[R] distribution overlap - how to quantify?

From: Doktor, Daniel <d.doktor03_at_imperial.ac.uk>
Date: Thu 25 Jan 2007 - 14:53:03 GMT


Dear R-Users,  

my objective is to measure the overlap/divergence of two probability density functions, p1(x) and p2(x). One could apply the chi-square test or determine the potential mixture components and then compare the respective means and sigmas. But I was rather looking for a simple measure of similarity.
Therefore, I used the concept of 'intrinsic discrepancy' which is defined as:  

    \delta{p_{1},p_{2}} = min
     \left\{ \int_{\chi}p_{1}(x)\log \frac{p_{1}(x)}{p_{2}(x)}dx, \int_{\chi}p_{2}(x)\log\frac{p_{2}(x)}{p_{1}(x)}dx \right\}

The smaller the delta the more similar are the distributions (0 when identical). I implemented this in 'R' using an adaptation of the Kullback-Leibler divergence. The function works, I get the expected results.  

The question is how to interpret the results. Obviously a delta of 0.5 reflects more similarity than a delta of 2.5. But how much more? Is there some kind of a statistical test for such an index (other than a simulation based evaluation)?  

Thanks in advance,  

Daniel      

Daniel Doktor
PhD Student
Imperial College
Royal School of Mines Building, DEST, RRAG Prince Consort Road
London SW7 2BP, UK
tel: 0044-(0)20-7589-5111-59276(ext)  

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