Re: Multivariate pdf of a normal distribution

[Paul Kienzle]

On Nov 7, 2005, at 8:51 AM, Mike Miller wrote:

> On Mon, 7 Nov 2005, Michael Creel wrote:
>
> > Paul Kienzle wrote:
> >
> > (snip)
> >
> > > Note: Google found a cdf here:
> > >     http://alex.strashny.org/a/Multivariate-normal-cumulative-  
> > > distribution-function-(cdf)-in-MATLAB.html
> >
> > On Monday morning, this was a broken link. In general, evaluating the  
> > mvn cdf for a p-vector will require monte carlo integration or  
> > similar techniques when p is at all large.
>
> I think the problem might be that there is a space in that URL when  
> there shouldn't be a space.  Try this:
>
> http://alex.strashny.org/a/Multivariate-normal-cumulative- 
> distribution-function-(cdf)-in-MATLAB.html
>
> That page says the following:
>
>   In statistics, the multivariate normal (mvn) is a popular  
> distribution.
>   Unfortunately, its cumulative distribution function (cdf) does not  
> have
>   a closed form. There are, however, a number of algorithms that
>   numerically estimate the value of the cdf. Here is an implementation  
> of
>   one such algorithm in MATLAB [URL below]. For the algorithm itself,  
> see
>
>      Genz, A. (1992) Numerical Computation of Multivariate Normal
>      Probabilities, J. Comp. Graph. Stat.
>
> The linked URL is this one:
>
> http://alex.strashny.org/b/mvncdf.m
>
> It is under the GPL.  There is also a link to the Genz paper in  
> postscript format:
>
> http://alex.strashny.org/local/Genz-1992-multivariate-normal- 
> probabilities.ps

The manual for NIST dataplot has some examples:

http://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/ 
mvncdf.htm

and some references:

"Comparison of Methods for the Computation ofMultivariate Normal  
Probabilities", Alan Genz, Computing Science and Statistics, 25, 1993,  
pp. 400-405.

"Numerical Computation of Multivariate Normal Probabilities", Alan  
Genz, Journal of Computational and Graphical Statistics, 1, 1992, pp.  
141-149.


Plugging the parameters and limits into the mvncdf code above I did not  
get the same values.  I didn't have a chance to figure out where I went  
wrong.

Here's the test I used if anybody wants to correct it:

%!test
%! sigma = [1.0        -0.707107  0.0  0.0 0.0;
%!      -0.707107    1.0       0.5  0.5 0.5;
%!       0.0         0.5       1.0  0.5 0.5;
%!       0.0         0.5       0.5  1.0 0.5;
%!       0.0         0.5       0.5  0.5 1.0 ];
%! lo = [0.0      0.0   1.7817   1.4755   -inf];
%! hi = [inf 1.5198 inf inf 1.5949];
%! accuracy = 0.00005;
%! Alo = mvncdf(lo,0,sigma,accuracy);
%! Ahi = mvncdf(hi,0,sigma,accuracy);
%! assert(Ahi-Alo, 0.2865, 0.0005);





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