Multivariate statistics

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Wishart distribution

If for , ..., has a multivariate normal distribution with mean vector and covariance matrix , and denotes the matrix composed of the row vectors , then the matrix has a Wishart distribution with scale matrix and degrees of freedom parameter . The Wishart distribution is most typically used when describing the covariance matrix of multinormal samples. The Wishart distribution is implemented as WishartDistribution[sigma, m] in the Wolfram Language package MultivariateStatistics` .

Trivariate normal distribution

A multivariate normal distribution in three variables. It has probability density function(1)where(2)The standardized trivariate normal distribution takes unit variances and . The quadrant probability in this special case is then given analytically by(3)(Rose and Smith 1996; Stuart and Ord 1998; Rose and Smith 2002, p. 231).

Multivariate normal distribution

A -variate multivariate normal distribution (also called a multinormal distribution) is a generalization of the bivariate normal distribution. The -multivariate distribution with mean vector and covariance matrix is denoted . The multivariate normal distribution is implemented as MultinormalDistribution[mu1, mu2, ..., sigma11, sigma12, ..., sigma12, sigma22, ..., ..., x1, x2, ...] in the Wolfram Language package MultivariateStatistics` (where the matrix must be symmetric since ).In the case of nonzero correlations, there is in general no closed-form solution for the distribution function of a multivariate normal distribution. As a result, such computations must be done numerically.

Cluster analysis

Cluster analysis is a technique used for classification of data in which data elements are partitioned into groups called clusters that represent collections of data elements that are proximate based on a distance or dissimilarity function.Cluster analysis is implemented as FindClusters[data] or FindClusters[data, n].The Season 1 pilot (2005) and Season 2 episode "Dark Matter" of the television crime drama NUMB3RS feature clusters and cluster analysis. In "Dark Matter," math genius Charlie Eppes runs a cluster analysis to find connections between the students that seemed to be systematically singled out by the anomalous third shooter. In Season 4 episode"Black Swan," characters Charles Eppes and Amita Ramanujan adjust cluster radii in their attempt to do a time series analysis of overlapping Voronoi regions to track the movements of a suspect. ..

Bivariate normal distribution

The bivariate normal distribution is the statistical distribution with probabilitydensity function(1)where(2)and(3)is the correlation of and (Kenney and Keeping 1951, pp. 92 and 202-205; Whittaker and Robinson 1967, p. 329) and is the covariance.The probability density function of the bivariate normal distribution is implemented as MultinormalDistribution[mu1, mu2, sigma11, sigma12, sigma12, sigma22] in the Wolfram Language package MultivariateStatistics` .The marginal probabilities are then(4)(5)and(6)(7)(Kenney and Keeping 1951, p. 202).Let and be two independent normal variates with means and for , 2. Then the variables and defined below are normal bivariates with unit variance and correlation coefficient :(8)(9)To derive the bivariate normal probability function, let and be normally and independently distributed variates with mean 0 and variance 1, then define(10)(11)(Kenney and Keeping..

Kendall operator

The operator that can be used to derive multivariate formulas for moments and cumulants from corresponding univariate formulas.For example, to derive the expression for the multivariate central moments in terms of multivariate cumulants, begin with(1)Now rewrite each variable as to obtain(2)Now differentiate each side with respect to , where(3)and wherever there is a term with a derivative , remove the derivative and replace the argument with times itself, so(4)Now set any s appearing as coefficients to 1, so(5)Dividing through by 4 gives(6)Finally, set any coefficients powers of appearing as term coefficients to 1 and interpret the resulting terms as , so that the above gives(7)This procedure can be repeated up to times, where is the subscript of the univariate case.Iterating the above procedure gives(8)(9)(10)(11)(12)giving the identities(13)(14)(15)(16)(17)..

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