comparison core/tools/kldiv.m @ 0:e9a9cd732c1e tip

first hg version after svn

+function KL = kldiv(varValue,pVect1,pVect2,varargin)
+%KLDIV Kullback-Leibler or Jensen-Shannon divergence between two distributions.
+%   KLDIV(X,P1,P2) returns the Kullback-Leibler divergence between two
+%   distributions specified over the M variable values in vector X.  P1 is a
+%   length-M vector of probabilities representing distribution 1, and P2 is a
+%   length-M vector of probabilities representing distribution 2.  Thus, the
+%   probability of value X(i) is P1(i) for distribution 1 and P2(i) for
+%   distribution 2.  The Kullback-Leibler divergence is given by:
+%
+%       KL(P1(x),P2(x)) = sum[P1(x).log(P1(x)/P2(x))]
+%
+%   If X contains duplicate values, there will be an warning message, and these
+%   values will be treated as distinct values.  (I.e., the actual values do
+%   not enter into the computation, but the probabilities for the two
+%   duplicate values will be considered as probabilities corresponding to
+%   two unique values.)  The elements of probability vectors P1 and P2 must 
+%   each sum to 1 +/- .00001.
+%
+%   A "log of zero" warning will be thrown for zero-valued probabilities.
+%   Handle this however you wish.  Adding 'eps' or some other small value 
+%   to all probabilities seems reasonable.  (Renormalize if necessary.)
+%
+%   KLDIV(X,P1,P2,'sym') returns a symmetric variant of the Kullback-Leibler
+%   divergence, given by [KL(P1,P2)+KL(P2,P1)]/2.  See Johnson and Sinanovic
+%   (2001).
+%
+%   KLDIV(X,P1,P2,'js') returns the Jensen-Shannon divergence, given by
+%   [KL(P1,Q)+KL(P2,Q)]/2, where Q = (P1+P2)/2.  See the Wikipedia article
+%   for "Kullback–Leibler divergence".  This is equal to 1/2 the so-called
+%   "Jeffrey divergence."  See Rubner et al. (2000).
+%
+%   EXAMPLE:  Let the event set and probability sets be as follow:
+%                X = [1 2 3 3 4]';
+%                P1 = ones(5,1)/5;
+%                P2 = [0 0 .5 .2 .3]' + eps;
+%  
+%             Note that the event set here has duplicate values (two 3's). These 
+%             will be treated as DISTINCT events by KLDIV. If you want these to
+%             be treated as the SAME event, you will need to collapse their
+%             probabilities together before running KLDIV. One way to do this
+%             is to use UNIQUE to find the set of unique events, and then
+%             iterate over that set, summing probabilities for each instance of
+%             each unique event.  Here, we just leave the duplicate values to be
+%             treated independently (the default):
+%                 KL = kldiv(X,P1,P2); 
+%                 KL =  
+%                      19.4899
+%
+%             Note also that we avoided the log-of-zero warning by adding 'eps' 
+%             to all probability values in P2.  We didn't need to renormalize
+%             because we're still within the sum-to-one tolerance. 
+%  
+%   REFERENCES:
+%   1) Cover, T.M. and J.A. Thomas. "Elements of Information Theory," Wiley, 
+%      1991.
+%   2) Johnson, D.H. and S. Sinanovic. "Symmetrizing the Kullback-Leibler 
+%      distance." IEEE Transactions on Information Theory (Submitted).
+%   3) Rubner, Y., Tomasi, C., and Guibas, L. J., 2000. "The Earth Mover's 
+%      distance as a metric for image retrieval." International Journal of 
+%      Computer Vision, 40(2): 99-121.
+%   4) <a href="matlab:web('http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence','-browser')">Kullback–Leibler divergence</a>. Wikipedia, The Free Encyclopedia.
+%
+%   See also: MUTUALINFO, ENTROPY
+
+if ~isequal(unique(varValue),sort(varValue)),
+    warning('KLDIV:duplicates','X contains duplicate values. Treated as distinct values.')
+end
+if ~isequal(size(varValue),size(pVect1)) || ~isequal(size(varValue),size(pVect2)),
+    error('All inputs must have same dimension.')
+end
+% Check probabilities sum to 1:
+if (abs(sum(pVect1) - 1) > .00001) || (abs(sum(pVect2) - 1) > .00001),
+    error('Probablities don''t sum to 1.')
+end
+
+if ~isempty(varargin),
+    switch varargin{1},
+        case 'js',
+            logQvect = log2((pVect2+pVect1)/2);
+            KL = .5 * (sum(pVect1.*(log2(pVect1)-logQvect)) + ...
+                sum(pVect2.*(log2(pVect2)-logQvect)));
+
+        case 'sym',
+            KL1 = sum(pVect1 .* (log2(pVect1)-log2(pVect2)));
+            KL2 = sum(pVect2 .* (log2(pVect2)-log2(pVect1)));
+            KL = (KL1+KL2)/2;
+            
+        otherwise
+            error(['Last argument' ' "' varargin{1} '" ' 'not recognized.'])
+    end
+else
+    KL = sum(pVect1 .* (log2(pVect1)-log2(pVect2)));
+end
+
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