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mvnrnd.m

## Copyright (C) 2003 Iain Murray
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; If not, see <http://www.gnu.org/licenses/>.

function s = mvnrnd(mu,Sigma,K)
% s = mvnrnd (mu, Sigma)
%   Draw n random d-dimensional vectors from a multivariate Gaussian distribution
%   with mean mu(nxd) and covariance matrix Sigma(dxd).
%
% s = mvnrnd (mu, Sigma, n)
%   Draw n vectors from distributions with the same mean mu(1xd).
%

% Iain Murray 2003 -- I got sick of this simple thing not being in Octave and
%                     locking up a stats-toolbox license in Matlab for no good
%                     reason.
% May 2004 take a third arg, cases. Makes it more compatible with Matlab's.

% Paul Kienzle <pkienzle@users.sf.net>
% * Add GPL notice.
% * Add docs for argument K

% If mu is column vector and Sigma not a scalar then assume user didn't read
% help but let them off and flip mu. Don't be more liberal than this or it will
% encourage errors (eg what should you do if mu is square?).
if ((size(mu,2)==1)&(size(Sigma)~=[1,1]))
      mu=mu';
end

if nargin==3
      mu=repmat(mu,K,1);
end

[n,d]=size(mu);

if (size(Sigma)~=[d,d])
      error('Sigma must have dimensions dxd where mu is nxd.');
end

try
      U=chol(Sigma);
catch
      [E,Lambda]=eig(Sigma);
      if (min(diag(Lambda))<0),error('Sigma must be positive semi-definite.'),end
      U = sqrt(Lambda)*E';
end

s = randn(n,d)*U + mu;

% {{{ END OF CODE --- Guess I should provide an explanation:
% 
% We can draw from axis aligned unit Gaussians with randn(d)
%     x ~ A*exp(-0.5*x'*x)
% We can then rotate this distribution using
%     y = U'*x
% Note that
%     x = inv(U')*y
% Our new variable y is distributed according to:
%     y ~ B*exp(-0.5*y'*inv(U'*U)*y)
% or
%     y ~ N(0,Sigma)
% where
%     Sigma = U'*U
% For a given Sigma we can use the chol function to find the corresponding U,
% draw x and find y. We can adjust for a non-zero mean by just adding it on.
% 
% But the Cholsky decomposition function doesn't always work...
% Consider Sigma=. Now inv(Sigma) doesn't actually exist, but Matlab's
% mvnrnd provides samples with this covariance st x(1)~N(0,1) x(2)=x(1). The
% fast way to deal with this would do something similar to chol but be clever
% when the rows aren't linearly independent. However, I can't be bothered, so
% another way of doing the decomposition is by diagonalising Sigma (which is
% slower but works).
% if
%     [E,Lambda]=eig(Sigma)
% then
%     Sigma = E*Lambda*E'
% so
%     U = sqrt(Lambda)*E'
% If any Lambdas are negative then Sigma just isn't even positive semi-definite
% so we can give up.
%
% Paul Kienzle adds:
%   Where it exists, chol(Sigma) is numerically well behaved.  chol(hilb(12)) 
%   for doubles and for 100 digit floating point differ in the last digit.
%   Where chol(Sigma) doesn't exist, X*sqrt(Lambda)*E' will be somewhat
%   accurate.  For example, the elements of sqrt(Lambda)*E' for hilb(12),
%   hilb(55) and hilb(120) are accurate to around 1e-8 or better.  This was
%   tested using the TNT+JAMA for eig and chol templates, and qlib for
%   100 digit precision.
% }}}


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