## Multivariate *t* Distribution

### Definition

The probability density function of the *d*-dimensional
multivariate Student's *t* distribution is given by

$$f(x,\Sigma ,\nu )=\frac{1}{{\left|\Sigma \right|}^{1/2}}\frac{1}{\sqrt{{(\nu \pi )}^{d}}}\frac{\Gamma ((\nu +d)/2)}{\Gamma (\nu /2)}{\left(1+\frac{{x}^{\prime}\text{}{\Sigma}^{-1}x}{\nu}\right)}^{-(\nu +d)/2}.$$

where *x* is a 1-by-*d*
vector, Σ is a *d*-by-*d* symmetric, positive
definite matrix, and *ν* is a positive scalar. While it is possible
to define the multivariate Student's *t* for singular Σ, the
density cannot be written as above. For the singular case, only random number
generation is supported. Note that while most textbooks define the multivariate
Student's *t* with *x* oriented as a column
vector, for the purposes of data analysis software, it is more convenient to orient
*x* as a row vector, and Statistics and Machine Learning Toolbox™ software uses that orientation.

### Background

The multivariate Student's *t* distribution is a generalization
of the univariate Student's *t* to two or more variables. It is a
distribution for random vectors of correlated variables, each element of which has a
univariate Student's *t* distribution. In the same way as the
univariate Student's *t* distribution can be constructed by
dividing a standard univariate normal random variable by the square root of a
univariate chi-square random variable, the multivariate Student's
*t* distribution can be constructed by dividing a
multivariate normal random vector having zero mean and unit variances by a
univariate chi-square random variable.

The multivariate Student's *t* distribution is parameterized
with a correlation matrix, Σ, and a positive scalar degrees of freedom parameter,
*ν*. *ν* is analogous to the degrees of
freedom parameter of a univariate Student's *t* distribution. The
off-diagonal elements of Σ contain the correlations between variables. Note that
when Σ is the identity matrix, variables are uncorrelated; however, they are not
independent.

The multivariate Student's *t* distribution is often used as a
substitute for the multivariate normal distribution in situations where it is known
that the marginal distributions of the individual variables have fatter tails than
the normal.

### Example

#### Plot PDF and CDF of Multivariate *t*-Distribution

Plot the pdf of a bivariate Student's *t* distribution. You can use this distribution for a higher number of dimensions as well, although visualization is not easy.

Rho = [1 .6; .6 1]; nu = 5; x1 = -3:.2:3; x2 = -3:.2:3; [X1,X2] = meshgrid(x1,x2); F = mvtpdf([X1(:) X2(:)],Rho,nu); F = reshape(F,length(x2),length(x1)); surf(x1,x2,F); caxis([min(F(:))-.5*range(F(:)),max(F(:))]); axis([-3 3 -3 3 0 .2]) xlabel('x1'); ylabel('x2'); zlabel('Probability Density');

Plot the cdf of a bivariate Student's *t* distribution.

F = mvtcdf([X1(:) X2(:)],Rho,nu); F = reshape(F,length(x2),length(x1)); surf(x1,x2,F); caxis([min(F(:))-.5*range(F(:)),max(F(:))]); axis([-3 3 -3 3 0 1]) xlabel('x1'); ylabel('x2'); zlabel('Cumulative Probability');

Since the bivariate Student's *t* distribution is defined on the plane, you can also compute cumulative probabilities over rectangular regions. For example, this contour plot illustrates the computation that follows, of the probability contained within the unit square shown in the figure.

contour(x1,x2,F,[.0001 .001 .01 .05:.1:.95 .99 .999 .9999]); xlabel('x'); ylabel('y'); line([0 0 1 1 0],[1 0 0 1 1],'linestyle','--','color','k');

Compute the value of the probability contained within the unit square.

F = mvtcdf([0 0],[1 1],Rho,nu)

F = 0.1401

Computing a multivariate cumulative probability requires significantly more work than computing a univariate probability. By default, the `mvtcdf`

function computes values to less than full machine precision and returns an estimate of the error, as an optional second output.

[F,err] = mvtcdf([0 0],[1 1],Rho,nu)

F = 0.1401

err = 1.0000e-08