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1 % CONF2MAHAL - Translates a confidence interval to a Mahalanobis
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2 % distance. Consider a multivariate Gaussian
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3 % distribution of the form
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4 %
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5 % p(x) = 1/sqrt((2 * pi)^d * det(C)) * exp((-1/2) * MD(x, m, inv(C)))
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6 %
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7 % where MD(x, m, P) is the Mahalanobis distance from x
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8 % to m under P:
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9 %
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10 % MD(x, m, P) = (x - m) * P * (x - m)'
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11 %
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12 % A particular Mahalanobis distance k identifies an
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13 % ellipsoid centered at the mean of the distribution.
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14 % The confidence interval associated with this ellipsoid
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15 % is the probability mass enclosed by it. Similarly,
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16 % a particular confidence interval uniquely determines
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17 % an ellipsoid with a fixed Mahalanobis distance.
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18 %
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19 % If X is an d dimensional Gaussian-distributed vector,
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20 % then the Mahalanobis distance of X is distributed
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21 % according to the Chi-squared distribution with d
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22 % degrees of freedom. Thus, the Mahalanobis distance is
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23 % determined by evaluating the inverse cumulative
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24 % distribution function of the chi squared distribution
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25 % up to the confidence value.
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26 %
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27 % Usage:
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28 %
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29 % m = conf2mahal(c, d);
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30 %
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31 % Inputs:
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32 %
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33 % c - the confidence interval
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34 % d - the number of dimensions of the Gaussian distribution
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35 %
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36 % Outputs:
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37 %
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38 % m - the Mahalanobis radius of the ellipsoid enclosing the
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39 % fraction c of the distribution's probability mass
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40 %
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41 % See also: MAHAL2CONF
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42
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43 % Copyright (C) 2002 Mark A. Paskin
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44 %
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45 % This program is free software; you can redistribute it and/or modify
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46 % it under the terms of the GNU General Public License as published by
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47 % the Free Software Foundation; either version 2 of the License, or
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48 % (at your option) any later version.
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49 %
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50 % This program is distributed in the hope that it will be useful, but
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51 % WITHOUT ANY WARRANTY; without even the implied warranty of
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52 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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53 % General Public License for more details.
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54 %
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55 % You should have received a copy of the GNU General Public License
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56 % along with this program; if not, write to the Free Software
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57 % Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
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58 % USA.
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59 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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60 function m = conf2mahal(c, d)
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61
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62 m = chi2inv(c, d); |
