Bhattacharyya distance

In statistics, the Bhattacharyya distance measures the similarity of two discrete or continuous probability distributions. It is closely related to the Bhattacharyya coefficient which is a measure of the amount of overlap between two statistical samples or populations. Both measures are named after Anil Kumar Bhattacharya, a statistician who worked in the 1930s at the Indian Statistical Institute.

The coefficient can be used to determine the relative closeness of the two samples being considered. It is used to measure the separability of classes in classification and it is considered to be more reliable than the Mahalanobis distance, as the Mahalanobis distance is a particular case of the Bhattacharyya distance when the standard deviations of the two classes are the same. Therefore, when two classes have similar means but different standard deviations, the Mahalanobis distance would tend to zero, however, the Bhattacharyya distance would grow depending on the difference between the standard deviations.

For probability distributions p and q over the same domain X, the Bhattacharyya distance is defined as:

where:

is the Bhattacharyya coefficient for discrete probability distributions.

For continuous probability distributions, the Bhattacharyya coefficient is defined as:

In either case, ${\displaystyle 0\leq BC\leq 1}$ and ${\displaystyle 0\leq D_{B}\leq \infty }$. ${\displaystyle D_{B}}$ does not obey the triangle inequality, but the Hellinger distance ${\displaystyle {\sqrt {1-BC}}}$ does obey the triangle inequality.

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