Daniel@0: % CONF2MAHAL - Translates a confidence interval to a Mahalanobis
Daniel@0: %              distance.  Consider a multivariate Gaussian
Daniel@0: %              distribution of the form
Daniel@0: %
Daniel@0: %   p(x) = 1/sqrt((2 * pi)^d * det(C)) * exp((-1/2) * MD(x, m, inv(C)))
Daniel@0: %
Daniel@0: %              where MD(x, m, P) is the Mahalanobis distance from x
Daniel@0: %              to m under P:
Daniel@0: %
Daniel@0: %                 MD(x, m, P) = (x - m) * P * (x - m)'
Daniel@0: %
Daniel@0: %              A particular Mahalanobis distance k identifies an
Daniel@0: %              ellipsoid centered at the mean of the distribution.
Daniel@0: %              The confidence interval associated with this ellipsoid
Daniel@0: %              is the probability mass enclosed by it.  Similarly,
Daniel@0: %              a particular confidence interval uniquely determines
Daniel@0: %              an ellipsoid with a fixed Mahalanobis distance.
Daniel@0: %
Daniel@0: %              If X is an d dimensional Gaussian-distributed vector,
Daniel@0: %              then the Mahalanobis distance of X is distributed
Daniel@0: %              according to the Chi-squared distribution with d
Daniel@0: %              degrees of freedom.  Thus, the Mahalanobis distance is
Daniel@0: %              determined by evaluating the inverse cumulative
Daniel@0: %              distribution function of the chi squared distribution
Daniel@0: %              up to the confidence value.
Daniel@0: %
Daniel@0: % Usage:
Daniel@0: % 
Daniel@0: %   m = conf2mahal(c, d);
Daniel@0: %
Daniel@0: % Inputs:
Daniel@0: %
Daniel@0: %   c    - the confidence interval
Daniel@0: %   d    - the number of dimensions of the Gaussian distribution
Daniel@0: %
Daniel@0: % Outputs:
Daniel@0: %
Daniel@0: %   m    - the Mahalanobis radius of the ellipsoid enclosing the
Daniel@0: %          fraction c of the distribution's probability mass
Daniel@0: %
Daniel@0: % See also: MAHAL2CONF
Daniel@0: 
Daniel@0: % Copyright (C) 2002 Mark A. Paskin
Daniel@0: %
Daniel@0: % This program is free software; you can redistribute it and/or modify
Daniel@0: % it under the terms of the GNU General Public License as published by
Daniel@0: % the Free Software Foundation; either version 2 of the License, or
Daniel@0: % (at your option) any later version.
Daniel@0: %
Daniel@0: % This program is distributed in the hope that it will be useful, but
Daniel@0: % WITHOUT ANY WARRANTY; without even the implied warranty of
Daniel@0: % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
Daniel@0: % General Public License for more details.
Daniel@0: %
Daniel@0: % You should have received a copy of the GNU General Public License
Daniel@0: % along with this program; if not, write to the Free Software
Daniel@0: % Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
Daniel@0: % USA.
Daniel@0: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Daniel@0: function m = conf2mahal(c, d)
Daniel@0: 
Daniel@0: m = chi2inv(c, d);