SEPARABILITY

Bhattacharya Distance

The Bhattacharya distance is a measure of the separation between two probability distributions.

b = ∫ sqrt( p1(x) p2(x) ) dx

b ranges between 0 and 1. If the two distributions coincide b has value 1. If they are completely separated b = 0.

If the distributions are gaussian the integration can be explicitly carried out and b=exp(-B) where

B = ¼ (m1 - m2)t ( S1 + S2 )-1 (m1 - m2) + ½ ln( |S1 + S2|/2 sqrt( |S1| |S2| )

For a Bayes classificator Prob(error)<=b/2.


Intraset and interset distances

These distances are used as meausre of the separability of a binary (two classes) classification. The interset distance, S, is the average of the distances between samples of class 1 and samples of class 2. This is the distance between the mean vectors of the two classes. The intraset distance is the average distance between the samples of one class, so there are two of them, R1 and R2.

The separability is

Q = ( R1 + R2 ) / S

The smaller Q is the better separated are the two classes.



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