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multivariate student t distribution pdf in matlab?

per — Tue, 24 Feb 2009 14:38:43 -0500

hi all,

i am trying to evaluate the pdf for various points that are
distributed according to a multivariate Student t distribution. all
the stat texts tell me that the multivariate t distribution pdf takes
three parameters: a mean mu and a correlation matrix C, and a degrees-
of-freedom parameter (and the point to evaluate the pdf on obviously.)

the 'mvtpdf' function in matlab takes only two arguments: a degrees-of-
freedom parameter and a correlation matrix C. what happened to the mu
(mean) parameter? is there a way to evaluate this pdf in matlab
according to the parametrization i have above?

thank you.

Re: multivariate student t distribution pdf in matlab?

Peter Perkins — Tue, 24 Feb 2009 15:56:54 -0500

per wrote:

> i am trying to evaluate the pdf for various points that are
> distributed according to a multivariate Student t distribution. all
> the stat texts tell me that the multivariate t distribution pdf takes
> three parameters: a mean mu and a correlation matrix C, and a degrees-
> of-freedom parameter (and the point to evaluate the pdf on obviously.)
>
> the 'mvtpdf' function in matlab takes only two arguments: a degrees-of-
> freedom parameter and a correlation matrix C. what happened to the mu
> (mean) parameter? is there a way to evaluate this pdf in matlab
> according to the parametrization i have above?

Strictly speaking, the multivariate t has two parameters, but it's often the case that people add scale and location to that, and a simple transformation does the trick. It's easy to compute the PDF for such a thing by unscaling and unshifting your data, then using MVTPDF and the usual rules for the PDF of a transformed variable.

Hope this helps.

Re: multivariate student t distribution pdf in matlab?

per — Fri, 27 Feb 2009 13:49:43 -0500

On Feb 24, 10:56=A0am, Peter Perkins
<Peter.PerkinsRemoveT...@mathworks.com> wrote:
> per wrote:
> > i am trying to evaluate the pdf for various points that are
> > distributed according to a multivariate Student t distribution. all
> > the stat texts tell me that the multivariate t distribution pdf takes
> > three parameters: a mean mu and a correlation matrix C, and a degrees-
> > of-freedom parameter (and the point to evaluate the pdf on obviously.)
>
> > the 'mvtpdf' function in matlab takes only two arguments: a degrees-of-
> > freedom parameter and a correlation matrix C. what happened to the mu
> > (mean) parameter? is there a way to evaluate this pdf in matlab
> > according to the parametrization i have above?
>
> Strictly speaking, the multivariate t has two parameters, but it's often =
the case that people add scale and location to that, and a simple transform=
ation does the trick. =A0It's easy to compute the PDF for such a thing by u=
nscaling and unshifting your data, then using MVTPDF and the usual rules fo=
r the PDF of a transformed variable.
>
> Hope this helps.

hi Peter,
can you please say more about the transformation? i'm not sure i am
following. what kind of transformation of the data would make it so i
don't have to give the location (mu) parameter, but just the
covariance matrix/correlation matrix (parameter C)?

Re: multivariate student t distribution pdf in matlab?

Peter Perkins — Fri, 27 Feb 2009 14:38:24 -0500

per wrote:

> can you please say more about the transformation? i'm not sure i am
> following. what kind of transformation of the data would make it so i
> don't have to give the location (mu) parameter, but just the
> covariance matrix/correlation matrix (parameter C)?

No transformation will do that; you will always have to specify the degrees of freedom parameter as well.

All you need to do to use MVT as a location/scale family is, separately for each column of your data, subtract the corresponding mean and divide by the corresponding scale factor. It sounds like you have the Statistics Toolbox. Take a look in stats/private/addtls.m; that code the handles the univariate t-location-scale distribution for the Distribution Fitting Tool is in there. Look at tlspdf and tlscdf. You need to do the same thing coordinate-wise.

Hope this helps.

Re: multivariate student t distribution pdf in matlab?

Kevin Murphy — Wed, 04 Nov 2009 19:21:04 -0500

You can just write your own function to compute it - see below.
Note that the mvtpdf function in the stats toolbox
first converts Sigma to a correlation matrix,
which is nonstandard (as far as I know).
Thus these two methods only
give the same results if Sigma has 1 on all the diagonals.

HTH
Kevin

function logp = mvtLogpdf(X, mu, Sigma, nu)
% Multivariate student T distribution, log pdf
% X(i,:) is i'th case
[N d] = size(X);
M = repmat(mu(:)', N, 1); % replicate the mean across rows
X = X-M;
mahal = sum((X*inv(Sigma)).*X,2); %#ok
logc = gammaln(nu/2 + d/2) - gammaln(nu/2) - 0.5*logdet(Sigma) ...
   - (d/2)*log(nu) - (d/2)*log(pi);
logp = logc -(nu+d)/2*log1p(mahal/nu);

if 0
   % this check only works if Sigma is a correlation matrix
  logp2 = log(mvtpdf(X, Sigma, nu));
  assert(approxeq(logp, logp2))
end

Peter Perkins <Peter.PerkinsRemoveThis@mathworks.com> wrote in message <go8tt0$rma$1@fred.mathworks.com>...
> per wrote:
>
> > can you please say more about the transformation? i'm not sure i am
> > following. what kind of transformation of the data would make it so i
> > don't have to give the location (mu) parameter, but just the
> > covariance matrix/correlation matrix (parameter C)?
>
> No transformation will do that; you will always have to specify the degrees of freedom parameter as well.
>
> All you need to do to use MVT as a location/scale family is, separately for each column of your data, subtract the corresponding mean and divide by the corresponding scale factor. It sounds like you have the Statistics Toolbox. Take a look in stats/private/addtls.m; that code the handles the univariate t-location-scale distribution for the Distribution Fitting Tool is in there. Look at tlspdf and tlscdf. You need to do the same thing coordinate-wise.
>
> Hope this helps.