Re: Multivariate student t: normalization in mvtrnd

[Daniel J Sebald]

On 11/26/2012 04:06 PM, Iñigo Urteaga wrote:
> Dear all,
>
> I am using the mvtrnd function provided in the statistics package for
> octave and I have encountered some inconsistent results that I do not
> comprehend.
>
> > From theory, we know that if Z is a standard Student's t random vector
> and we define X as a linear combination (i.e. X=Sigma * Z), the
> resulting variable X is a multivariate Student's t distribution with
> covariance matrix Var(X)=nu/(nu-2)*V, where V=Sigma * Sigma'.
>
> In fact, when we define a V and sample from the following expression,
> where Sigma=chol(V), n is the dimensionality, nu the degrees of
> freedom and M the number of samples:
>
> samples=Sigma' * mvtrnd( eye(n), nu, M)'; (1)
>
> the covariance computed from those samples (i.e. cov(samples) )
> matches the expected variance nu/(nu-2)*V.
>
> However, if we directly apply:
>
> samples=mvtrnd(V, nu); (2)

Just a quick response on first thoughts without diving too deep...

http://octave.sourceforge.net/statistics/function/mvtrnd.html describes 
the function where it appears "sigma" is correlation coefficients. 
Sigma is typically a variable associated with standard deviation and 
sigma^2 variance.  You've defined V=Sigma * Sigma' which is more along 
the lines of variance, not standard deviation.  If that first input is 
to be scaling factor, perhaps what you want is

samples=mvtrnd(sqrt(V), nu);

See if that produces something more in line with nu/(nu-2) scaling 
factor for the variance.

Dan


> then, the covariance computed does not match  nu/(nu-2)*V, but the
> alternative expression using the correlation matrix R: nu/(nu-2)*R.
> Here, R is the correlation matrix as computed by
> R=V./(sqrt(diag(V)*diag(V))). The mvtrnd code indicates that the
> provided V is normalized if diagonal elements are non-unit, but I
> don't understand why this is required. In fact, sampling from the
> multivariate normal (mvnrnd called within the function) actually
> performs Cholesky decomposition of the provided V to accommodate non
> unitary covariance matrices.
>
> As I understand it, both approaches (1) and (2) should provide the
> same result (and that is the case if no normalization is performed).
> Since this is not the case, I was wondering what is the underlying
> reason for that normalization step. Am I missing something here or is
> this normalization step in mvtrnd unnecessary?
>
> I would truly appreciate some help/hints here.
>
> Thank you very much in advance,
>
> Iñigo Urteaga

--

Dan Sebald
email: daniel(DOT)sebald(AT)ieee(DOT)org
URL: http://www(DOT)dansebald(DOT)com