mvnpdf.m

%% Author: Paul Kienzle <pkienzle@users.sf.net>
%% This program is granted to the public domain.

%% -*- texinfo -*-
%% @deftypefn {Function File} {@var{y} =} mvnpdf (@var{x})
%% @deftypefnx{Function File} {@var{y} =} mvnpdf (@var{x}, @var{mu})
%% @deftypefnx{Function File} {@var{y} =} mvnpdf (@var{x}, @var{mu}, @var{sigma})
%% Compute multivariate normal pdf for @var{x} given mean @var{mu} and covariance matrix 
%% @var{sigma}.  The dimension of @var{x} is @var{d} x @var{p}, @var{mu} is
%% @var{1} x @var{p} and @var{sigma} is @var{p} x @var{p}. The normal pdf is
%% defined as
%%
%% @example
%% @iftex
%% @tex
%% $$ 1/y^2 = (2 pi)^p |\Sigma| \exp \{ (x-\mu)^T \Sigma^{-1} (x-\mu) \} $$
%% @end tex
%% @end iftex
%% @ifnottex
%% 1/@var{y}^2 = (2 pi)^@var{p} |@var{Sigma}| exp @{ (@var{x}-@var{mu})' inv(@var{Sigma})@
%% (@var{x}-@var{mu}) @}
%% @end ifnottex
%% @end example
%%
%% @strong{References}
%% 
%% NIST Engineering Statistics Handbook 6.5.4.2
%% http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc542.htm
%%
%% @strong{Algorithm}
%%
%% Using Cholesky factorization on the positive definite covariance matrix:
%%
%% @example
%% @var{r} = chol (@var{sigma});
%% @end example
%%
%% where @var{r}'*@var{r} = @var{sigma}. Being upper triangular, the determinant
%% of @var{r} is  trivially the product of the diagonal, and the determinant of
%% @var{sigma} is the square of this:
%%
%% @example
%% @var{det} = prod (diag (@var{r}))^2;
%% @end example
%%
%% The formula asks for the square root of the determinant, so no need to 
%% square it.
%%
%% The exponential argument @var{A} = @var{x}' * inv (@var{sigma}) * @var{x}
%%
%% @example
%% @var{A} = @var{x}' * inv (@var{sigma}) * @var{x}
%%   = @var{x}' * inv (@var{r}' * @var{r}) * @var{x}
%%   = @var{x}' * inv (@var{r}) * inv(@var{r}') * @var{x}
%% @end example
%%
%% Given that inv (@var{r}') == inv(@var{r})', at least in theory if not numerically,
%%
%% @example
%% @var{A}  = (@var{x}' / @var{r}) * (@var{x}'/@var{r})' = sumsq (@var{x}'/@var{r})
%% @end example
%%
%% The interface takes the parameters to the multivariate normal in columns rather than 
%% rows, so we are actually dealing with the transpose:
%%
%% @example
%% @var{A} = sumsq (@var{x}/r)
%% @end example
%%
%% and the final result is:
%%
%% @example
%% @var{r} = chol (@var{sigma})
%% @var{y} = (2*pi)^(-@var{p}/2) * exp (-sumsq ((@var{x}-@var{mu})/@var{r}, 2)/2) / prod (diag (@var{r}))
%% @end example
%%
%% @seealso{mvncdf, mvnrnd}
%% @end deftypefn

function pdf = mvnpdf (x, mu, sigma)
  %% Check input
  %mu = 0;
  %sigma = 1; 
  if (~ismatrix (x))
    error ('mvnpdf: first input must be a matrix');
  end
  
  if (~isvector (mu) && ~isscalar (mu))
    error ('mvnpdf: second input must be a real scalar or vector');
  end
  
  if (~ismatrix (sigma) || ~issquare (sigma))
    error ('mvnpdf: third input must be a square matrix');
  end
  
  [ps, ps] = size (sigma);
  [d, p] = size (x);
  if (p ~= ps)
    error ('mvnpdf: dimensions of data and covariance matrix does not match');
  end
  
  if (numel (mu) ~= p && numel (mu) ~= 1)
    error ('mvnpdf: dimensions of data does not match dimensions of mean value');
  end

  mu = mu (:).';
  if (all (size (mu) == [1, p]))
    mu = repmat (mu, [d, 1]);
  end
  
  if (nargin < 3)
    pdf = (2*pi)^(-p/2) * exp (-sumsq (x-mu, 2)/2);
  else
    r = chol (sigma);
    pdf = (2*pi)^(-p/2) * exp (-sumsq ((x-mu)/r, 2)/2) / prod (diag (r));
  end
  end

%~demo
%~ mu = [0, 0];
%~ sigma = [1, 0.1; 0.1, 0.5];
%~ [X, Y] = meshgrid (linspace (-3, 3, 25));
%~ XY = [X(:), Y(:)];
%~ Z = mvnpdf (XY, mu, sigma);
%~ mesh (X, Y, reshape (Z, size (X)));
%~ colormap jet

%~test
%~ mu = [1,-1];
%~ sigma = [.9 .4; .4 .3];
%~ x = [ 0.5 -1.2; -0.5 -1.4; 0 -1.5];
%~ p = [   0.41680003660313; 0.10278162359708; 0.27187267524566 ];
%~ q = mvnpdf (x, mu, sigma);
%~ assert (p, q, 10*eps);