MATLAB Examples

# 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 ```