomena in which independent variables affecting the system can show complex and synergetic nonlinear effects. Contrary to traditional modelling methods, such as linear regression and basic statistical methods, nonlinear modelling can be utilized efficiently in a vast number of situations where traditional modelling is impractical or impossible. The newer nonlinear modelling approaches include non-parametric methods, such as feedforward neural networks, kernel regression, multivariate splines, etc. | |

In the early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency. In 1900, Hilbert posed a famous list of 23 problems for the next century. The first two of these were to resolve the continuum hypothesis and prove the consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution. | |

ANOVA is a relatively robust procedure with respect to violations of the normality assumption. The one-way ANOVA can be generalized to the factorial and multivariate layouts, as well as to the analysis of covariance. It is often stated in popular literature that none of these F-tests are robust when there are severe violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts. Furthermore, it is also claimed that if the underlying assumption of homoscedasticity is violated, the Type I error properties degenerate much more severely. | |

E. H. Bakraji, M. Ahmad, N. Salman, D. Haloum, N. Boutros, R. Abboud., Dating and classification of Syrian excavated pottery from Tell Saka Site, by means of thermoluminescence analysis, and multivariate statistical methods, based on PIXE analysis, Journal of Radioanalytical and Nuclear Chemistry, Akadémiai Kiadó, co-published with Springer Science+Business Media B.V., 1999 A courtyard was excavated measuring by . Columns marked the entrance to the south and four large columns were positioned in a square in the centre of the courtyard. Tempera or perhaps Fresco technique Paintings were found on the walls showing ancient Egyptian style and motifs. | |

A function that can be utilized to evaluate any Boolean output in relation to its Boolean input by logical type of calculations. Such functions play a basic role in questions of complexity theory as well as the design of circuits and chips for digital computers. The properties of Boolean functions play a critical role in cryptography, particularly in the design of symmetric key algorithms (see substitution box). Boolean functions are often represented by sentences in propositional logic, and sometimes as multivariate polynomials over GF(2), but more efficient representations are binary decision diagrams (BDD), negation normal forms, and propositional directed acyclic graphs (PDAG). | |

Brain morphometry is a subfield of both morphometry and the brain sciences, concerned with the measurement of brain structures and changes thereof during development, aging, learning, disease and evolution. Since autopsy-like dissection is generally impossible on living brains, brain morphometry starts with noninvasive neuroimaging data, typically obtained from magnetic resonance imaging (MRI). These data are born digital, which allows researchers to analyze the brain images further by using advanced mathematical and statistical methods such as shape quantification or multivariate analysis. This allows researchers to quantify anatomical features of the brain in terms of shape, mass, volume (e.g. | |

Atenello et al., in 2016, studied 99,472 US paediatric patients with shunted hydrocephalus, 16% of whom were admitted on a weekend. After adjustment for disease severity, time to procedure, and admission acuity, weekend admission was not associated with an increase in the inpatient mortality rate (p=0.46) or a change in the percentage of routine discharges (p=0.98) after ventricular shunt procedures. In addition, associations were unchanged after an evaluation of patients who underwent shunt revision surgery. High-volume centres were incidentally noted in multivariate analysis to have increased rates of routine discharge (OR = 1.04; 95% CI 1.01-1.07; p=0.02). | |

In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. They are heavily used in survey research, business intelligence, engineering, and scientific research. They provide a basic picture of the interrelation between two variables and can help find interactions between them. The term contingency table was first used by Karl Pearson in "On the Theory of Contingency and Its Relation to Association and Normal Correlation", part of the Drapers' Company Research Memoirs Biometric Series I published in 1904. | |

An early interest in social science applications of statistics had already begun to show through in Cambridge (such as extensive experiments into the reliability of trained taste-testers for quality assessments and into price subsidies in the food industry). Also developed were two early aversions, the first to multivariate techniques imposed on simple data, and the second to mathematics for its own sake in applied statistics. Ehrenberg's belief that the methods of physical science are applicable to social science was expressed in an article in the hard science journal Nature.Ehrenberg, A. (1993a), Even the social sciences have laws, Nature, 365, 30 September 385. | |

Although at first the choice of the solution to this regularized problem may look artificial, and indeed the matrix \Gamma seems rather arbitrary, the process can be justified from a Bayesian point of view. Note that for an ill-posed problem one must necessarily introduce some additional assumptions in order to get a unique solution. Statistically, the prior probability distribution of x is sometimes taken to be a multivariate normal distribution. For simplicity here, the following assumptions are made: the means are zero; their components are independent; the components have the same standard deviation \sigma _x. | |

Symbolically, it can do multivariate polynomial arithmetic, factor polynomials, compute GCDs, expand series, and compute with matrices. It is equipped to handle certain noncommutative algebras which are extensively used in theoretical high energy physics: Clifford algebras, SU(3) Lie algebras, and Lorentz tensors. Due to this, it is extensively used in dimensional regularization computations – but it is not restricted to physics. GiNaC is the symbolic foundation in several open-source projects: there is a symbolic extension for GNU Octave, a simulator for magnetic resonance imaging, and since May 2009, Pynac, a fork of GiNaC, provides the backend for symbolic expressions in SageMath. | |

Therefore, PCA or L1-PCA are commonly employed for dimensionality reduction for the purpose of data denoising or compression. Among the advantages of standard PCA that contributed to its high popularity are low-cost computational implementation by means of singular-value decomposition (SVD) and statistical optimality when the data set is generated by a true multivariate Normal data source. However, in modern big data sets, data often include corrupted, faulty points, commonly referred to as outliers. Standard PCA is known to be sensitive to outliers, even when they appear as a small fraction of the processed data. | |

Significant improvements can sometimes be seen through testing elements like copy text, layouts, images and colors, but not always. In these tests, users only see one of two versions, as the goal is to discover which of the two versions is preferable. Multivariate testing or multinomial testing is similar to A/B testing, but may test more than two versions at the same time or use more controls. Simple A/B tests are not valid for observational, quasi-experimental or other non-experimental situations, as is common with survey data, offline data, and other, more complex phenomena. | |

These algorithms are examples of multivariate interpolation on a uniform grid, using relatively straightforward mathematical operations on nearby instances of the same color component. The simplest method is nearest- neighbor interpolation which simply copies an adjacent pixel of the same color channel. It is unsuitable for any application where quality matters, but can be useful for generating previews given limited computational resources. Another simple method is bilinear interpolation, whereby the red value of a non-red pixel is computed as the average of the two or four adjacent red pixels, and similarly for blue and green. | |

There are examples of algorithms that do not have polynomial-time complexity. For example, a generalization of Gaussian elimination called Buchberger's algorithm has for its complexity an exponential function of the problem data (the degree of the polynomials and the number of variables of the multivariate polynomials). Because exponential functions eventually grow much faster than polynomial functions, an exponential complexity implies that an algorithm has slow performance on large problems. Several algorithms for linear programming—Khachiyan's ellipsoidal algorithm, Karmarkar's projective algorithm, and central-path algorithms—have polynomial time-complexity (in the worst case and thus on average). | |

Wold's thesis, A Study in the analysis of stationary time series, was an important contribution. The main result was the "Wold decomposition" by which a stationary series is expressed as a sum of a deterministic component and a stochastic component which can itself be expressed as an infinite moving average. Beyond this, the work brought together for the first time the work on individual processes by English statisticians, principally Udny Yule, and the theory of stationary stochastic processes created by Russian mathematicians, principally A. Ya. Khinchin. Wold's results on univariate time series were generalized to multivariate time series by his student Peter Whittle. | |

In multivariate statistics, exploratory factor analysis (EFA) is a statistical method used to uncover the underlying structure of a relatively large set of variables. EFA is a technique within factor analysis whose overarching goal is to identify the underlying relationships between measured variables. It is commonly used by researchers when developing a scale (a scale is a collection of questions used to measure a particular research topic) and serves to identify a set of latent constructs underlying a battery of measured variables. It should be used when the researcher has no a priori hypothesis about factors or patterns of measured variables. | |

Correspondence analysis of species by environmental variable matrices. Journal of Vegetation Science 1(4): 453–460., were devoted to exploring new and innovative applications for multivariate analysis, including the analysis of time-series data and the exploration of the niche preference of species within a community. In the first two decades of the 21st Century, with the advent of powerful computers and large datasets from molecular genetics studies, the methods of numerical classification and eigenvector ordination, promoted by Greig-Smith half a century earlier, exploded in their use to become indispensable tools in the new molecular approaches to taxonomy, genomics, phylogeography, and anthropology. | |

There is no one specifically defined method for conducting these field assessments, however the different multivariate analysis typically produces results identifying relationships between variables when a robust correlation exists. Knowledge of the site-specific ecosystem and the ecological roles of dominant species within that ecosystem are critical to producing biological evidence of alteration in benthic community resultant of contaminant exposure. When possible, it is recommended to observe changes in community structure that directly relate to the test species used during the sediment toxicity portion of the triad approach in order to produce the most reliable evidence. | |

His work on multivariate inference has led to collaborations with Parashkev Nachev and work in high- dimensional neurology and with DeepMind. He co-edited a large reference book entitled the Neurobiology of Attention with Laurent Itti and John Tsotsos, and is the author of numerous articles and invited reviews on the functional imaging of consciousness. He was a member of the board of the Association for the Scientific Study of Consciousness until stepping down in 2007, and with Patrick Wilken organised its tenth annual meeting that was held at St. Anne's College, Oxford in June 2006. | |

In probability theory, although simple examples illustrate that linear uncorrelatedness of two random variables does not in general imply their independence, it is sometimes mistakenly thought that it does imply that when the two random variables are normally distributed. This article demonstrates that assumption of normal distributions does not have that consequence, although the multivariate normal distribution, including the bivariate normal distribution, does. To say that the pair (X,Y) of random variables has a bivariate normal distribution means that every linear combination aX+bY of X and Y for constant (i.e. not random) coefficients a and b has a univariate normal distribution. | |

Christian Genest is best known for developing models and statistical inference techniques for studying the dependence between variables through the concept of copula. He has designed, among others, various techniques for selecting, estimating and validating copula-based models through rank-based methods. His methodological contributions in multivariate analysis and extreme-value theory found numerous practical applications in finance, insurance, and hydrology. Throughout his career, Christian Genest also made significant contributions to the development of techniques for the reconciliation and use of expert opinions and pairwise comparison methods used to establish priorities in multiple-criteria decision analysis. | |

PCA of the multivariate Gaussian distribution centered at (1, 3) with a standard deviation of 3 in roughly the (0.878, 0.478) direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) covariance matrix scaled by the square root of the corresponding eigenvalue. (Just as in the one-dimensional case, the square root is taken because the standard deviation is more readily visualized than the variance. The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. | |

Anil Kumar Bhattacharya (also spelled Anil Kumar Bhattacharyya, Bengali: অনিল কুমার ভট্টাচার্য) (1 April 1915 - 17 July 1996) was an Indian statistician who worked at the Indian Statistical Institute in the 1930s and early 40s. He made fundamental contributions to multivariate statistics, particularly for his measure of similarity between two multinomial distributions, known as the Bhattacharya coefficient, based on which he defined a metric, the Bhattacharya distance. This measure is widely used in comparing statistical samples in biology, physics, computer science, etc. Distance between statistical distributions had been addressed in 1936 by Mahalanobis, who proposed the D2 metric, now known as Mahalanobis distance. | |

The authors recommend to use a version of QUAD with an 80-bit key, 80-bit IV and an internal state of n = 160 bits. It outputs 160 keystream bits (m = 320) at each iteration until 240 bits of keystream have been produced. At Eurocrypt 2006, speed reports were presented for QUAD instances with 160-bit state and output block over the fields GF(2), GF(16), and GF(256). These speed reports were part of an analysis of "Efficient Implementations of Multivariate Quadratic Systems" which was published by Berbain, Billet, and Gilbert at SAC 2006. | |

Schimd, Monika (2008) Rijksuniversiteit Groningen, The Netherlands "Defining Language Attrition" A positive attitude towards the potentially attriting language or its speech community and motivation to retain the language are other factors which may reduce attrition. These factors are too difficult to confirm by research.Dusseldorp, Elise., Schimd, Monika (2010) Rijksuniversiteit Groningen, The Netherlands/TNO, Quality of Life & Leiden University, The Netherlands "Quantitative analyses in a multivariate study of language attrition: The impact of extralinguistic factors" However, a person's age can well predict the likelihood of attrition; children are demonstrably more likely to lose their first language than adults. | |

However, the information provided may be adequate to eliminate a suspect as an author or narrow down an author from a small group of suspects. Authorship measures that analysts use include word length average, average number of syllables per word, article frequency, type-token ratio, punctuation (both in terms of overall density and syntactic boundaries) and the measurements of hapax legomena (unique words in a text). Statistical approaches include factor analysis, Bayesian statistics, Poisson distribution, multivariate analysis, and discriminant function analysis of function words. The Cusum (Cumulative Sum) method for text analysis has also been developed. | |

The MSSA forecasting results can be used in examining the efficient market hypothesis controversy (EMH). The EMH suggests that the information contained in the price series of an asset is reflected “instantly, fully, and perpetually” in the asset’s current price. Since the price series and the information contained in it are available to all market participants, no one can benefit by attempting to take advantage of the information contained in the price history of an asset by trading in the markets. This is evaluated using two series with different series length in a multivariate system in SSA analysis (Hassani et al. 2010). | |

This is in contrast to a technical system approach to soil classification, where soils are grouped according to their fitness for a specific use and their edaphic characteristics. Natural system approaches to soil classification, such as the French Soil Reference System (Référentiel pédologique français) are based on presumed soil genesis. Systems have developed, such as USDA soil taxonomy and the World Reference Base for Soil Resources, which use taxonomic criteria involving soil morphology and laboratory tests to inform and refine hierarchical classes. Another approach is numerical classification, also called ordination, where soil individuals are grouped by multivariate statistical methods such as cluster analysis. | |

Ramanathan Gnanadesikan (2 November 1932 – 6 July 2015) was an Indian statistician, known for his work in multivariate data analysis and leadership in the field. He received his Ph.D. at the University of North Carolina and headed research groups in statistics at Bell Laboratories and Bellcore. He was a fellow of American Association for the Advancement of Science, American Statistical Association, Institute of Mathematical Statistics and Royal Statistical Society, and elected member of the International Statistical Institute. He served as President of Institute of Mathematical Statistics and the International Association for Statistical Computing, the latter of which he helped found. | |

The key reason for studentizing is that, in regression analysis of a multivariate distribution, the variances of the residuals at different input variable values may differ, even if the variances of the errors at these different input variable values are equal. The issue is the difference between errors and residuals in statistics, particularly the behavior of residuals in regressions. Consider the simple linear regression model : Y = \alpha_0 + \alpha_1 X + \varepsilon. \, Given a random sample (Xi, Yi), i = 1, ..., n, each pair (Xi, Yi) satisfies : Y_i = \alpha_0 + \alpha_1 X_i + \varepsilon_i,\, where the errors \varepsilon_i, are independent and all have the same variance \sigma^2. | |

In computer science, Eppstein's research has included work on minimum spanning trees, shortest paths, dynamic graph data structures, graph coloring, graph drawing and geometric optimization. He has published also in application areas such as finite element meshing, which is used in engineering design, and in computational statistics, particularly in robust, multivariate, nonparametric statistics. Eppstein served as the program chair for the theory track of the ACM Symposium on Computational Geometry in 2001, the program chair of the ACM- SIAM Symposium on Discrete Algorithms in 2002, and the co-chair for the International Symposium on Graph Drawing in 2009. | |

Women with GDM have an insulin resistance that they cannot compensate for with increased production in the β-cells of the pancreas. Placental hormones, and, to a lesser extent, increased fat deposits during pregnancy, seem to mediate insulin resistance during pregnancy. Cortisol and progesterone are the main culprits, but human placental lactogen, prolactin and estradiol contribute, too. Multivariate stepwise regression analysis reveals that, in combination with other placental hormones, leptin, tumor necrosis factor alpha, and resistin are involved in the decrease in insulin sensitivity occurring during pregnancy, with tumor necrosis factor alpha named as the strongest independent predictor of insulin sensitivity in pregnancy. | |

For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation . Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. | |

Samarendra Nath Roy is known for his pioneering contributions in multivariate statistics. Among colleagues of Mahalanobis, other notable contributors were K. R. Nair in Design of experiments, Jitendra Mohan Sengupta in Sample Survey, Ajit Dasgupta in Demography and Ramkrishna Mukherjea in Quantitative Sociology. C. R. Rao's contributions during his association with ISI include two theorems of Statistical Inference known as Cramér–Rao inequality and Rao-Blackwell Theorem, and introduction of orthogonal arrays in Design of Experiments. Anil Kumar Gain is known for his contributions to the Pearson product-moment correlation coefficient with his colleague Sir Ronald Fisher at the University of Cambridge. | |

Degeneracy of a multivariate distribution in n random variables arises when the support lies in a space of dimension less than n. This occurs when at least one of the variables is a deterministic function of the others. For example, in the 2-variable case suppose that Y = aX + b for scalar random variables X and Y and scalar constants a ≠ 0 and b; here knowing the value of one of X or Y gives exact knowledge of the value of the other. All the possible points (x, y) fall on the one-dimensional line y = ax + b. | |

The multivariate nature of SA significantly complicates its quantification and measurement, as it is conceivable that a metric may only tap into one aspect of the operator's SA. Further, studies have shown that different types of SA measures do not always correlate strongly with each other (cf. Durso, Truitt, Hackworth, Crutchfield, Nikolic, Moertl, Ohrt, & Manning, 1995; Endsley, Selcon, Hardiman, & Croft, 1998; Vidulich, 2000). Accordingly, rather than rely on a single approach or metric, valid and reliable measurement of SA should utilize a battery of distinct yet related measures that complement each other (e.g., Harwood, Barnett, & Wickens, 1988). | |

A homogeneous polynomial defines a homogeneous function. This means that, if a multivariate polynomial P is homogeneous of degree d, then :P(\lambda x_1, \ldots, \lambda x_n)=\lambda^d\,P(x_1,\ldots,x_n)\,, for every \lambda in any field containing the coefficients of P. Conversely, if the above relation is true for infinitely many \lambda then the polynomial is homogeneous of degree d. In particular, if P is homogeneous then :P(x_1,\ldots,x_n)=0 \quad\Rightarrow\quad P(\lambda x_1, \ldots, \lambda x_n)=0, for every \lambda. This property is fundamental in the definition of a projective variety. | |

Neuropsychological tests, such as the Wechsler scales and Wisconsin Card Sorting Test, are mostly questionnaires or simple tasks used which assess a specific type of mental function in the respondent. These can be used in experiments, as in the case of lesion experiments evaluating the results of damage to a specific part of the brain.Russell M. Bauer, Elizabeth C. Leritz, & Dawn Bowers, "Neuropsychology", in Weiner (ed.), Handbook of Psychology (2003), Volume 2: Research Methods in Psychology. Observational studies analyze uncontrolled data in search of correlations; multivariate statistics are typically used to interpret the more complex situation. | |

In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal. The terminology arises from the connection with algebraic geometry. If R = k[x0, ..., xn] (a multivariate polynomial ring in n+1 variables over an algebraically closed field k) graded with respect to degree, there is a bijective correspondence between projective algebraic sets in projective n-space over k and homogeneous, radical ideals of R not equal to the irrelevant ideal. | |

Group concept mapping integrates qualitative group processes with multivariate analysis to help a group organize and visually represent its ideas on any topic of interest through a series of related maps. It combines the ideas of diverse participants to show what the group thinks and values in relation to the specific topic of interest. It is a type of structured conceptualization used by groups to develop a conceptual framework, often to help guide evaluation and planning efforts. Group concept mapping is participatory in nature, allowing participants to have an equal voice and to contribute through various methods. | |

Muusoctopus levis It is known to inhabit shallow depths between . It is predatory, living and feeding in the benthic zone, where it feeds heavily on brittle stars. In a study in which the stomach contents of 70 specimens were examined, around 50 were shown to have fed only on brittle stars. A study on the reproductive strategies of coleoid cephalopods concluded, while "a simultaneous terminal spawning strategy is most likely" for M. levis, "the egg-length frequency graphs and multivariate analysis also suggest a greater variation in egg-lengths which could lead to spawning over an extended period". | |

In 1994, Gelfand was presented with a dataset that he had previously not encountered: scallop catches on the Atlantic Ocean. Intrigued by the challenges associated with analyzing data with structured spatial correlation, Gelfand, along with colleagues Sudipto Banerjee and Brad Carlin, created an inferential paradigm for analyzing spatial data. Gelfand’s contributions to spatial statistics include spatially-varying coefficient models, linear models of coregionalization for multivariate spatial processes, predictive processes for analysis of large spatial data and non-parametric approaches to the analysis of spatial data. Gelfand's research in spatial statistics spans application areas of ecology, disease and the environment. | |

The latter sense of the term causes some overlap with the concept of complications. For example, in longstanding diabetes mellitus, the extent to which coronary artery disease is an independent comorbidity versus a diabetic complication is not easy to measure, because both diseases are quite multivariate and there are likely aspects of both simultaneity and consequence. The same is true of intercurrent diseases in pregnancy. In other examples, the true independence or relation is not ascertainable because syndromes and associations are often identified long before pathogenetic commonalities are confirmed (and, in some examples, before they are even hypothesized). | |

Multivariate statistical analyses such as principal component analysis of a range of lipid biomarkers (e.g., other sterols, fatty acids, and fatty alcohols) enable identification of compounds that have similar origins or behaviour. An example can be seen in the loadings plot for sediment samples from the Mawddach Estuary, Wales. Principal Component Analysis of several lipid biomarkers from the Mawddach Esturay - brassicasterol is highlighted in red The location of brassicasterol in this figure (shown in red) indicates that the distribution of this compound is similar to that of the short-chain fatty acids and alcohols, which are known to be of marine origin. | |

Illustration of approximate non-negative matrix factorization: the matrix is represented by the two smaller matrices and , which, when multiplied, approximately reconstruct . Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix is factorized into (usually) two matrices and , with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms or muscular activity, non-negativity is inherent to the data being considered. | |

In this classification method, the identity and location of some of the land-cover types are obtained beforehand from a combination of fieldwork, interpretation of aerial photography, map analysis, and personal experience. The analyst would locate sites that have similar characteristics to the known land-cover types. These areas are known as training sites because the known characteristics of these sites are used to train the classification algorithm for eventual land-cover mapping of the remainder of the image. Multivariate statistical parameters (means, standard deviations, covariance matrices, correlation matrices, etc.) are calculated for each training site. | |

In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations. The classical elimination theory culminated with the work of Macaulay on multivariate resultants, and its description in chapter Elimination theory of the first editions (1930) of van der Waerden's Moderne Algebra. After that, elimination theory was ignored by most algebraic geometers for almost thirty years, until the introduction of new methods for solving polynomial equations, such as Gröbner bases, which were needed for computer algebra. | |

Since 2007 he has been Scientific director of the Interuniversity Graduate School for Psychometrics and Sociometrics, and since 2008 also scientific director of the Institute of Psychology at the University of Leiden. From 2002 to 2015 he served as Editor-in-Chief of the Journal of Classification. Heiser's research interests are in the fields of "multivariate categorical data using multidimensional scaling and classification techniques... advanced clustering and classification methodology for FMRI data." At his farewell speech on 31 January 2014, Heiser was knighted in the Order of the Dutch Lion by the mayor of Leiden, Henri Lenferink.Prof.dr. | |

Multivariate analysis of Raman spectra has enabled development of a quantitative measure for wound healing progress. Spatially offset Raman spectroscopy (SORS), which is less sensitive to surface layers than conventional Raman, can be used to discover counterfeit drugs without opening their packaging, and to non-invasively study biological tissue. A huge reason why Raman spectroscopy is so useful in biological applications is because its results often do not face interference from water molecules, due to the fact that they have permanent dipole moments, and as a result, the Raman scattering cannot be picked up on. This is a large advantage, specifically in biological applications. | |

For factoring a multivariate polynomial over a field or over the integers, one may consider it as a univariate polynomial with coefficients in a polynomial ring with one less indeterminate. Then the factorization is reduced to factorizing separately the primitive part and the content. As the content has one less indeterminate, it may be factorized by applying the method recursively. For factorizing the primitive part, the standard method consists of substituting integers to the indeterminates of the coefficients in a way that does not change the degree in the remaining variable, factorizing the resulting univariate polynomial, and lifting the result to a factorization of the primitive part. | |

Jean Piaget (Vuyk, 1980) came to agree that there were adult postformal stages beyond the stage of formal operations; his earlier theory had located an endpoint to the development of cognitive structures in the adolescent's acquisition of formal operations. John L. Horn (1970, 1979) found that crystallized intelligence, represented by such things as vocabulary size, increased in adulthood. Robert Kegan (1982) combined a Piagetian and an existential-phenomenological approach to create what he called constructive-developmental psychology. Lawrence Kohlberg (1984) found that in early adulthood, some people come to think of moral, ethical and societal issues in multivariate terms (Systematic stage 11, the first postformal stage). | |

Google Website Optimizer was a free website optimization tool that helped online marketers and webmasters increase visitor conversion rates and overall visitor satisfaction by continually testing different combinations of website content. The Google Website Optimizer could test any element that existed as HTML code on a page including calls to action, fonts, headlines, point of action assurances, product copy, product images, product reviews, and forms. It allowed webmasters to test alternative versions of an entire page, called A/B testing — or test multiple combinations of page elements such as headings, images, or body copy; known as Multivariate testing. It could be used at multiple stage in the conversion funnel. | |

On the other hand, the frequentist multivariate methods involve approximations and assumptions that are not stated explicitly or verified when the methods are applied (see discussion on meta-analysis models above). For example, the mvmeta package for Stata enables network meta-analysis in a frequentist framework. However, if there is no common comparator in the network, then this has to be handled by augmenting the dataset with fictional arms with high variance, which is not very objective and requires a decision as to what constitutes a sufficiently high variance.van Valkenhoef G, Lu G, de Brock B, Hillege H, Ades AE, Welton NJ. Automating network meta-analysis. | |

In the univariate module single variables are evaluated (by a t-test) and ranked for their separation performance (i.e. the AUC of the ROC), including confidence intervals (CI) and a computed optimal threshold. In the multivariate module one can choose between three different techniques – SVM (support vector machine), PLS-DA (partial least squares discriminant analysis) and Random Forests for classifying and selecting metabolites or clinical variables for an optimal ROC performance. The resulting analysis produces the top-performing multi-variable model(s) based on their ROC curve characteristics. This module also presents the significant variables (clinical variables and/or metabolites) contributing to the model (via “ROC explorer”). | |

Conway graduated in 1971 from the University of Wisconsin–Madison, with a bachelor's degree in mathematics, earning a second bachelor's degree in computer methods and statistics in 1972. She went to Stanford University for graduate study in statistics, earning a master's degree in 1975 and completing her Ph.D. in 1979. Her dissertation, Multivariate Distribution with Specified Marginals, was supervised by Ingram Olkin. She became an assistant professor of business at the University of Chicago in 1979, as the first female faculty member in the University of Chicago Booth School of Business, and moved to the USC Marshall School of Business as an associate professor in 1985. | |

Recent studies have revealed extensive network of cortical regions that contribute to individual face recognition, including face-selective regions such as fusiform face area (FFA). Nemrodov et al. (2016) conducted multivariate analyses of EEG signals that might be involved in identity related information and applied pattern classification to ERP signals both in time and in space. Main target of the study was as follows: 1) evaluating whether previously known ERP components such as N170 and others are involved in individual face recognition or not, 2) locating temporal landmarks of individual level recognition from ERP signals, and 3) figuring out the spatial profile of individual face recognition. | |

As an undergraduate at the University of Illinois at Urbana–Champaign, Bock earned a bachelor's degree in the German language in 1967. She switched to mathematics for her graduate studies at the same university, completing her PhD in 1974 under the supervision of Robert B. Ash with a dissertation on Certain Minimax Estimators of the Mean of a Multivariate Normal Distribution. As chair of statistics at Purdue from 1995 to 2010, Bock led the department through a period of growth, and took a multidisciplinary approach to the subject that included computational statistics as well as application areas including biostatistics, statistical finance, and environmental statistics. | |

Despite this, a recent study has sought to confirm the applicability of a perceptual decision model to free will decisions. When shown a masked and therefore invisible stimulus, participants were asked to either guess between a category or make a free decision for a particular category. Multivariate pattern analysis using fMRI could be trained on "free-decision" data to successfully predict "guess decisions", and trained on "guess data" in order to predict "free decisions" (in the precuneus and cuneus region). Contemporary voluntary decision prediction tasks have been criticised based on the possibility the neuronal signatures for pre-conscious decisions could actually correspond to lower- conscious processing rather than unconscious processing. | |

Anatoly Aleksandrovich Zhigljavsky (born 19 November 1953) is a professor of statistics in the school of mathematics at Cardiff University. He has authored over 100 publications.Publications list His research interests include time series analysis, multivariate statistical analysis, statistical modeling in market research, stochastic global optimisation, probabilistic methods in search, and dynamical system approach for studying convergence of search algorithms and number theory.Research Interests at Cardiff University He is the Director of the Centre for Optimisation and its Applications, an interdisciplinary centre which encourages joint research and applied projects among members of the Schools of Mathematics, Computer Science and Business and Manufacturing Engineering Centre at Cardiff University. | |

An extensive theoretical literature has grown up around these algorithms, concerning conditions for convergence, rates of convergence, multivariate and other generalizations, proper choice of step size, possible noise models, and so on.Stochastic Approximation and Recursive Estimation, Mikhail Borisovich Nevel'son and Rafail Zalmanovich Has'minskiĭ, translated by Israel Program for Scientific Translations and B. Silver, Providence, RI: American Mathematical Society, 1973, 1976. . These methods are also applied in control theory, in which case the unknown function which we wish to optimize or find the zero of may vary in time. In this case, the step size a_n should not converge to zero but should be chosen so as to track the function. | |

He served as president of the APA's Division 20 from 1982 to 1983, and of the Society of Multivariate Experimental Psychology from 1999 to 2000. Nesselroade studied with Raymond Cattell at the University of Illinois at Urbana–Champaign in the 1960s. This later proved controversial when Nesselroade served on the NCAA's Data Analysis Working Group in the 1990s and Congresswoman Cardiss Collins wrote a letter to the NCAA criticizing him and two other panelists (John L. Horn and John J. McArdle) for their links to Cattell. Collins, as well as the Black Coaches Association, accused the panelists of sympathizing with Cattell's support for eugenics. | |

In 2012 Ruscio and Roche introduced the comparative data (CD) procedure in an attempt improve upon the PA method. The authors state that "rather than generating random datasets, which only take into account sampling error, multiple datasets with known factorial structures are analyzed to determine which best reproduces the profile of eigenvalues for the actual data" (p. 258). The strength of the procedure is its ability to not only incorporate sampling error, but also the factorial structure and multivariate distribution of the items. Ruscio and Roche's (2012) simulation study determined that the CD procedure outperformed many other methods aimed at determining the correct number of factors to retain. | |

The gaussian correlation inequality states that probability of hitting both circle and rectangle with a dart is greater than or equal to the product of the individual probabilities of hitting the circle or the rectangle. The Gaussian correlation inequality (GCI), formerly known as the Gaussian correlation conjecture (GCC), is a mathematical theorem in the fields of mathematical statistics and convex geometry. A special case of the inequality was published as a conjecture in a paper from 1955;Dunnett, C. W.; Sobel, M. Approximations to the probability integral and certain percentage points of a multivariate analogue of Student's t-distribution. Biometrika 42, (1955). 258–260. | |

One use for the probability integral transform in statistical data analysis is to provide the basis for testing whether a set of observations can reasonably be modelled as arising from a specified distribution. Specifically, the probability integral transform is applied to construct an equivalent set of values, and a test is then made of whether a uniform distribution is appropriate for the constructed dataset. Examples of this are P-P plots and Kolmogorov-Smirnov tests. A second use for the transformation is in the theory related to copulas which are a means of both defining and working with distributions for statistically dependent multivariate data. | |

Suppose that for some population of 1000 people, each person is asked to provide their age, height, weight, and number of nose hairs. Thus to each member of the population there is associated an ordered quadruple of numbers. Since n-dimensional Euclidean space is defined as all ordered n-tuples of numbers, this means that the data on 1000 people maybe be thought of as 1000 points in 4-dimensional Euclidean space. The grand tour converts the spatial complexity of the multivariate data set into temporal complexity by using the relatively simple 2-dimensional views of the projected data as the individual frames of the movie. | |

The security of the keystream generation of QUAD is provably reducible to the conjectured intractability of the MQ problem, namely solving a multivariate system of quadratic equations. The first proof was done over field GF(2) for an old-fashioned stream cipher (where the key is the initial state). It was later extended by Berbain and Gilbert in order to take into account the set-up procedure of a modern cipher (with a setup stage deriving the initial state from the key). The security of the whole cipher as a Pseudo Random Function can be related to the conjectured intractability of the MQ problem. | |

Finally, the use of multivariate methods that probe for the latent structure of the data, such as factor analysis and cluster analysis, have proven useful as analytic approaches that go well beyond the bi-variate approaches (cross- tabs) typically employed with smaller data sets. In health and biology, conventional scientific approaches are based on experimentation. For these approaches, the limiting factor is the relevant data that can confirm or refute the initial hypothesis. A new postulate is accepted now in biosciences: the information provided by the data in huge volumes (omics) without prior hypothesis is complementary and sometimes necessary to conventional approaches based on experimentation. | |

A primary advantage is that the mathematical problem to be solved in the algorithm is quantum-resistant. So when a quantum computer is built that can handle enough states to break commercial signature schemes like RSA or ElGamal, the unbalanced oil and vinegar signature scheme should remain secure, as no algorithm currently exists that gives a quantum computer a great advantage in solving these multivariate systems. The second advantage is that the operations used in the equations are relatively simple. Signatures get created and validated only with addition and multiplication of "small" values, making this signature viable for low-resource hardware as found in smart cards. | |

The giant sphinx moth is a rare stray in west Texas and has been collected in Big Bend National Park near long-spur columbine populations; however, the common pollinators are likely large hawkmoths in the genera Manduca and Agrius with tongue lengths from 9–14 cm long. Hybridization is common in the genus Aquilegia and populations with intermediate spur lengths from 7–9 cm are found near some long-spur columbine populations. In a multivariate analyses of floral characteristics, the intermediate plants with spurs 7–9 cm long cluster with the more common golden columbine. One population with intermediate spur lengths is found at Cattail Falls in Big Bend National Park. | |

Reading-writing relationships The impact of reading on writing is obvious, since writers use the letters, punctuation, grammatical constructions, and discourse organization with which readers would necessarily be familiar. However, empirical study of these relations was sporadic, simple (usually examining only two variables), and were not particularly influential of theory or practice. This situation changed with Shanahan’s multivariate investigations of reading-writing relationships during the 1980s; a period that could accurately be described as the beginning of the modern age of reading-writing research. These early studies have been widely cited , and have been replicated and extended with students with learning disabilities Berninger,V. | |

Jones thought the study would provide important comparisons with the findings of climate modeling, which showed a "pretty reasonable" fit to proxy evidence. A commentary on MBH98 by Jones was published in Science on 24 April 1998. He noted that it used almost all the available long term proxy climate series, "and if the new multivariate method of relating these series to the instrumental data is as good as the paper claims, it should be statistically reliable." He discussed some of the difficulties, and emphasised that "Each paleoclimatic discipline has to come to terms with its own limitations and must unreservedly admit to problems, warts and all.". | |

In her multivariate study of these regions, however, Davidson-Schmich narrowed these factors down even further to the most significant variables of: Catholicism and agricultural economics (Davidson- Schmich, 2006, p. 228). This is very intriguing, and as she explains, "the success of voluntary gender quotas in the German states hinged not on the political structure of these Lander, but rather the willingness of within the system to act on the opportunities inherent in these structures" (Davidson- Schmich, 2006, p. 228). Social factors and inherent gender discrimination are more important in the success of a female political quota than the structure of the quota itself. | |

In 1974, Rajendran Raja immigrated to the United States and became a physicist at Fermilab, a U.S. Department of Energy national laboratory near Batavia, Illinois specializing in high-energy particle physics. During his forty years at Fermilab, Raja's important contributions to the lab's scientific work included playing a key role in the design of the hermetic DZero detector. He served as head of the top quark analysis group, and the multivariate algorithm he developed was a crucial tool in the discovery of this particle at Fermilab in 1995. During his last years at Fermilab, he served as spokesperson for the Main Injector Particle Production experiment. | |

Some software packages such as Markerview include multivariate statistical analysis (for example, principal component analysis) and these will be helpful for the identification of correlations in lipid metabolites that are associated with a physiological phenotype, in particular for the development of lipid-based biomarkers.Another objective of the information technology side of lipidomics involves the construction of metabolic maps from data on lipid structures and lipid-related protein and genes. Some of these lipid pathways are extremely complex, for example the mammalian glycosphingolipid pathway.SphingoMAP The establishment of searchable and interactive databases of lipids and lipid-related genes/proteins is also an extremely important resource as a reference for the lipidomics community. | |

He wrote the text How to Multiply Matrices Faster (Springer, 1984) surveying early developments in this area. In 1998, with his student Xiaohan Huang, Pan showed that matrix multiplication algorithms can take advantage of rectangular matrices with unbalanced aspect ratios, multiplying them more quickly than the time bounds one would obtain using square matrix multiplication algorithms. Since that work, Pan has returned to symbolic and numeric computation and to an earlier theme of his research, computations with polynomials. He developed fast algorithms for the numerical computation of polynomial roots, and, with Bernard Mourrain, algorithms for multivariate polynomials based on their relations to structured matrices. | |

When performing non- selective measurements, a sum signal from several analytes is measured which means that multivariate data analyses such as neural networks have to be used for quantification. However, it is also possible to use selectively measuring polymers, so-called molecular imprinted polymers (MIPs) which provide artificial recognition elements. When using biosensors, polymers such as polyethylene glycols or dextrans are applied onto the layer system, and on these recognition elements for biomolecules are immobilized. Basically, any molecule can be used as recognition element (proteins such as antibodies, DNA/RNA such as aptamers, small organic molecules such as estrone, but also lipids such as phospholipid membranes). | |

In statistics, a bivariate random vector (X, Y) is jointly elliptically distributed if its iso-density contours—loci of equal values of the density function—are ellipses. The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are ellipsoids. A special case is the multivariate normal distribution. The elliptical distributions are important in finance because if rates of return on assets are jointly elliptically distributed then all portfolios can be characterized completely by their mean and variance—that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return. | |

During its 35-year history, the Psychophysiology Research Group was a centre for multivariate psychophysiological research on personality, research on cardiovascular rehabilitation, illness behaviour, and life satisfaction. The laboratory was generously supported by the Volkswagen Foundation (with eight scientific and technical staff, two computer-based electrophysiological labs and a clinical-chemistry lab). The research group also developed and promoted both methodology and techniques of ambulatory monitoring (ambulatory assessment) to assist behavioural research in everyday situations. A number of tests and personality scales were developed, one of which, the Freiburg Personality Inventory (FPI), comparable to the 16 PF Questionnaire, is the most frequently used in German-speaking countries. | |

While the SA construct has been widely researched, the multivariate nature of SA poses a considerable challenge to its quantification and measurement (for a detailed discussion on SA measurement, see Endsley & Garland, 2000; Fracker, 1991a; 1991b). In general, techniques vary in terms of direct measurement of SA (e.g., objective real- time probes or subjective questionnaires assessing perceived SA) or methods that infer SA based on operator behavior or performance. Direct measures are typically considered to be "product-oriented" in that these techniques assess an SA outcome; inferred measures are considered to be "process-oriented," focusing on the underlying processes or mechanisms required to achieve SA (Graham & Matthews, 2000). | |

By determining what are called "concordance" rates for a disease or trait among identical and fraternal twin pairs, researchers can estimate whether contributing factors for that disease or trait are more likely to be hereditary, environmental, or some combination of these. A concordance rate is a statistical measure of probability - if one twin has a specific trait or disease, what is the probability that the other twin has (or will develop) that same trait or disease. In addition, with structural equation modeling and multivariate analyses of twin data, researchers can offer estimates of the extent to which allelic variants and environment may influence phenotypic traits. | |

A significant small increase in short term perception by individuals that they maintained control over their lives was observed - this is referred to in psychology as internal locus of control. The researchers found that subjects had some minor short-term positive effects perceived from the Large Group Awareness Training, but no noticeable longer- term positive effects, stating: "In fact, with the exception of the short-term multivariate results for Perceived Control, there was no appreciable effect on any dimension which could reflect positive change." After the participants returned for the 18-month follow-up analysis, the results revealed that the small increase in perception of control by the individuals had disappeared. | |

A natural example of a problem in co-RP currently not known to be in P is Polynomial Identity Testing, the problem of deciding whether a given multivariate arithmetic expression over the integers is the zero-polynomial. For instance, is the zero-polynomial while is not. An alternative characterization of RP that is sometimes easier to use is the set of problems recognizable by nondeterministic Turing machines where the machine accepts if and only if at least some constant fraction of the computation paths, independent of the input size, accept. NP on the other hand, needs only one accepting path, which could constitute an exponentially small fraction of the paths. | |

RHAMM is also one of 3 biomarkers associated with aggressiveness in a multivariate analysis of human prostate tumors and elevated levels of RHAMM are associated with both androgen deprivation therapy and castration resistant disease. RHAMM has also been identified as one of 4 gene products identified in circulating tumor cells in patients with lung adenocarcinoma. While RHAMM has been less studied than CD44 in the process of cancer metastasis, it is likely just as important in this process and can act in concert with, or independently of CD44 to promote cell motility. Increased RHAMM expression is correlated with metastases in colorectal cancer, among others. | |

However, because of the integer coefficients resulting of the derivation, this multivariate resultant may be divisible by a power of , and it is better to take, as a discriminant, the primitive part of the resultant, computed with generic coefficients. The restriction on the characteristic is needed, as, otherwise, a common zero of the partial derivative is not necessarily a zero of the polynomial (see Euler's identity for homogeneous polynomials). In the case of a homogeneous bivariate polynomial of degree , this general discriminant is d^{d-2} times the discriminant defined in . Several other classical types of discriminants, that are instances of the general definition are described in next sections. | |

The research and application of inducing plant system resistance have been encouraging but are not yet a major factor in controlling plant pathogens. Incorporation into integrated pest management programs have shown some promising results. There is research regarding defense against leaf chewing insect pests, by the activation of jasmonic acid signalling triggered by root- associated microorganisms. Some ongoing research into ISR includes (1) how to systematically improve the selection of induction factors; (2) the injury of induced factors; (3) the phenomenon of multi-effect of induced factors; (4) the effects of chemical induction factors on environmental factors; (5) Establishment of population stability of multivariate biological inducible factor. | |

In the case of multivariate normal distributions, the parameters would be n − 1 correlations and (n − 1)(n − 2)/2 partial correlations, which were noted to be algebraically independent in (−1, 1). An entirely different motivation underlay the first formal definition of vines in Cooke. Uncertainty analyses of large risk models, such as those undertaken for the European Union and the US Nuclear Regulatory Commission for accidents at nuclear power plants, involve quantifying and propagating uncertainty over hundreds of variables. Dependence information for such studies had been captured with Markov trees, which are trees constructed with nodes as univariate random variables and edges as bivariate copulas. | |

In the field of bioinformatics and computational biology, many statistical methods have been proposed and used to analyze codon usage bias. Methods such as the 'frequency of optimal codons' (Fop), the relative codon adaptation (RCA) or the codon adaptation index (CAI) are used to predict gene expression levels, while methods such as the 'effective number of codons' (Nc) and Shannon entropy from information theory are used to measure codon usage evenness. Multivariate statistical methods, such as correspondence analysis and principal component analysis, are widely used to analyze variations in codon usage among genes. There are many computer programs to implement the statistical analyses enumerated above, including CodonW, GCUA, INCA, etc. | |

Raman Tool Set is a free software package for processing and analysis of Raman spectroscopy datasets. It has been developed mainly aiming to Raman spectra analysis, but since it works with 2-columns datafiles (Intensity vs Frequency) it can deal with the results of many spectroscopy techniques. Beyond the spectra preprocessing steps, such as baseline subtraction, normalization of spectra, smoothing and scaling, Raman Tool Set allows the user for chemometric analysis by means of principal component analysis (PCA), extended multiplicative signal correction (EMSC) and cluster analysis. Chemometric and multivariate data analysis can also be applied to hyperspectral maps, using PCA, independent component analysis (ICA) and cluster analysis. | |

The multivariate mutual information may be positive, negative or zero. The positivity corresponds to relations generalizing the pairwise correlations, nullity corresponds to a refined notion of independence, and negativity detects high dimensional "emergent" relations and clusterized datapoints ). For the simplest case of three variables X, Y, and Z, knowing, say, X yields a certain amount of information about Z. This information is just the mutual information I(X;Z) (yellow and gray in the Venn diagram above). Likewise, knowing Y will also yield a certain amount of information about Z, that being the mutual information I(Y;Z) (cyan and gray in the Venn diagram above). | |

In the multivariate case, the consideration of complex zeros is also needed, but not sufficient: one must also consider zeros at infinity. For example, Bézout's theorem asserts that the intersection of two plane algebraic curves of respective degrees and consists of exactly points if one consider complex points in the projective plane, and if one counts the points with their multiplicity. Another example is the genus–degree formula that allows computing the genus of a plane algebraic curve form its singularities in the complex projective plane. So a projective variety is the set of points in a projective space, whose homogeneous coordinates are common zeros of a set of homogeneous polynomials. | |

Paul David Polly is an American paleontologist and the Robert R. Shrock Professor in the Department of Earth and Atmospheric Sciences at Indiana University. Polly's research focuses on quantitative evolution, phylogeny, and paleoecology of vertebrates. Much of his work has been on the phylogenetics and functional evolution of mammals, especially Carnivora and Creodonta, on the correspondence between phenotypic and genetic differentiation, on the role of functional traits in structuring mammalian communities, and on the evolution of multivariate quantitative morphological traits. With lead author Jason Head and other co-authors, he helped describe the giant fossil snake Titanoboa and the associated methods for estimating paleotemperature from the size of extinct reptiles. | |

Johannes Petrus "John" van de Geer (21 June 1926, Rotterdam – 9 February 2008, aged 81) was a Dutch psychologist, and Professor Experimental Psychology at the Leiden University, particularly known for his "Introduction to multivariate analysis for the social sciences".G. David Garson (1976) Political science methods. p. 167 His daughter is Sara van de Geer. Van de Geer received his PhD in 1957 at the Leiden University with a thesis entitled "A Psychological Study of Problem Solving" advised by Alfons Chorus.Johannes Petrus van de Geer in Mathematics Genealogy Project Van de Geer was appointed Professor Experimental Psychology at the Leiden University Department of Data Theory in 1961. | |

Quantitative Pharmacology (QP), or Quantitative Systems Pharmacology (QSP), is an organized approach integrating individuals from different disciplines in a combined effort to develop quantitative models in order to solve specific, complex, and multivariate problems in drug development. QP translates the relationship(s) between disease, drug action, and individual variability into improved patient outcomes by leveraging improved teamwork and collaboration to enable quantitative decision-making processes from early discovery to late- stage development. QP encourages more transparent and objective study designs, as well as more data-driven risk-taking to optimize timelines, analyses, and decision-making, resulting in greater efficiency in the drug development process. | |

The original description of the three slug species in the subgenus Carinarion, Arion (Carinarion) fasciatus, Arion (Carinarion) silvaticus and Arion (Carinarion) circumscriptus, was based on small differences in body pigmentation and details of the genital anatomy. A 2006 study of these morphospecies (typological species) claims that previous studies had shown that body colour in these slugs may be influenced by their diet, and that the putative genital differences were not confirmed by subsequent multivariate morphometric analyses. Analysis of alloenzyme and albumen gland proteins gave conflicting results. Also there was evidence of interspecific hybridization in places where these predominantly self- fertilizing slugs apparently outcross, contradicting their status as biological species. | |

The original description of the three slugs in the subgenus Carinarion, Arion (Carinarion) fasciatus, Arion (Carinarion) silvaticus and Arion (Carinarion) circumscriptus was based on small differences in body pigmentation and details of the genital anatomy. A recent study of these morphospecies (typological species) claims that previous studies had shown that body colour in these slugs may be influenced by their diet, and the genital differences were not confirmed by subsequent multivariate morphometric analyses. Analysis of alloenzyme and albumen gland proteins had given conflicting results. Also that evidence of interspecific hybridization in places where these predominantly self- fertilizing slugs apparently outcross contradicted their status as biological species. | |

The original description of the three slugs in the subgenus Carinarion, Arion (Carinarion) fasciatus, Arion (Carinarion) silvaticus and Arion (Carinarion) circumscriptus was based on small differences in body pigmentation and details of the genital anatomy. A recent study of these morphospecies (typological species) claims that previous studies had shown that body colour in these slugs may be influenced by their diet, and the genital differences were not confirmed by subsequent multivariate morphometric analyses. Analysis of alloenzyme and albumen gland proteins had given conflicting results. Also that evidence of interspecific hybridization in places where these predominantly self-fertilizing slugs apparently outcross contradicted their status as biological species. | |

The distribution was independently rediscovered by the English mathematician Karl Pearson in the context of goodness of fit, for which he developed his Pearson's chi-square test, published in 1900, with computed table of values published in , collected in . The name "chi-square" ultimately derives from Pearson's shorthand for the exponent in a multivariate normal distribution with the Greek letter Chi, writing −½χ2 for what would appear in modern notation as −½xTΣ−1x (Σ being the covariance matrix). R. L. Plackett, Karl Pearson and the Chi-Squared Test, International Statistical Review, 1983, 61f. See also Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics. | |

They found the holotype specimen to have been equal to Tyrannosaurus in size at (marginally smaller than "Sue"), but that the larger dentary might have represented an animal of , if geometrically similar to the holotype specimen. By using multivariate regression equations, these authors also suggested an alternative weight of for the holotype and for the larger specimen, and that the latter was therefore the largest known terrestrial carnivore. Frankfurt Hauptbahnhof. In 2005, the paleontologist Cristiano Dal Sasso and colleagues described new skull material (a snout) of Spinosaurus (the original fossils of which were also destroyed during World War II), and concluded this dinosaur would have been long with a weight , exceeding the maximum size of all other theropods. | |

In other words, an integer GCD computation reduces the factorization of a polynomial over the rationals to the factorization of a primitive polynomial with integer coefficients, and the factorization over the integers to the factorization of an integer and a primitive polynomial. Everything that precedes remains true if Z is replaced by a polynomial ring over a field F and Q is replaced by a field of rational functions over F in the same variables, with the only difference that "up to a sign" must be replaced by "up to the multiplication by an invertible constant in F". This reduces the factorization over a purely transcendental field extension of F to the factorization of multivariate polynomials over F. | |

This was true both of those rejected boys who had earlier antisocial contacts and those with a normal level of earlier involvement. Neglected children had the next highest level of antisocial contact, suggesting (insofar as neglect and withdrawal co-occur) that more withdrawn children also turn to unpopular peers, though less to the point of delinquency. The results also showed a strong correlation between both harsh parental discipline and lack of monitoring and antisocial associations in the sixth grade. Consistent with the researchers' hypothesis, however, a multivariate test showed that stable levels of antisocial involvement and poor parenting co-occurred; only academic skills and popularity were able to account for significant increases in antisocial contacts. | |

In DBM, highly non-linear registration algorithms are used, and the statistical analyses are not performed on the registered voxels but on the deformation fields used to register them (which requires multivariate approaches) or derived scalar properties thereof, which allows for univariate approaches. One common variant—sometimes referred to as Tensor-based morphometry (TBM)--is based on the Jacobian determinant of the deformation matrix. Of course, multiple solutions exist for such non-linear warping procedures, and to balance appropriately between the potentially opposing requirements for global and local shape fit, ever more sophisticated registration algorithms are being developed. Most of these, however, are computationally expensive if applied with a high-resolution grid. | |

For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as basically the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point. | |

It is important to note that prognostic gene signatures are not a target of therapy; they offer additional information to consider when discussing details such as duration or dosage or drug sensitivity etc. in therapeutic intervention. The criteria a gene signature must meet to be deemed a prognostic marker include demonstration of its association with the outcomes of the condition, reproducibility and validation of its association in an independent group of patients and lastly, the prognostic value must demonstrate independence from other standard factors in a multivariate analysis. The applications of these prognostic signatures include prognostic assays for breast cancer, hepatocellular carcinoma, leukaemia and are continually being developed for other types of cancers and disorders as well. | |

Wiggins was one of the most prominent advocates of circular representations of personality and he formalized the circular model (also called circumplex model) with modern statistical techniques. Wiggins started with the lexical assumption - (the idea that all important individual differences are encoded within the natural language). But he went further in his effort at taxonomy by arguing that trait terms specify different kinds of ways in which individuals differ. Wiggins was most concerned primarily with interpersonal traits and carefully separated these from other categories of traits. In 2004, the journal Multivariate Behavioral Research published a special edition entitled ‘Personality Topics in Honor of Jerry S. Wiggins’ which was guest edited by Lewis R Goldberg. | |

As light passes from a sample through the element, the normalized intensity, which is detected by a broad band detector, is proportional to the dot product of the regression vector with that spectrum, i.e. is proportional to the concentration of the analyte for which the regression vector was designed. The quality of the analysis is then equal to the quality of the regression vector which is encoded. If the resolution of the regression vector is encoded to the resolution of the laboratory instrument from which that regression vector was designed and the resolution of the detector is equivalent, then the measurement made by Multivariate Optical Computing will be equivalent to that laboratory instrument by conventional means. | |

Center for Health Transformation (CHT) was re-envisioned in 2012 by WellStar Health System president and CEO, Reynold Jennings, as a means to leverage the collective talent of various healthcare systems, which would collaboratively approach significant problems in healthcare. Jennings believed that transforming healthcare requires multiple, independent viewpoints that analyze case studies from dissimilar clinical silos. The reinvented CHT focused on a multivariate approach to specific clinical challenges, and created adoptable, repeatable solutions for all of its member institutions. In a transparent format, participating systems identified workable solutions for duplication at member hospitals, within the model of "operationalize sustainable best practices℠", thus attempting to improve the quality of care while simultaneously lowering costs. | |

For example, as you zoom into a text object it may be represented as a small dot, then a thumbnail of a page of text, then a full-sized page and finally a magnified view of the page. ZUIs use zooming as the main metaphor for browsing through hyperlinked or multivariate information. Objects present inside a zoomed page can in turn be zoomed themselves to reveal further detail, allowing for recursive nesting and an arbitrary level of zoom. When the level of detail present in the resized object is changed to fit the relevant information into the current size, instead of being a proportional view of the whole object, it's called semantic zooming. | |

This study reported a statistically significant 60% accuracy rate, which may be limited by experimental setup; machine-learning data limitations (time spent in fMRI) and instrument precision. Another version of the fMRI multivariate pattern analysis experiment was conducted using an abstract decision problem, in an attempt to rule out the possibility of the prediction capabilities being product of capturing a built-up motor urge. Each frame contained a central letter like before, but also a central number, and 4 surrounding possible "answers numbers". The participant first chose in their mind whether they wished to perform an addition or subtraction operation, and noted the central letter on the screen at the time of this decision. | |

The proof technique involves constructing an auxiliary multivariate polynomial in an arbitrarily large number of variables depending upon \varepsilon, leading to a contradiction in the presence of too many good approximations. More specifically, one finds a certain number of rational approximations to the irrational algebraic number in question, and then applies the function over each of these simultaneously (i.e. each of these rational numbers serve as the input to a unique variable in the expression defining our function). By its nature, it was ineffective (see effective results in number theory); this is of particular interest since a major application of this type of result is to bound the number of solutions of some diophantine equations. | |

The geographical distribution of many species of invertebrate in the river reflects the geology of the catchment area. Viviparid snails and water scorpions (of the genus Nepidae) are commonly found where the river runs over the London Clay. Crayfish are common in areas associated with high alkalinity, particularly around Brockham, and the tributaries which run over the Weald Clay provide an excellent habitat for stoneflies, caddisflies, fast swimming mayflies and riffle beetles.Ruse LP (1996) Multivariate techniques relating macroinvertebrate and environmental data from a river catchment Water Research 30 (12) 3017–3024 The beautiful demoiselle (Calopteryx virgo) disappeared from the River Mole during the 1960s owing to deteriorating water quality, but has since recolonised. | |

Mahalanobis's definition was prompted by the problem of identifying the similarities of skulls based on measurements in 1927.Mahalanobis, Prasanta Chandra (1927); Analysis of race mixture in Bengal, Journal and Proceedings of the Asiatic Society of Bengal, 23:301–333 Mahalanobis distance is widely used in cluster analysis and classification techniques. It is closely related to Hotelling's T-square distribution used for multivariate statistical testing and Fisher's Linear Discriminant Analysis that is used for supervised classification. In order to use the Mahalanobis distance to classify a test point as belonging to one of N classes, one first estimates the covariance matrix of each class, usually based on samples known to belong to each class. | |

The DES takes form of self-report, where individuals are asked to rank their emotions within the discrete categories of fundamental emotions. Due to the subjective-experience component of this system, this therefore leads to the many concerns and criticism as to whether or not this will hinder the reliability and validity of the results attained. DES is different from other multivariate measures of mood states as it is based on the principle that characteristic patterns of fundamental emotions are involved in the mood states such as anxiety and depressed feelings. Many studies have been carried out on large samples, these factor analyses have supported at least eight of the suggested fundamental emotions. | |

If a factorial moment measure is absolutely continuous, then with respect to the Lebesgue measure it is said to have a density (which is a generalized form of a derivative), and this density is known by a number of names such as factorial moment density and product density, as well as coincidence density, joint intensity , correlation function or multivariate frequency spectrumK. Handa. The two-parameter {Poisson-Dirichlet} point process. Bernoulli, 15(4):1082–1116, 2009. The first and second factorial moment densities of a point process are used in the definition of the pair correlation function, which gives a way to statistically quantify the strength of interaction or correlation between points of a point process. | |

TPF is complicated by the fact that campaigns are described by both quantitative (such as price and discount) and qualitative (such as display space and support by sales representatives) variables. New approaches are being developed to address this and other challenges. Most of these approaches attempt to incorporate large amounts of heterogeneous data in the forecasting process. One researcher validated the ability of multivariate regression models to forecast the impact on sales of a product of many variables including price, discount, visual merchandizing, etc.Balasubramanian Kanagasabapathi, K. Antony Arokia Durai Raj, B. Shoban Babu, Mitul Shah, “Forecasting volumes for trade promotions in CPG industry using market drivers,” International Journal of Business Forecasting and Marketing Intelligence, 2009 Vol. | |

In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is difficult. This sequence can be used to approximate the joint distribution (e.g., to generate a histogram of the distribution); to approximate the marginal distribution of one of the variables, or some subset of the variables (for example, the unknown parameters or latent variables); or to compute an integral (such as the expected value of one of the variables). Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled. | |

In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex valued functions may be easily reduced to the study of the real valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real valued functions will be considered in this article. | |

A recent study was conducted involving the planktonic foraminifer Turborotalia. The authors extracted "51 stratigraphically ordered samples from a site within the oceanographically stable tropical North Pacific gyre". Two hundred individual species were examined using ten specific morphological traits (size, compression index, chamber aspect ratio, chamber inflation, aperture aspect ratio, test height, test expansion, umbilical angle, coiling direction, and the number of chambers in the final whorl). Utilizing multivariate statistical clustering methods, the study found that the species continued to evolve non-directionally within the Eocene from 45 Ma to about 36 Ma. However, from 36 Ma to approximately 34 Ma, the stratigraphic layers showed two distinct clusters with significantly defining characteristics distinguishing one another from a single species. | |

Example star plot from NASA, with some of the most desirable design results represented in the center. This spider chart represents the allocated budget versus actual spending for a given organization. A radar chart is a graphical method of displaying multivariate data in the form of a two-dimensional chart of three or more quantitative variables represented on axes starting from the same point. The relative position and angle of the axes is typically uninformative, but various heuristics, such as algorithms that plot data as the maximal total area, can be applied to sort the variables (axes) into relative positions that reveal distinct correlations, trade-offs, and a multitude of other comparative measures. | |

In the multivariate correlation analysis, high market share was associated with high profits, but high profits could have been associated with high market share, or a third factor common to both could have caused the correlation. Many analysts believe that it is possible to use a statistical causality test to determine causation, but if the whole problem is that correlation is insufficient to determine causation in the first place, then how can using another correlation, which is what is used in the tests, determine causation. → In connection with the market share, already indicated and frequent allegations that correlations are used in the PIMS investigations to draw conclusions about causal relationships, i.e. correlation is equated with causality. | |

He was noted for his work on multivariate statistics. He also conducted work on unit-weighted regression, proving the idea that under a wide variety of common conditions, almost all sets of weights will yield composites that are very highly correlated (Wilks, 1938), a result that has been dubbed Wilks's theorem (Ree, Carretta, & Earles, 1998). Another result, also called “Wilks' theorem” occurs in the theory of likelihood ratio tests, where Wilks showed the distribution of log likelihood ratios is asymptotically \chi^2. From the start of his career, Wilks favored a strong focus on practical applications for the increasingly abstract field of mathematical statistics; he also influenced other researchers, notably John Tukey, in a similar direction. | |

The basic idea of kriging is to predict the value of a function at a given point by computing a weighted average of the known values of the function in the neighborhood of the point. The method is mathematically closely related to regression analysis. Both theories derive a best linear unbiased estimator, based on assumptions on covariances, make use of Gauss–Markov theorem to prove independence of the estimate and error, and make use of very similar formulae. Even so, they are useful in different frameworks: kriging is made for estimation of a single realization of a random field, while regression models are based on multiple observations of a multivariate data set. | |

The use of the phrase "arc diagram" for this kind of drawings follows the use of a similar type of diagram by to visualize the repetition patterns in strings, by using arcs to connect pairs of equal substrings. However, this style of graph drawing is much older than its name, dating back to the work of and , who used arc diagrams to study crossing numbers of graphs. An older but less frequently used name for arc diagrams is linear embeddings. write that arc diagrams "may not convey the overall structure of the graph as effectively as a two-dimensional layout", but that their layout makes it easy to display multivariate data associated with the vertices of the graph. | |

In the case of multivariate Dirichlet distributions, there is some confusion over how to define the concentration parameter. In the topic modelling literature, it is often defined as the sum of the individual Dirichlet parameters, when discussing symmetric Dirichlet distributions (where the parameters are the same for all dimensions) it is often defined to be the value of the single Dirichlet parameter used in all dimensions. This second definition is smaller by a factor of the dimension of the distribution. A concentration parameter of 1 (or k, the dimension of the Dirichlet distribution, by the definition used in the topic modelling literature) results in all sets of probabilities being equally likely, i.e. | |

In the first year of course offerings, more than 100 students applied to the two programs, with an initial enrollment of 80 students. Both programs are designated to achieve accreditation by the Commission on Accreditation of Healthcare Management Education (CAHME) by 2020. The master of public informatics program was created in 2019 to provide a vehicle for educating students in the competencies needed in the field of big data: context, statistics, programming, data management, data analytics, visualization, spatial analysis, applications and the integration of these skills. The school's curriculum has always required intensive study of data analysis and multivariate methods, and as students mastered these skills, more challenging applications of data analysis and interpretation have been added. | |

In March 2001 Tapio Schneider published his regularized expectation–maximization (RegEM) technique for analysis of incomplete climate data. The original MBH98 and MBH99 papers avoided undue representation of large numbers of tree ring proxies by using a principal component analysis step to summarise these proxy networks, but from 2001 Mann stopped using this method and introduced a multivariate Climate Field Reconstruction (CFR) technique based on the RegEM method which did not require this PCA step. In May 2002 Mann and Scott Rutherford published a paper on testing methods of climate reconstruction which discussed this technique. By adding artificial noise to actual temperature records or to model simulations they produced synthetic datasets which they called "pseudoproxies". | |

We may use an audit problem to illustrate the three types of variables as follows. Suppose we want to audit the ending balance of accounts receivable (E). As we saw earlier, E is equal to the beginning balance (B) plus the sales (S) for the period minus the cash receipts (C) on the sales plus a residual (R) that represents insignificant sales returns and cash discounts. Thus, we can represent the logical relation as a linear equation: : E=B+S-C+R Furthermore, if the auditor believes E and B are 100 thousand dollars on the average with a standard deviation 5 and the covariance 15, we can represent the belief as a multivariate normal distribution. | |

Solving multivariate quadratic equations (MQ) over a finite set of numbers is an NP-hard problem (in the general case) with several applications in cryptography. The XSL attack requires an efficient algorithm for tackling MQ. In 1999, Kipnis and Shamir showed that a particular public key algorithm, known as the Hidden Field Equations scheme (HFE), could be reduced to an overdetermined system of quadratic equations (more equations than unknowns). One technique for solving such systems is linearization, which involves replacing each quadratic term with an independent variable and solving the resultant linear system using an algorithm such as Gaussian elimination. To succeed, linearization requires enough linearly independent equations (approximately as many as the number of terms). | |

Ashley Montagu said modern human skulls (left) are more neotenized than Neanderthal skulls (right). Delbert D. Thiessen said that Homo sapiens are more neotenized than Homo erectus, Homo erectus was more neotenized than Australopithecus, Great Apes are more neotenized than Old World monkeys and Old World monkeys are more neotenized than New World monkeys. Nancy Lynn Barrickman said that Brian T. Shea concluded by multivariate analysis that Bonobos are more neotenized than the common chimpanzee, taking into account such features as the proportionately long torso length of the Bonobo. Montagu said that part of the differences seen in the morphology of "modernlike types of man" can be attributed to different rates of "neotenous mutations" in their early populations. | |

MammaPrint is based on the Amsterdam 70-gene breast cancer gene signature and uses formalin-fixed-paraffin-embedded (FFPE) or fresh tissue for microarray analysis. It is a laboratory developed test (LDT) which falls into the class of In Vitro Diagnostic Multivariate Index Assays (IVDMIA). MammaPrint was the first (2007) IVDMIA to be cleared by the Food and Drug Administration (FDA) in a De Novo Classification Process (Evaluation of Automatic Class III Designation) and is the only molecular diagnostic test with a randomized prospective clinical trial validating clinical utility. The test uses RNA isolated from tumor samples and run on custom glass microarray slides in order to determine the expression of a 70-gene signature. | |

Given the definition of the permanent of a matrix, it is clear that PERM(M) for any n-by-n matrix M is a multivariate polynomial of degree n over the entries in M. Calculating the permanent of a matrix is a difficult computational task--PERM has been shown to be #P-complete (proof). Moreover, the ability to compute PERM(M) for most matrices implies the existence of a random program that computes PERM(M) for all matrices. This demonstrates that PERM is random self-reducible. The discussion below considers the case where the matrix entries are drawn from a finite field Fp for some prime p, and where all arithmetic is performed in that field. | |

In statistics, path analysis is used to describe the directed dependencies among a set of variables. This includes models equivalent to any form of multiple regression analysis, factor analysis, canonical correlation analysis, discriminant analysis, as well as more general families of models in the multivariate analysis of variance and covariance analyses (MANOVA, ANOVA, ANCOVA). In addition to being thought of as a form of multiple regression focusing on causality, path analysis can be viewed as a special case of structural equation modeling (SEM) - one in which only single indicators are employed for each of the variables in the causal model. That is, path analysis is SEM with a structural model, but no measurement model. | |

In 1869, 10 years after Darwin's On the Origin of Species, Galton published his results in Hereditary Genius. In this work, Galton found that the rate of "eminence" was highest among close relatives of eminent individuals, and decreased as the degree of relationship to eminent individuals decreased. While Galton could not rule out the role of environmental influences on eminence, a fact which he acknowledged, the study served to initiate an important debate about the relative roles of genes and environment on behavioural characteristics. Through his work, Galton also "introduced multivariate analysis and paved the way towards modern Bayesian statistics" that are used throughout the sciences—launching what has been dubbed the "Statistical Enlightenment". | |

Under suitable differentiability conditions, any multivariate density f1…n on n variables, with univariate densities f1,…,fn, may be represented in closed form as a product of univariate densities and (conditional) copula densities on any R-vine : where edges with conditioning set are in the edge set of any regular vine . The conditional copula densities in this representation depend on the cumulative conditional distribution functions of the conditioned variables, , and, potentially, on the values of the conditioning variables. When the conditional copulas do not depend on the values of the conditioning variables, one speaks of the simplifying assumption of constant conditional copulas. Though most applications invoke this assumption, exploring the modelling freedom gained by discharging this assumption has begun . | |

Marketing mix modeling (MMM) is statistical analysis such as multivariate regressions on sales and marketing time series data to estimate the impact of various marketing tactics (marketing mix) on sales and then forecast the impact of future sets of tactics. It is often used to optimize advertising mix and promotional tactics with respect to sales revenue or profit. The techniques were developed by econometricians and were first applied to consumer packaged goods, since manufacturers of those goods had access to accurate data on sales and marketing support. Improved availability of data, massively greater computing power, and the pressure to measure and optimize marketing spend has driven the explosion in popularity as a marketing tool. | |

Born in Voorburg, De Leeuw attended the HBS-B in Voorburg and Alphen aan den Rijn from 1957 to 1963. He studied at Leiden University, where he received his propedeutic examination psychology summa cum laude in 1964; his candidate examination psychology summa cum laude in 1967; and his doctoral examination psychology summa cum laude in 1969. In 1973 he received his PhD cum laude with a thesis entitled "Canonical Analysis of Categorical Data" advised by John P. van de Geer.Jan de Leeuw in Mathematics Genealogy Project The thesis described an alternative organization of multivariate data analysis techniques, which formed the basis for the Gifi group in Leiden and the Gifi system more broadly. | |

In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them. All factorization algorithms, including the case of multivariate polynomials over the rational numbers, reduce the problem to this case; see polynomial factorization. | |

The journal also publishes articles on best-practice and innovation in modelling, for example in multivariate statistics or multi-level modelling. A feature of the journal is the inclusion in articles of complete code by which readers can reproduce results and examples. The journal is indexed in the ISI Web of Knowledge. Despite including many non-citable news articles, it had a 2017 impact factor of 1.371 and a 5-year IF of 2.522. In 2017 the journal was ranked 79th out of 105 journals in computer science and 55th out of 124 journals in statistics and probability. Google Scholar Metrics gave The R Journal an h5-index of 24 and an h5-median of 63 (April 2020). | |

His academic paper, "an integrated approach to the parallel calibration of multivariate linear systems", was published in Acta Automatica (Volume 3, Issue 1), a journal by Chinese Association of Automation and the Institute of Automation, Chinese Academy of Sciences, which was later summarized by the Soviet Academy of Sciences at the time. He had been to Aerospace Valley (France), University of Maryland (USA), National Central University (Taiwan) and Tatung University (Taiwan) for delivering lectures on Grey System Theory. His works has been cited by more than 30,000 times, according to CNKI database. His highly cited paper, his first paper on Grey Systems, received more than 3500 citations till February 2019, according to Google Scholar. | |

Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations, not for all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations). | |

As well as exploring patterns of variation, Multivariate statistical methods can be used to test statistical hypotheses about factors that affect shape and to visualize their effects, although PCA is not needed for this purpose unless the method requires inverting the variance-covariance matrix. Landmark data allow the difference between population means, or the deviation an individual from its population mean, to be visualized in at least two ways. One depicts vectors at landmarks that show the magnitude and direction in which that landmark is displaced relative to the others. The second depicts the difference via the thin plate splines, an interpolation function that models change between landmarks from the data of changes in coordinates of landmarks. | |

Where once there had been peat digging, there were now economically important fisheries. She collaborated with Jennings and Smith on a further study of the Broads; their results were published in 1960 as The Making of the Broads: a reconsideration of their origin in the light of new evidence. In 1950 Lambert had been appointed lecturer in botany at Southampton University. At Southampton, Lambert made a pioneering contribution to the use of computers in botanical science in her collaboration with her head of department, Bill Williams, on the multivariate analysis of plant communities. The Norfolk Record Office holds a large collection of Dr Lambert’s papers from the 1920s-2005, which includes drawings, maps, photographs and written works. | |

Born in The Hague, Meulman received her Master Degree in Mathematical Psychology and Data Theory at Leiden University in 1981, and obtained her PhD in Data Theory in 1986 with the thesis entitled "A distance approach to nonlinear multivariate analysis" advised by Jan de Leeuw and John P. van de Geer. Jacqueline Jacinthe Meulman in Mathematics Genealogy Project She was a consultant for Bell Telephone Laboratories in Murray Hill, NJ, from 1982-1983. In addition to being an Associate Professor in the Department of Data Theory in Leiden, she was an Adjunct Professor at the University of Illinois at Urbana–Champaign from 1993-1999. In 1998, she was appointed Professor of Applied Data Theory at Leiden University. | |

In total, 12,763 males, 40–59 years of age, were enrolled as 16 cohorts, in seven countries, in four regions of the world (United States, Northern Europe, Southern Europe, Japan). One cohort is in the United States, two cohorts in Finland, one in the Netherlands, three in Italy, five in Yugoslavia (two in Croatia, and three in Serbia), two in Greece, and two in Japan. The entry examinations were performed between 1958 and 1964 with an average participation rate of 90%, lowest in the US, with 75% and highest in one of the Japanese cohorts, with 100%.Ancel Keys (ed), Seven Countries: A multivariate analysis of death and coronary heart disease, 1980. | |

Lustig, Robert (July 30, 2009). "Sugar: The Bitter Truth", University of California Television (UCTV) via YouTube, retrieved February 20, 2015 Lustig gave details in his book Fat Chance: that Keys cherry-picked seven of 22 countries; consumption of trans-fat peaked in the 1960s and Keys failed to separate them out; results for Japan and Italy could be explained by either low saturated fat consumption or by low sugar consumption; and Keys wrote that sucrose and saturated fat were intercorrelated but failed to perform the sucrose half of his multivariate correlation analysis.Lustig, Robert, M.D., M.S.L. (2012). Fat Chance: Beating the Odds Against Sugar, Processed Food, Obesity, and Disease, Plume (Penguin), , pp. 110-111. | |

Researchers have not been able to explain and exploit them satisfactorily, but never together. Unfortunately, there is no worldwide repository for such data, and those databases are most often under- exploited using too simplistic analyses, or neglecting cross-correlations among them (most often because such data are acquired and possessed by distinct and competing institutions). The GEFS stands as a revolutionary initiative with the following goals: (i) initiate collaborations with many datacenters across the world to unify competences; (ii) propose a collaborative platform (InnovWiki, developed at ETH Zürich) to develop a mega repository of data and tools of analysis; (iii) develop and test rigorously real-time, high-dimension multivariate algorithms to predict earthquakes (location, time and magnitude) using all available data. | |

However, Nandyala et al., in the following year, in a US study of 34,122 patients who had undergone cervical fusion for cervical spine trauma, found the mortality rate was not significantly different among the weekend patients. Desai et al., in the US, in 2015, investigated 580 children undergoing emergency neurosurgical procedures. After multivariate analysis, children undergoing procedures during a weekday after hours or weekends were more likely to experience complications (p=0.0227), and had an increased mortality. In 2016, Tanenbaum et al. in the US, studied 8,189 patients who had had atlantoaxial fusion. Significant predictors of in-hospital mortality included increased age, emergent or urgent admission, weekend admission, congestive heart failure, coagulopathy, depression, electrolyte disorder, metastatic cancer, neurologic disorder, paralysis, and non-bleeding peptic ulcer. | |

1977 had been the driest year in state history to date. According to the Los Angeles Times, "Drought in the 1970s spurred efforts at urban conservation and the state's Drought Emergency Water Bank came out of drought in the 1980s.". Additionally as drought prediction was essentially random and in response to recent severe drought years, in 1977 the U.S. Department of the Interior, Office of Water Research and Technology contracted Entropy Limited for an exploratory study of the applicability of the entropy minimax entropy minimax method of statistical analysis of multivariate data to the problem of determining the conditional probability of drought one or two years into the future, with the area of special interest being California. Christensen et al. | |

PCA of a multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.866, 0.5) direction and of 1 in the orthogonal direction. The vectors shown are the eigenvectors of the covariance matrix scaled by the square root of the corresponding eigenvalue, and shifted so their tails are at the mean. The principal components of a collection of points in a real p-space are a sequence of p direction vectors, where the i^{th} vector is the direction of a line that best fits the data while being orthogonal to the first i-1 vectors. Here, a best-fitting line is defined as one that minimizes the average squared distance from the points to the line. | |

Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (Golub and Van Loan, 1983), eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's Principal Component Analysis),Jolliffe I.T. Principal Component Analysis, Series: Springer Series in Statistics, 2nd ed., Springer, NY, 2002, XXIX, 487 p. 28 illus. Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science, empirical eigenfunction decomposition (Sirovich, 1987), empirical component analysis (Lorenz, 1956), quasiharmonic modes (Brooks et al. | |

Thus a recursion on the number of variables shows that if GCDs exist and may be computed in R, then they exist and may be computed in every multivariate polynomial ring over R. In particular, if R is either the ring of the integers or a field, then GCDs exist in R[x1,..., xn], and what precedes provides an algorithm to compute them. The proof that a polynomial ring over a unique factorization domain is also a unique factorization domain is similar, but it does not provide an algorithm, because there is no general algorithm to factor univariate polynomials over a field (there are examples of fields for which there does not exist any factorization algorithm for the univariate polynomials). | |

A probability distribution is a function that assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference. Examples are found in experiments whose sample space is non-numerical, where the distribution would be a categorical distribution; experiments whose sample space is encoded by discrete random variables, where the distribution can be specified by a probability mass function; and experiments with sample spaces encoded by continuous random variables, where the distribution can be specified by a probability density function. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures. A probability distribution can either be univariate or multivariate. | |

The Grand Tour is a technique developed by Daniel Asimov in 1985, which is used to explore multivariate statistical data by means of an animation. The animation, or "movie", consists of a series of distinct views of the data as seen from different directions, displayed on a computer screen, that appear to change continuously and that get closer and closer to all possible views. This allows a human- or computer-based evaluation of these views, with the goal of detecting patterns that will convey useful information about the data. This technique is like what many museum visitors do when they encounter a complicated abstract sculpture: They walk around it to view it from all directions, in order to understand it better. | |

He was among the first few investigators who, in working with his students, introduced advanced biostatistical methods from western countries to China. A few examples include: multiple linear regression, logistic regression, Cox regression, proportional hazards models, multi-stage survival model, structural equation modeling, generalized linear model, and epidemic model, which are now widely used by epidemiologists and other medical researchers in China. Considering China's precarious political atmosphere, when the nation was heavily ravaged by seemingly incessant political turmoils, these accomplishments were quite a feat. At Shanghai Medical University, he designed, introduced, and lectured many courses for medical students including "Introduction to Biostatistics" and "Clinical Trials", "Design of Experiments" and "Multivariate Analysis" to graduate students in public health, and "Quality Control" to students in health administration. | |

Phytosociological data contain information collected in relevés (or plots) listing each species cover- abundance values and the measured environmental variables. This data is conveniently databanked in a program like TURBOVEG allowing for editing, storage and export to other applications. Data is usually classified and sorted using TWINSPANHill MO (1979) TWINSPAN: A FORTRAN Programme for arranging multivariate data in an ordered two-way table by classification of the individuals and attributes. Ecology and Systematics, Cornell University, Ithaca, NY in host programs like JUICE to create realistic species-relevé associations. Further patterns are investigated using clustering and resemblance methods, and ordination techniques available in software packages like CANOCOter Braak CJF, Šmilauer P (2002) CANOCO Reference manual and CanoDraw for Windows User’s guide: Software for Canonical Community Ordination (version 4.5). | |

Data-driven prognostics usually use pattern recognition and machine learning techniques to detect changes in system states. The classical data-driven methods for nonlinear system prediction include the use of stochastic models such as the autoregressive (AR) model, the threshold AR model, the bilinear model, the projection pursuit, the multivariate adaptive regression splines, and the Volterra series expansion. Since the last decade, more interests in data- driven system state forecasting have been focused on the use of flexible models such as various types of neural networks (NNs) and neural fuzzy (NF) systems. Data-driven approaches are appropriate when the understanding of first principles of system operation is not comprehensive or when the system is sufficiently complex such that developing an accurate model is prohibitively expensive. | |

In recent years, Korea has experienced an increase in the number of international marriages and multicultural families, and the treatment of these families has become an important social policy issue for the Korean government. Comparison of social exclusion showed that multicultural families tend to experience a higher level of social exclusion in general than Korean families do. A multivariate analysis using a zero-inflated Poisson regression model revealed that females, elderly marriage migrants and those with lower social status were exposed to particularly higher risks of exclusion, whereas those who spoke fluent Korean or possessed fertile social networks were less likely to face social exclusion. Children with multicultural backgrounds face discrimination at school, reflecting the prejudices against biracial people in the wider Korean society. | |

Several skulls of fossil large canids from sites in Belgium, the Ukraine and Russia were examined using multivariate analysis to look for evidence of the presence of Paleolithic dogs that were separate from Pleistocene wolves. Reference groups included the Eliseevichi-1 prehistoric dogs, recent dogs and wolves. The osteometric analysis of the skulls indicated that the Paleolithic dogs fell outside the skull ranges of the Pleistocene wolf group and the modern wolf group, and were closer related to those of the Eliseevichi-1 prehistoric dog group. The fossil large canid from Goyet, Belgium dated at 36,000 YBP was clearly different from the recent wolves, resembling most closely the Eliseevichi-1 prehistoric dogs and suggesting that dog domestication had already started during the Aurignacian. | |

Jørgensen identified a number of other classes of dispersion models which included the multivariate dispersion models, the dispersion models for extremes and the dispersion models for geometric sums. He had an interest in a class of exponential dispersion models identified by Maurice Tweedie characterized by closure under additive and reproductive convolution as well as under transformations of scale that are now called the Tweedie distributions. These models express a power law relationship between the variance and the mean which manifests in ecological systems where it is known as Taylor's law and in physical systems where it is known as fluctuation scaling. Jørgensen proved a number of convergence theorems, related to the central limit theorem, that specified the asymptotic behavior of the variance functions of the Tweedie models. | |

Non-white and pregnant patients were over- represented. 45% of patients had at least one underlying condition, mainly asthma, and 13% received antiviral drugs before admission. Of 349 with documented chest x-rays on admission, 29% had evidence of pneumonia, but bacterial co-infection was uncommon. Multivariate analyses showed that physician-recorded obesity on admission and pulmonary conditions other than asthma or chronic obstructive pulmonary disease (COPD) were associated with a severe outcome, as were radiologically confirmed pneumonia and a raised C-reactive protein (CRP) level (≥. 59% of all in-hospital deaths occurred in previously healthy people. Fulminant (sudden-onset) myocarditis has been linked to infection with H1N1, with at least four cases of myocarditis confirmed in patients also infected with A/H1N1. | |

Prior to this work, the inclusion of writing within American reading curricula was rare but today most commercial core reading instruction programs include a writing component. Shanahan’s work on reading-writing relationships conceptualized the connections as being multivariate and developmental (changing in nature as students progressed). Although at the time this line of research began it was common to claim that reading and writing were closely related, Shanahan found the relations to be moderate in scope—meaning that reading and writing could influence each other, but also that they would differ in important ways. This means that to accomplish high levels of reading and writing ability, it is essential that both be taught, rather than simply teaching one to accomplish the other. | |

In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. | |

In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean. The various moments form one set of values by which the properties of a probability distribution can be usefully characterized. Central moments are used in preference to ordinary moments, computed in terms of deviations from the mean instead of from zero, because the higher-order central moments relate only to the spread and shape of the distribution, rather than also to its location. Sets of central moments can be defined for both univariate and multivariate distributions. | |

Tennessee began using the system in 1993, and it since has been adopted by a number of other school districts across the United States. Sanders' approach has been used to support the theory that the quality of teachers is central to educational achievement. The Pennsylvania and New Hampshire Departments of Education sponsor pilots, and the Iowa School Board Association sponsors his value-added work in that state. Battelle for Kids provides interpretation and use trainings for the SAS EVAAS services for the participating districts in Ohio. “Using mixed model equations, TVAAS uses the covariance matrix from this multivariate, longitudinal data set to evaluate the impact of the educational system on student progress in comparison to national norms, with data reports at the district, school, and teacher levels. | |

RCM II, Reliability Centered Maintenance, Second edition 2008, page 250-260, the role of Actuarial analysis in Reliability Reliability engineering can be achieved through process and reliability testing. "Nearly all teaching and literature on the subject emphasize these aspects, and ignore the reality that the ranges of uncertainty involved largely invalidate quantitative methods for prediction and measurement."O'Connor, Patrick D. T. (2002), Practical Reliability Engineering (Fourth Ed.), John Wiley & Sons, New York. . For example, it is easy to represent "probability of failure" as a symbol or value in an equation, but it is almost impossible to predict its true magnitude in practice, which is massively multivariate, so having the equation for reliability does not begin to equal having an accurate predictive measurement of reliability. | |

Storm chasing is chiefly a recreational endeavor, with motives usually given toward photographing or videoing the storm and for multivariate personal reasons. These can include the beauty of views afforded by the sky and land, the mystery of not knowing precisely what will unfold and the quest to undetermined destination on the open road, intangible experiences such as feeling one with a much larger and powerful natural world, the challenge of correctly forecasting and intercepting storms with the optimal vantage points, and pure thrill seeking. Pecuniary interests and competition may also be components; in contrast, camaraderie is common. Although scientific work is sometimes cited as a goal, direct participation in such work is almost always impractical except for those collaborating in an organized university or government project. | |

She researched adolescent alcohol norms as well as family, teacher, and peer processes. In one of Horne's study examining legal recognition of same-sex couple relationships, she found that participants in committed or legally recognized relationships reported less psychological distress and higher well-being than single participants. She used an online survey sample of 2,677 lesbian, gay, and bisexual individuals, and participants were placed in 4 groups: single, dating, in a committed relationship, and in a legally recognized relationship. Analyses revealed that Significant group differences and multivariate analyses indicated that participants in a legally recognized relationship reported less internalized homophobia, fewer depressive symptoms, lower levels of stress, and more meaning in their lives than those in committed relationships, even after controlling for other factors. | |

Thus, face recognition is transformed to a multivariate, statistical pattern recognition problem. In a similar fashion to appearance-based face recognition, an appearance-based gait recognition approach considers gait as a holistic pattern and uses a full-body representation of a human subject as silhouettes or contours. Gait video sequences are naturally three-dimensional objects, formally named tensor objects, and they are very difficult to deal with using traditional vector-based learning algorithms. In order to deal with these tensor objects effectively, Venetsanopoulos and his research team developed a framework of multilinear subspace learning, so that computation and memory demands are reduced, natural structure and correlation in the original data are preserved, and more compact and useful features can be obtained. | |

As the speckle pattern is random, any region of the myocardium has a unique speckle pattern: Within the picture, a defined area "kernel" can be defined, and as this speckle pattern is relatively stable, the kernel can be recognised in the next frame, within a larger search area, by a "best match" search algorithm. There are different search algorithms, the most commonly used is "sum of absolute differences", shown to be similarly accurate as cross-correlation, which is an alternative.Insana MF, Wagner RF, Garra BS, Momenan R, Shawker TH. Pattern recognition methods for optimizing multivariate tissue signatures in diagnostic ultrasound. Ultrason Imaging. 1986 Jul;8(3):165-80Bohs LN, Friemel BH, Trahey GE. Experimental velocity profiles and volumetric flow via two- dimensional speckle tracking. | |

Define f to be holomorphic if it is analytic at each point in its domain. Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function f, this is equivalent to f being holomorphic in each variable separately (meaning that if any coordinates are fixed, then the restriction of f is a holomorphic function of the remaining coordinate). The much deeper Hartogs' theorem proves that the continuity hypothesis is unnecessary: f is holomorphic if and only if it is holomorphic in each variable separately. More generally, a function of several complex variables that is square integrable over every compact subset of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions. | |

Likewise, a change in x_j is not necessary to change y, because a change in y could be caused by something implicit in the error term (or by some other causative explanatory variable included in the model). Regression analysis controls for other relevant variables by including them as regressors (explanatory variables). This helps to avoid mistaken inference of causality due to the presence of a third, underlying, variable that influences both the potentially causative variable and the potentially caused variable: its effect on the potentially caused variable is captured by directly including it in the regression, so that effect will not be picked up as a spurious effect of the potentially causative variable of interest. In addition, the use of multivariate regression helps to avoid wrongly inferring that an indirect effect of, say x1 (e.g. | |

Despite criticisms, experimenters are still trying to gather data that may support the case that conscious "will" can be predicted from brain activity. fMRI machine learning of brain activity (multivariate pattern analysis) has been used to predict the user choice of a button (left/right) up to 7 seconds before their reported will of having done so. Brain regions successfully trained for prediction included the frontopolar cortex (anterior medial prefrontal cortex) and precuneus/posterior cingulate cortex (medial parietal cortex). In order to ensure report timing of conscious "will" to act, they showed the participant a series of frames with single letters (500 ms apart), and upon pressing the chosen button (left or right) they were required to indicate which letter they had seen at the moment of decision. | |

Optimization is viewed as a series of incremental updates of a probabilistic model, starting with the model encoding an uninformative prior over admissible solutions and ending with the model that generates only the global optima. EDAs belong to the class of evolutionary algorithms. The main difference between EDAs and most conventional evolutionary algorithms is that evolutionary algorithms generate new candidate solutions using an implicit distribution defined by one or more variation operators, whereas EDAs use an explicit probability distribution encoded by a Bayesian network, a multivariate normal distribution, or another model class. Similarly as other evolutionary algorithms, EDAs can be used to solve optimization problems defined over a number of representations from vectors to LISP style S expressions, and the quality of candidate solutions is often evaluated using one or more objective functions. | |

Christian Genest was the first recipient of the CRM-SSC Prize in 1999. He received the SUMMA Research Award from Université Laval the same year. In 2011, the Statistical Society of Canada awarded him its most prestigious distinction, the Gold Medal, "in recognition of his remarkable contributions to multivariate analysis and nonparametric statistics, notably through the development of models and methods of inference for studying stochastic dependence, synthesizing expert judgments and multi-criteria decision making, as well as for his applications thereof in various fields such as insurance, finance, and hydrology." Christian Genest is a fellow of the American Statistical Association since 1996, a fellow of the Institute of Mathematical Statistics since 1997, and an honorary member of the Association des statisticiennes et statisticiens du Québec since 2012. | |

Christian Genest has served the mathematical and statistical communities in many ways. Among others, he was director of the Institut des sciences mathématiques du Québec (2012–15), president of the Statistical Society of Canada (2007–08) and president of the Association des statisticiennes et statisticiens du Québec (2005–08). He served on Statistics Canada's Advisory Committee on Statistical Methods for several years, and on the editorial board of various peer-review journals, including The Canadian Journal of Statistics (1988–2003), the Journal de la Société française de statistique (1999–2008) and the Journal of Multivariate Analysis (2003–2015). He was also editor in chief of The Canadian Journal of Statistics (1998–2000) and guest editor for various books and special issues, including two for Insurance: Mathematics and Economics (2005, 2009). | |

Patrice Brun (7 July 1951, Koblenz) is a French archaeologist, a professor at Pantheon-Sorbonne University where he teaches European early history as well as theories and methods of archeology. His main focus encompasses the 6,500-year BCE in Europe, from the advent of the Agro-pastoral economy to the State in a major part of the continent, with an emphasis on recent Protohistory, i.e., Bronze and Iron Ages. He greatly contributed to large- scale data collected from region-level excavation campaigns. Patrice Brun’s research covers a broad range of aspects present in all Europe, namely trade and exchanges, the settlement patterns, and dynamics of identity. This multivariate and multidisciplinary approach allows shedding the light on Patrice Brun’s main focus: dynamics of social changes that led to the rise of State. | |

Revelle has previously served as the President (2005-2009) of the International Society for the Study of Individual Differences (ISSID), the President (2008-2009) of the Association for Research in Personality (ARP), and the President (1984) of the Society of Multivariate Experimental Psychology (SMEP). Currently, he is Vice- Chair of the Governing Board of the Bulletin of the Atomic Scientists, having previously served as Chair (2009-2012). He also serves as the President (2018–present) of the International Society for Intelligence Research (ISIR). Additionally, he is a Fellow of the American Association for the Advancement of Science (AAAS; 1996–present), the Association for Psychological Science (APS; 1994–present), the American Psychological Association (APA Division 5; 2011–present), and the Society for Personality and Social Psychology (SPSP; 2015–present). | |

The joint probabilistic data-association filter (JPDAF) is a statistical approach to the problem of plot association (target-measurement assignment) in a target tracking algorithm. Like the probabilistic data association filter (PDAF), rather than choosing the most likely assignment of measurements to a target (or declaring the target not detected or a measurement to be a false alarm), the PDAF takes an expected value, which is the minimum mean square error (MMSE) estimate for the state of each target. At each time, it maintains its estimate of the target state as the mean and covariance matrix of a multivariate normal distribution. However, unlike the PDAF, which is only meant for tracking a single target in the presence of false alarms and missed detections, the JPDAF can handle multiple target tracking scenarios. | |

Individuals making subjective assessments of their own SA are often unaware of information they do not know (the "unknown unknowns"). Subjective measures also tend to be global in nature, and, as such, do not fully exploit the multivariate nature of SA to provide the detailed diagnostics available with objective measures. Nevertheless, self- ratings may be useful in that they can provide an assessment of operators' degree of confidence in their SA and their own performance. Measuring how SA is perceived by the operator may provide information as important as the operator's actual SA, since errors in perceived SA quality (over-confidence or under-confidence in SA) may have just as harmful an effect on an individual's or team's decision-making as errors in their actual SA (Endsley, 1998). | |

Some authorities, such as Thorington (1976), posit that S. leucopus is very closely related to the cotton-top tamarin, Saguinus oedipus. Other analyses made by Hernandez-Camacho & Cooper (1976), and later Mittermeier and Coimbra-Filho in 1981, and finally Grooves (2001) This view is supported by Hanihara & Natoria's multivariate analysis of toothcomb dental morphology (1987) and by Skinner's work in 1991, which found high similarity between S. oedipus and S. leucopus in 16 out of 17 morphological traits considered. This is further supported by the transition from child to adult, during which the fur coloration changes take place and are similar between the two species. Philip Hershkovitz proposed that the divergence of the two species occurred in the Pleistocene at the height of the Atrato River, where it intersected the Cauca-Magdalena. | |

The technology, though still in the early stages of development, promises many applications, such as: quality control in food processing, detection and diagnosis in medicine, detection of drugs, explosives and other dangerous or illegal substances, disaster response, and environmental monitoring. One type of proposed machine olfaction technology is via gas sensor array instruments capable of detecting, identifying, and measuring volatile compounds. However, a critical element in the development of these instruments is pattern analysis, and the successful design of a pattern analysis system for machine olfaction requires a careful consideration of the various issues involved in processing multivariate data: signal-preprocessing, feature extraction, feature selection, classification, regression, clustering, and validation. Another challenge in current research on machine olfaction is the need to predict or estimate the sensor response to aroma mixtures. | |

She earned a master's degree in economics from Vanderbilt University in 1988, with V. Kerry Smith as her advisor, and completed her Ph.D. in statistics in 1992 at Colorado State University. Her dissertation, supervised by Ronald W. Butler, was Saddlepoint Approximations in Multivariate Analysis. She joined the department of statistics of the University of Georgia in 1992, moved to the University of Wyoming in 1995, was on leave there from 2012 to 2014 while working as deputy director of the Statistical and Applied Mathematical Sciences Institute in Research Triangle Park, North Carolina, and moved to West Virginia University as chair in 2017. Huzurbazar is the daughter of noted Indian statistician V. S. Huzurbazar and the sister of noted statistician Aparna V. Huzurbazar, whose husband, Brian J. Williams, is also a statistician. | |

However, in his later monograph of 1980, Keys included multivariate regressions in which sugar is added to the regression and saturated fat is controlled for. In this regression, Keys found that sugar was not statistically significantly related to incidence of heart disease when dietary saturated fat was controlled for. Today, sugar intake is known to increase the risk of diabetes mellitus, and increased dietary intake of sugar is known to be associated with higher blood pressure, unfavorable blood lipids and cardiometabolic risks. However, a 2010 conference debate of the American Dietetic Association expressed concern over the health risks of replacing saturated fats in the diet with refined carbohydrates, which carry a high risk of obesity and heart disease, particularly at the expense of polyunsaturated fats which may have health benefits. | |

These are the basic terms that the client uses to make sense of the elements, and are always expressed as a contrast. Thus the meaning of "good" depends on whether you intend to say "good versus poor", as if you were construing a theatrical performance, or "good versus evil", as if you were construing the moral or ontological status of some more fundamental experience. # A set of ratings of elements on constructs. Each element is positioned between the two extremes of the construct using a 5- or 7-point rating scale system; this is done repeatedly for all the constructs that apply; and thus its meaning to the client is modeled, and statistical analysis varying from simple counting, to more complex multivariate analysis of meaning, is made possible. | |

Weekend admissions were associated with significantly higher in-hospital mortality (43.4%) than weekday admissions (36.9%; p<0.001). Multivariate regression analysis showed that weekend admission was an independent risk factor for increased in- hospital mortality (OR = 1.32; 95% CI 1.14-1.52; p<0.001). Two years later, in the US, in a study of 5832 patients, Groves et al. found that patients admitted on the weekend had a statistically significant increase in mortality compared with those admitted on the weekdays (OR = 1.32; 95% CI 1.13-1.55; p=0.0004). There are also two studies on lower limb ischaemia, both performed in the US. In a study of 63,768 patients with an ischaemic lower limb in 2014, Orandi et al. found no statistically significant association between weekend admission and in-hospital mortality (OR = 1.15; 95% CI 1.06-1.25; p=0.10). | |

In multivariate quantitative genetics, a genetic correlation (denoted r_g or r_a) is the proportion of variance that two traits share due to genetic causes,Falconer 1960, Introduction to Quantitative Genetics, Ch. 19 "Correlated Characters"Lynch & Walsh 1998, Genetics and Analysis of Quantitative Traits, Ch21, "Correlations Between Characters", "Ch25, Threshold Characters"Neale & Maes 1996, Methodology for genetics studies of twins and families (6th ed.). Dordrecht, The Netherlands: Kluwer. the correlation between the genetic influences on a trait and the genetic influences on a different traitpg 123 of Plomin 2012Martin & Eaves 1977, "The Genetical Analysis of Covariance Structure" Eaves et al 1978, "Model-fitting approaches to the analysis of human behaviour"Loehlin & Vandenberg 1968, "Genetic and environmental components in the covariation of cognitive abilities: An additive model", in Progress in Human Behaviour Genetics, ed. S. G. Vandenberg, pp. 261–278. | |

In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized. The underlying random variables may be random real numbers, or they may be random vectors (each having the same dimension), in which case the mixture distribution is a multivariate distribution. In cases where each of the underlying random variables is continuous, the outcome variable will also be continuous and its probability density function is sometimes referred to as a mixture density. The cumulative distribution function (and the probability density function if it exists) can be expressed as a convex combination (i.e. | |

In the case of parametric model the number of data points kNT (k--number of channels, NT--number of points in the data window) has to be bigger (preferably by order of magnitude) than the number of parameters, which in case of MVAR is equal to k2p (p--model order). In order to evaluate dynamics of the process, a short data window has to be applied, which requires an increase of the number of the data points, which may be achieved by means of a repetition of the experiment. A non-stationary recording may be divided into shorter time windows, short enough to treat the data within a window as quasi-stationary. Estimation of MVAR coefficients is based on calculation of the correlation matrix between channels Rij of k signals Xi from multivariate set, separately for each trial. | |

Joseph F. Hair, Jr. is an American author, consultant, and professor. Currently he serves as Distinguished Professor of Marketing, is the holder of the Cleverdon Chair of Business and Director of the DBA program at the Mitchell College of Business at the University of South Alabama. Previously he held the positions of Senior Scholar, DBA program at the Michael J. Coles College of Business at ennesaw State University, and held the Copeland Endowed Chair of Entrepreneurship at the Ourso College of Business Administration at Louisiana Louisiana State University. He has authored over 60 books, including Multivariate Data Analysis (7th edition, 2010) (cited 125,000+ times), Essentials of Business Research Methods (2016), A Primer on Partial least Squares Structural Equation Modeling - PLS (2nd edition, 2016), and Essentials of Marketing Research (4th edition, 2017), and MKTG (12th edition, 2019). | |

Paige’s 1977 The Scientific Study of Political Leadership has been considered, jointly with James MacGregor Burns’ 1978 Leadership, a landmark for the foundation and institutionalization of political leadership as discipline, following Harold Lasswell's challenge to study this field as a subject for multidisciplinary research grounded in social science theory. In this essay, Paige presents a conceptual framework through which the study of political leadership, and leadership in general, can be organized and developed following scientific bases. This framework, presented as an "multivariate, multidimensional linkage approach" considers six main factors that impact the behavior of political leaders: personality, role, organization, tasks, values, and setting. At the same time, these factors also generate patterns of behavior that can affect or be affected by 18 societal political dimensions as the extent of conflict, the use of violence, the presence of consensus, and the practice of compromise. | |

Speed-and-feed selection is analogous to other examples of applied science, such as meteorology or pharmacology, in that the theoretical modeling is necessary and useful but can never fully predict the reality of specific cases because of the massively multivariate environment. Just as weather forecasts or drug dosages can be modeled with fair accuracy, but never with complete certainty, machinists can predict with charts and formulas the approximate speed and feed values that will work best on a particular job, but cannot know the exact optimal values until running the job. In CNC machining, usually the programmer programs speeds and feedrates that are as maximally tuned as calculations and general guidelines can supply. The operator then fine-tunes the values while running the machine, based on sights, sounds, smells, temperatures, tolerance holding, and tool tip lifespan. | |

Thus later studies rather than focusing on subjects in groups, focus more on individual differences in the neural bases by jointly looking at behavioural analyses and neuroimaging. Neuroimaging studies on loss aversion involves measuring brain activity with functional magnetic resonance imaging (fMRI) to investigate whether individual variability in loss aversion were reflected in differences in brain activity through bidirectional or gain or loss specific responses, as well as multivariate source-based morphometry (SBM) to investigate a structural network of loss aversion and univariate voxel-based morphometry (VBM) to identify specific functional regions within this network. Brain activity in a right ventral striatum cluster increases particularly when anticipating gains. This involves the ventral caudate nucleus, pallidum, putamen, bilateral orbitofrontal cortex, superior frontal and middle gyri, posterior cingulate cortex, dorsal anterior cingulate cortex, and parts of the dorsomedial thalamus connecting to temporal and prefrontal cortex. | |

Let X be a random n-by-n matrix with entries from Fp. Since all the entries of any matrix M + kX are linear functions of k, by composing those linear functions with the degree n multivariate polynomial that calculates PERM(M) we get another degree n polynomial on k, which we will call p(k). Clearly, p(0) is equal to the permanent of M. Suppose we know a program that computes the correct value of PERM(A) for most n-by-n matrices with entries from Fp\---specifically, 1 − 1/(3n) of them. Then with probability of approximately two-thirds, we can calculate PERM(M + kX) for k = 1,2,...,n + 1. Once we have those n + 1 values, we can solve for the coefficients of p(k) using interpolation (remember that p(k) has degree n). | |

The origins of geodemographics are often identified as Charles Booth and his studies of deprivation and poverty in early twentieth century London, and the Chicago School of sociology. Booth developed the idea of 'classifying neighborhoods', exemplified by his multivariate classification of the 1891 UK Census data to create a generalized social index of London's (then) registration districts. Research at the Chicago School – though generally qualitative in nature – strengthened the idea that such classifications could be meaningful by developing the idea of 'natural areas' within cities: conceived as geographical units with populations of broadly homogenous social-economic and cultural characteristics. The idea that census outputs could serve to identify and to characterize the geographies of cities gathered momentum with the increased availability of national census data and the computational ability to look for patterns in such data. | |

The book has seven chapters. The first is introductory; it describes simple linear regression (in which there is only one independent variable), discusses the possibility of outliers that corrupt either the dependent or the independent variable, provides examples in which outliers produce misleading results, defines the breakdown point, and briefly introduces several methods for robust simple regression, including repeated median regression. The second and third chapters analyze in more detail the least median of squares method for regression (in which one seeks a fit that minimizes the median of the squared residuals) and the least trimmed squares method (in which one seeks to minimize the sum of the squared residuals that are below the median). These two methods both have breakdown point 50% and can be applied for both simple regression (chapter two) and multivariate regression (chapter three). | |

The more mathematical approach uses an index-free notation, emphasizing the geometric and algebraic structure of the gauge theory and its relationship to Lie algebras and Riemannian manifolds; for example, treating gauge covariance as equivariance on fibers of a fiber bundle. The index notation used in physics makes it far more convenient for practical calculations, although it makes the overall geometric structure of the theory more opaque. The physics approach also has a pedagogical advantage: the general structure of a gauge theory can be exposed after a minimal background in multivariate calculus, whereas the geometric approach requires a large investment of time in the general theory of differential geometry, Riemannian manifolds, Lie algebras, representations of Lie algebras and principle bundles before a general understanding can be developed. In more advanced discussions, both notations are commonly intermixed. | |

A variable rules analysis computes a multivariate statistical model, on the basis of observed token counts, such that each determining factor is assigned a numerical factor weight that describes how it influences the probabilities of choice of either form. This is done by means of stepwise logistic regression, using a maximum likelihood algorithm. Although the necessary computations required for a variable rules analysis can be carried out with the help of mainstream general-purpose statistics software packages such as SPSS, it is more often done by means of a specialised software dedicated to the needs of linguists, called Varbrul. It was originally written by David Sankoff and currently exists in freeware implementations for Mac OS and Microsoft Windows, under the title of Goldvarb X. There are also versions implemented in the statistical language R and therefore available on most platforms. | |

An overview of PETS in action He is a programmer in several computer languages and has developed software for music composition, text transformation and sonification.SoniPy website His PhD dissertation was on the development of a software framework for the sonification of information in large or high-frequency multivariate data sets, such as those from securities trading engines.Sonification and information: Concepts, instruments and techniques Worrall was appointed to the Faculty of Music at the University of Melbourne in 1979 to teach musical composition and undertake research in computer music. In addition to composition, orchestration, twentieth-century techniques and free improvisation, in 1981, before the advent of the personal computer, he designed and taught the first undergraduate course in computer music in Australia using a mainframe computer, and MusicC (a development of CMusic) and Gary Lee Nelson's Music Programming Library (written in APL). | |

As such permanent magnet technology offers the potential to extend the accessibility and availability of NMR to institutions that do not have access to super-conducting spectrometers (e.g., beginning undergraduates or small-businesses). Many automated applications utilizing multivariate statistical analyses (chemometrics) approaches to derive structure-property and chemical and physical property correlations between 60 MHz 1H NMR spectra and primary analysis data particularly for petroleum and petrochemical process control applications have been developed over the past decade.“Process NMR Spectroscopy: Technology and On-line Applications” John C. Edwards, and Paul J. Giammatteo, Chapter 10 in Process Analytical Technology: Spectroscopic Tools and Implementation Strategies for the Chemical and Pharmaceutical Industries, 2nd Ed., Editor Katherine Bakeev, Blackwell-Wiley, 2010“A Review of Applications of NMR Spectroscopy in Petroleum Chemistry” John C. Edwards, Chapter 16 in Monograph 9 on Spectroscopic Analysis of Petroleum Products and Lubricants, Editor: Kishore Nadkarni, ASTM Books, 2011. | |

Bernardo, J. M. & Smith, A.F.M. (2000) Bayesian Theory, Wiley For instance, if both the prior and the likelihood have Gaussian form, and the precision matrix of both of these exist (because their covariance matrix is full rank and thus invertible), then the precision matrix of the posterior will simply be the sum of the precision matrices of the prior and the likelihood. As the inverse of a Hermitian matrix, the precision matrix of real-valued random variables, if it exists, is positive definite and symmetrical. Another reason the precision matrix may be useful is that if two dimensions i and j of a multivariate normal are conditionally independent, then the ij and ji elements of the precision matrix are 0. This means that precision matrices tend to be sparse when many of the dimensions are conditionally independent, which can lead to computational efficiencies when working with them. | |

Douglas Paul Wiens is a Canadian statistician; he is a professor in the Department of Mathematical and Statistical Sciences at the University of Alberta. Wiens earned a B.Sc. in mathematics (1972), two master's degrees in mathematical logic (1974) and statistics (1979), and a Ph.D. in statistics (1982), all from the University of Calgary.Education and Professional Experience from Wiens' web site at Alberta, retrieved 2010-02-07. As part of his work on mathematical logic, in connection with Hilbert's tenth problem, Wiens helped find a diophantine formula for the primes: that is, multivariate polynomial with the property that the positive values of this polynomial, over integer arguments, are exactly the prime numbers.. Wiens and his co-authors won the Lester R. Ford award of the Mathematical Association of America in 1977 for their paper describing this result.The Mathematical Association of America's The Lester R. Ford Award, retrieved 2010-02-07. | |

In the late 1980s and 1990s, the company introduced software-based audience analysis techniques for radio broadcasters, by seeking to commercialize techniques derived from the field of Multivariate statistics; procedures like Cluster Analysis and Factor Analysis, which were drawn from academic experimentation and market research being performed for Fortune 500 firms, such as Procter & Gamble. During the 1980s, the Arbitron Company was developing the Portable People Meter, or PPM, technology to replace its self-administered, seven-day radio diary method to collect radio listening data from Arbitron survey participants. The radio diary had been the most generally accepted method of measuring radio listening since 1964. Rantel became an early evangelist for the new PPM method, because Rantel researchers had performed many audits of Arbitron radio diaries during its early years and were keenly aware of the weaknesses of the seven-day radio diary method. | |

Brinkley attended Texas A&M; University, graduating with Bachelor's and Master's degrees in industrial engineering. He later completed part-time doctoral work in operations research at North Carolina State University, before choosing to focus on his professional career. Brinkley has published research on mathematical modeling,"Multivariate Zero-Inflated Poisson Models and Their Applications.", Technometrics, February 1, 1999, Li, Chin-Shang; Lu, Jye-Chyi; Park, Jinho; Kim, Kyungmoo; Brinkley, Paul A.; Peterson, John PBrinkley, P.A., Meyer, K.P. and Lu, J.C. "Combined generalized linear modeling-nonlinear programming approach to robust process design—a case-study in circuit board quality improvement". Applied Statistics, Journal of the Royal Statistical Society (Series C), 45(1), 99-110 industrial statistics,"Nortel Redefines Factory Information Technology: An OR-Driven Approach" Paul A. Brinkley, David Stepto, Kristopher R. Haag, John Folger, Kui Wang, Kuanlian Liou and W. David Carr, Interfaces, January/February 1998 vol. | |

In computer algebra, the Faugère F4 algorithm, by Jean-Charles Faugère, computes the Gröbner basis of an ideal of a multivariate polynomial ring. The algorithm uses the same mathematical principles as the Buchberger algorithm, but computes many normal forms in one go by forming a generally sparse matrix and using fast linear algebra to do the reductions in parallel. The Faugère F5 algorithm first calculates the Gröbner basis of a pair of generator polynomials of the ideal. Then it uses this basis to reduce the size of the initial matrices of generators for the next larger basis: > If Gprev is an already computed Gröbner basis (f2, …, fm) and we want to > compute a Gröbner basis of (f1) + Gprev then we will construct matrices > whose rows are m f1 such that m is a monomial not divisible by the leading > term of an element of Gprev. | |

In Tanzania by 2010, breastfeeding was initiated within the first hour of birth in 46.1% of mothers. Over 97 percent of mothers in Tanzania do breastfeed, however, the prevalence of exclusive breastfeeding in infants aged 0–6 months is 50 percent. Although the national average reported to be 50% prevalence, one regional study focusing on Kilimanjaro region only revealed the general prevalence of 88.1% at one month, 65.5% at three months and 20.7% for an infant of six months of age, which is very low and did not vary between rural and urban. A multivariate analysis using 2010 TDHS data revealed that the risk of delayed initiation of breastfeeding within 1 hour after birth was significantly higher among young mothers aged <24 years, uneducated and employed mothers from rural areas who delivered by caesarean section and those who delivered at home and were assisted by traditional birth attendants or relatives. | |

LSI is also an application of correspondence analysis, a multivariate statistical technique developed by Jean-Paul Benzécri in the early 1970s, to a contingency table built from word counts in documents. Called " indexing" because of its ability to correlate related terms that are in a collection of text, it was first applied to text at Bellcore in the late 1980s. The method, also called latent semantic analysis (LSA), uncovers the underlying latent semantic structure in the usage of words in a body of text and how it can be used to extract the meaning of the text in response to user queries, commonly referred to as concept searches. Queries, or concept searches, against a set of documents that have undergone LSI will return results that are conceptually similar in meaning to the search criteria even if the results don’t share a specific word or words with the search criteria. | |

A crude version of this algorithm to find a basis for an ideal I of a polynomial ring R proceeds as follows: :Input A set of polynomials F that generates I :Output A Gröbner basis G for I :# G := F :# For every fi, fj in G, denote by gi the leading term of fi with respect to the given ordering, and by aij the least common multiple of gi and gj. :# Choose two polynomials in G and let Sij = (aij / gi) fi − (aij / gj) fj (Note that the leading terms here will cancel by construction). :# Reduce Sij, with the multivariate division algorithm relative to the set G until the result is not further reducible. If the result is non-zero, add it to G. :# Repeat steps 1-4 until all possible pairs are considered, including those involving the new polynomials added in step 4. | |

Considering the extensive application of Statistics in solving various problems in real life such as analyzing multivariate anthropometric data, applying sample surveys as a method of data collection, analyzing meteorological data, estimating crop yield etc., this group, particularly, Mahalanobis and his younger colleagues S. S. Bose and H. C. Sinha felt the necessity of forming a specialized institute to facilitate research and learning of Statistics. On 17 December 1931, Mahalonobis held a meeting with Pramatha Nath Banerji (Minto Professor of Economics), Nikhil Ranjan Sen (Khaira Professor of Applied Mathematics) and Sir R. N. Mukherjee. This meeting led to the establishment of the Indian Statistical Institute (ISI), which was formally registered on 28 April 1932, as a non-profit distributing learned society under the Societies Registration Act XXI of 1860. Later, the institute was registered under the West Bengal Societies Registration Act XXVI of 1961 amended in 1964. | |

The main part of the book is organized into three parts. The first part, covering three chapters and roughly the first quarter of the book, concerns the symbolic method in combinatorics, in which classes of combinatorial objects are associated with formulas that describe their structures, and then those formulas are reinterpreted to produce the generating functions or exponential generating functions of the classes, in some cases using tools such as the Lagrange inversion theorem as part of the reinterpretation process. The chapters in this part divide the material into the enumeration of unlabeled objects, the enumeration of labeled objects, and multivariate generating functions. The five chapters of the second part of the book, roughly half of the text and "the heart of the book", concern the application of tools from complex analysis to the generating function, in order to understand the asymptotics of the numbers of objects in a combinatorial class. | |

The GRE subject test in mathematics is a standardized test in the United States created by the Educational Testing Service (ETS), and is designed to assess a candidate's potential for graduate or post-graduate study in the field of mathematics. It contains questions from many fields of mathematics; about 50% of the questions come from calculus (including pre-calculus topics, multivariate calculus, and differential equations), 25% come from algebra (including linear algebra, abstract algebra, and number theory), and 25% come from a broad variety of other topics typically encountered in undergraduate mathematics courses, such as point-set topology, probability and statistics, geometry, and real analysis. Similar to all the GRE subject tests, the GRE Mathematics test is paper-based, as opposed to the GRE general test which is usually computer-based. It contains approximately 66 multiple-choice questions, which are to be answered within 2 hours and 50 minutes. | |

The faculty of the Department are holders of advanced academic degrees and are noted in the academe, such as Dr. Tereso Tullao, Jr., current executive director of the Center for Business and Economics Research and Development, past dean of the College of Business and Economics, and who was cited as one of the most outstanding teachers in the Philippines in 1993 by the Metrobank Foundation, Dr. Angelo Unite, whose paper, The Effect of Capital Market Liberalization Measures on the Integration of the Philippine Stock Market with International Markets: Evidence from Johansen’s Multivariate Cointegration Procedure, won an Outstanding Scientific Paper award in the 2005 National Academy of Science and Technology Awards, Dr. Ponciano Intal, Jr., who was a member of the editorial board of the Philippine Journal of Development and former executive director of the DLSU-Angelo King Institute for Economic and Business Studies, and Dr. Lawrence Dacuycuy, who is the current dean of the College and Mr. Marvin Raymond Castell as its new vice dean. | |

In The Democratic Horizon. Hyperpluralism and the Renewal of Political Liberalism, Ferrara argues that Rawls's “political liberalism” – which due to its anti-perfectionist thrust, its embedded sense of the contingency of justice, its openness to plurality still constitutes the best available paradigm for understanding what a complex democratic society free of oppression could look like – needs to be updated in order to improve it traction in a historical context rapidly become different from the original one. Four adjustments – conjectural arguments, an enriched notion of the democratic ethos, a decentering of it in several local varieties, as well as the remedial model of a multivariate democratic polity – are suggested in order to enable political liberalism to meet the challenge of hyperpluralism. The aesthetic sources of normativity that have formed the object of Ferrara's earlier work—exemplarity, judgment, the normativity of identity, and the imagination—are called on to supplement the conceptual resources of a revisited political liberalism. | |

This conclusion was reached following craniometric multivariate analyses, including measuring Mahalanobis distance, which suggested a strong affinity with two reference populations of African-American females from the 19th and 20th Centuries. Isotope analysis of oxygen and strontium isotopes suggest that she spent her childhood in the west of Britain or in coastal areas of Western Europe and the Mediterranean. However, a 2009 study found that FORDISC 3.0 "is only likely to be useful when an unidentified specimen is more or less complete and belongs to one of the populations represented in its reference samples", and even in such "favorable circumstances it can be expected to classify no more than 1 per cent of specimens with confidence." Immediately after the publication of this research and its discussion in the press the Ivory Bangle Lady became a focal point of a debate about immigration in the past, with public discussions focusing on her racial identity. | |

On July 17, 2014, a few years after his retirement, when brushing his teeth, Royen had a flash of insight: how to use the Laplace transform of the multivariate gamma distribution to achieve a relatively simple proof for the Gaussian correlation inequality, a conjecture on the intersection of geometry, probability theory and statistics, formulated after work by Dunnett and Sobel (1955) and the American statistician Olive Jean Dunn (1958), that had remained unsolved since then. He sent a copy of his proof to Donald Richards, an acquainted American mathematician, who worked on a proof of the GCI for 30 years. Richards immediately saw the validity of Royen's proof and subsequently helped him to transform the mathematical formulas into LaTeX. When Royen contacted other reputed mathematicians, though, they didn't bother to investigate his proof, because Royen was relatively unknown, and these mathematicians therefore estimated the chance that Royen's proof would be false as very high. | |

A widely used method for drawing (sampling) a random vector x from the N-dimensional multivariate normal distribution with mean vector μ and covariance matrix Σ works as follows: # Find any real matrix A such that . When Σ is positive-definite, the Cholesky decomposition is typically used, and the extended form of this decomposition can always be used (as the covariance matrix may be only positive semi-definite) in both cases a suitable matrix A is obtained. An alternative is to use the matrix A = UΛ½ obtained from a spectral decomposition Σ = UΛU−1 of Σ. The former approach is more computationally straightforward but the matrices A change for different orderings of the elements of the random vector, while the latter approach gives matrices that are related by simple re-orderings. In theory both approaches give equally good ways of determining a suitable matrix A, but there are differences in computation time. | |

In general such statistics arrive in the presence of heavy-tailed distributions, and the presence of dragon kings will augment the already oversized impact of extreme events. Despite the importance of extreme events, due to ignorance, misaligned incentives, and cognitive biases, there is often a failure to adequately anticipate them. Technically speaking, this leads to poorly specified models where distributions that are not heavy-tailed enough, and under-appreciate both serial and multivariate dependence of extreme events. Some examples of such failures in risk assessment include the use of Gaussian models in finance (Black–Scholes, the Gaussian copula, LTCM), the use of Gaussian processes and linear wave theory failing to predict the occurrence of rogue waves, the failure of economic models in general to predict the financial crisis of 2007–2008, and the under-appreciation of external events, cascades, and nonlinear effects in probabilistic risk assessment, leading to not anticipating the Fukushima Daiichi nuclear disaster in 2011. | |

Mann continued his interest in improving methodology to find patterns in high-resolution paleoclimate reconstructions: he was lead author with Bradley and Hughes on a study of long term variability in the El Niño southern oscillations and related teleconnections, published in 2000. His areas of research have included climate signal detection, attribution of climate change and coupled ocean-atmosphere modeling, developing and assessing methods of statistical and time series analysis and comparing the results of modelling against data. The original MBH98 and MBH99 papers avoided undue representation of large numbers of tree ring proxies by using a principal component analysis step to summarise these proxy networks, but from 2001 Mann stopped using this method and introduced a multivariate Climate Field Reconstruction (CFR) technique using a regularized expectation–maximization (RegEM) method which did not require this PCA step. In May 2002 Mann and Scott Rutherford published a paper on testing methods of climate reconstruction which discussed this technique. | |

Brown later focused his statistics research on developing signal processing algorithms and statistical methods for neuronal data analysis. He developed a state-space point process (SSPP) paradigm to study how neural systems maintain dynamic representations of information. For the analysis of neural spiking activity and binary behavioral tasks represented as multivariate or univariate point processes (0-1 events that occur in continuous time), his research produced analogs of the Kalman filter, Kalman smoothing, sequential Monte Carlo algorithms, and combined state and parameter estimation algorithms commonly applied to continuous-valued time series observations. Brown used the methods to: show that ensembles of neurons in the rodent hippocampus maintained a highly accurate representation of the animal’s spatial location; track the formation of neural receptive fields on a millisecond time scale; track concurrent changes in neural activity and behavior during learning experiments; decode how groups of motor neurons represent movement information; and track burst suppression in patients under general anesthesia. | |

Around 1890, David Hilbert introduced non-effective methods, and this was seen as a revolution, which led most algebraic-geometers of the first half of 20th century to try to "eliminate elimination". Nevertheless Hilbert's Nullstellensatz, may be considered to belong to elimination theory, as it asserts that a system of polynomial equations does not have any solution if and only one may eliminate all unknowns for getting 1. Elimination theory culminated with the work of Kronecker, and, finally, F.S. Macaulay, who introduced multivariate resultants and U-resultants, providing complete elimination methods for systems of polynomial equations, which have been described in chapter Elimination theory of the first editions (1930) of van der Waerden's Moderne Algebra. After that, elimination theory has been considered as old fashioned, removed from next editions of Moderne Algebra, and generally ignored, until the introduction of computers, and more specifically of computer algebra, which set the problem of designing elimination algorithms that are sufficiently efficient for being implemented. | |

In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.... For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the integers and the reals, it is reducible over the complex numbers as x^2+y^2 = (x+iy)(x-iy), and thus not absolutely irreducible. More generally, a polynomial defined over a field K is absolutely irreducible if it is irreducible over every algebraic extension of K,. and an affine algebraic set defined by equations with coefficients in a field K is absolutely irreducible if it is not the union of two algebraic sets defined by equations in an algebraically closed extension of K. In other words, an absolutely irreducible algebraic set is a synonym of an algebraic variety,. which emphasizes that the coefficients of the defining equations may not belong to an algebraically closed field. Absolutely irreducible is also applied, with the same meaning to linear representations of algebraic groups. | |

He also recruited J. A. Todd, Patrick du Val, Harold Davenport, L. C. Young, and invited distinguished visitors. Although Manchester was later to be known as the birthplace of the electronic computer, Douglas Hartree made an earlier contribution building a differential analyser in 1933. The machine was used for ballistics calculations as well calculating railway timetables. Mordell was succeeded by the famous topologist and cryptanalyst Max Newman in 1945 who, as head of department, transformed it into a centre of international renown.Walter Ledermann, Encounters of a Mathematician, 2009 Undergraduate numbers increased from eight per year to 40 and then 60. In 1948 Newman recruited Alan Turing as Reader in the department, and he worked there until his death in 1954, completing some of his profound work on the foundations of computer science including Computing Machinery and Intelligence. Newman retired in 1964. From 1949 to 1960 M. S. Bartlett held the first chair in mathematical statistics at VUM, he is known for his contribution to the analysis of data with spatial and temporal patterns, the theory of statistical inference and in multivariate analysis. | |

This has been interpreted as evidence that a strewn field from the Younger Dryas impact event may have affected at least 30% of Earth's radius. Also in 2019, analysis of age-dated sediments from a long-lived pond in South Carolina showed not just an overabundance of platinum but a platinum/palladium ratio inconsistent with a terrestrial origin, as well as an overabundance of soot and a decrease in fungal spores associated with the dung of large herbivores, suggesting large-scale regional wildfires and at least a local decrease in ice age megafauna. In 2019, a South African team consisting of Francis Thackeray, Louis Scott and Philip Pieterse announced the discovery of a platinum (Pt) spike in peat deposits at Wonderkrater, an artesian spring site in South Africa in the Limpopo Province, near the town of Mookgophong (formerly Naboomspruit) situated between Pretoria and Polokwane. The spike in platinum was documented in a sample dated at 12,744 years BP (calibrated) preceding a decline in a paleo-temperature index based on multivariate analysis of pollen spectra. | |

Given random variables X,Y,\ldots, that are defined on a probability space, the joint probability distribution for X,Y,\ldots is a probability distribution that gives the probability that each of X,Y,\ldots falls in any particular range or discrete set of values specified for that variable. In the case of only two random variables, this is called a bivariate distribution, but the concept generalizes to any number of random variables, giving a multivariate distribution. The joint probability distribution can be expressed either in terms of a joint cumulative distribution function or in terms of a joint probability density function (in the case of continuous variables) or joint probability mass function (in the case of discrete variables). These in turn can be used to find two other types of distributions: the marginal distribution giving the probabilities for any one of the variables with no reference to any specific ranges of values for the other variables, and the conditional probability distribution giving the probabilities for any subset of the variables conditional on particular values of the remaining variables. | |

Various studies have compared the predictive performance of PI-RADS v1 for detecting significant prostate cancer against either image-guided biopsy results (definitive pathology) and/or prostatectomy specimens (histopathology). In a 2015 articles in the Journal of Urology, Thompson reported multi-parametric MRI detection of significant prostate cancer had sensitivity of 96%, specificity of 36%, negative predictive value and positive predictive values of 92% and 52%; when PI-RADS was incorporated into a multivariate analysis (PSA, digital rectal exam, prostate volume, patient age) the area under the curve (AUC) improved from 0.776 to 0.879, p<0.001. A similar paper in European Radiology found that when correlated with histopathology, PI-RADS v2 correctly identified 94-95% of prostate cancer foci ≥0.5 mL, but was limited for the assessment of GS ≥4+3 (significant) tumors ≤0.5 mL; in their series, DCE-MRI offered limited added value to T2WI+DW-MRI. Other applications for which PI-RADS may be useful include prediction of termination of Active Surveillance due to tumor progression/aggressiveness, detection of extraprostatic extension of prostate cancer, and supplemental information when considering whether to re-biopsy patients with a history of previous negative biopsy. | |

Third, the principle that effects cannot precede causes can be invoked, by including on the right side of the regression only variables that precede in time the dependent variable; this principle is invoked, for example, in testing for Granger causality and in its multivariate analog, vector autoregression, both of which control for lagged values of the dependent variable while testing for causal effects of lagged independent variables. Regression analysis controls for other relevant variables by including them as regressors (explanatory variables). This helps to avoid false inferences of causality due to the presence of a third, underlying, variable that influences both the potentially causative variable and the potentially caused variable: its effect on the potentially caused variable is captured by directly including it in the regression, so that effect will not be picked up as an indirect effect through the potentially causative variable of interest. Given the above procedures, coincidental (as opposed to causal) correlation can be probabilistically rejected if data samples are large and if regression results pass cross-validation tests showing that the correlations hold even for data that were not used in the regression. | |

In 1983, again anticipating the trends in consumption of natural herbal infusions launched, starting with Chamomile (one of the most popular herbs originating in the flora of Europe and West Asia), Lemon Grass, Grass Candy (produced by the selection of the fruits of the plant, guaranteeing the aniseed-flavour and taste particularly sweet infusion), Boldo and Mint flavours. But in the summer of 1987, the launch of ready-to-drink infusion in cups of 300ml initially focussing its sales targeting the beaches of Rio saw its soar. The production of multivariate dimensional packages for the iced infusion namely 330ml, 500ml, 1.5 litres and the 300ml cups, took place in 2002 with a new factory opening in Rio de Janeiro. In 2004, Leão invested in functional infusions introducing a new line of infusions with mixed combinations of flavours and properties matching with various tailor-made herbs which had specific customer needs like "Boa Noite" (Good Night), with aroma and mild flavour being ideal for relaxation; "Silvestre", is the combination of aromas of strawberry, raspberry and black currant; "Orchard", a mixture of soft aromas of peach, apple, cherry and orange; and "Tropical", a delicate combination of flavours containing fruits like mango, pineapple, apple and banana. |

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