Bhattacharyya distance

In statistics, the Bhattacharyya distance measures the similarity of two 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.^[1]

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. Consequently, when two classes have similar means but different standard deviations, the Mahalanobis distance would tend to zero, whereas the Bhattacharyya distance grows depending on the difference between the standard deviations.

Definition[]

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

${\displaystyle D_{B}(p,q)=-\ln \left(BC(p,q)\right)}$

where

${\displaystyle BC(p,q)=\sum _{x\in X}{\sqrt {p(x)q(x)}}}$

is the Bhattacharyya coefficient for discrete probability distributions.

For continuous probability distributions, the Bhattacharyya coefficient is defined as

${\displaystyle BC(p,q)=\int {\sqrt {p(x)q(x)}}\,dx}$

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, which is given by ${\displaystyle {\sqrt {1-BC(p,q)}}}$ does obey the triangle inequality.

In its simplest formulation, the Bhattacharyya distance between two classes under the normal distribution can be calculated^[2] by extracting the mean and variances of two separate distributions or classes:

${\displaystyle D_{B}(p,q)={\frac {1}{4}}\ln \left({\frac {1}{4}}\left({\frac {\sigma _{p}^{2}}{\sigma _{q}^{2}}}+{\frac {\sigma _{q}^{2}}{\sigma _{p}^{2}}}+2\right)\right)+{\frac {1}{4}}\left({\frac {(\mu _{p}-\mu _{q})^{2}}{\sigma _{p}^{2}+\sigma _{q}^{2}}}\right)}$

where:

${\displaystyle \sigma _{p}^{2}}$	is the variance of the p-th distribution,
${\displaystyle \mu _{p}}$	is the mean of the p-th distribution, and
${\displaystyle p,q}$	are two different distributions.

The Mahalanobis distance used in Fisher's linear discriminant analysis is a particular case of the Bhattacharyya Distance.

For multivariate normal distributions ${\displaystyle p_{i}={\mathcal {N}}({\boldsymbol {\mu }}_{i},\,{\boldsymbol {\Sigma }}_{i})}$,

${\displaystyle D_{B}={1 \over 8}({\boldsymbol {\mu }}_{1}-{\boldsymbol {\mu }}_{2})^{T}{\boldsymbol {\Sigma }}^{-1}({\boldsymbol {\mu }}_{1}-{\boldsymbol {\mu }}_{2})+{1 \over 2}\ln \,\left({\det {\boldsymbol {\Sigma }} \over {\sqrt {\det {\boldsymbol {\Sigma }}_{1}\,\det {\boldsymbol {\Sigma }}_{2}}}}\right),}$

where ${\displaystyle {\boldsymbol {\mu }}_{i}}$ and ${\displaystyle {\boldsymbol {\Sigma }}_{i}}$ are the means and covariances of the distributions, and

${\displaystyle {\boldsymbol {\Sigma }}={{\boldsymbol {\Sigma }}_{1}+{\boldsymbol {\Sigma }}_{2} \over 2}.}$

Note that, in this case, the first term in the Bhattacharyya distance is related to the Mahalanobis distance.

Bhattacharyya coefficient[]

The Bhattacharyya coefficient is an approximate measurement of the amount of overlap between two statistical samples. The coefficient can be used to determine the relative closeness of the two samples being considered.

Calculating the Bhattacharyya coefficient involves a rudimentary form of integration of the overlap of the two samples. The interval of the values of the two samples is split into a chosen number of partitions, and the number of members of each sample in each partition is used in the following formula,

${\displaystyle BC(\mathbf {p} ,\mathbf {q} )=\sum _{i=1}^{n}{\sqrt {p_{i}q_{i}}},}$^[3]

where, considering the samples p and q, n is the number of partitions, and ${\displaystyle p_{i}}$, ${\displaystyle q_{i}}$ are the numbers of members of samples p and q in the i-th partition.

This formula is hence larger with each partition that has members from both samples, and larger with each partition that has a large overlap of the two sample's members within it. The choice of number of partitions depends on the number of members in each sample; too few partitions will lose accuracy by overestimating the overlap region, and too many partitions will lose accuracy by creating individual partitions with no members despite being in a densely populated sample space.

The Bhattacharyya coefficient will be 0 if there is no overlap at all due to the multiplication by zero in every partition. This means the distance between fully separated samples will not be exposed by this coefficient alone.

The Bhattacharyya coefficient is used in the construction of polar codes.^[4]

Applications[]

The Bhattacharyya distance is widely used in research of feature extraction and selection,^[5] image processing,^[6] speaker recognition,^[7] and phone clustering.^[8]

A "Bhattacharyya space" has been proposed as a feature selection technique that can be applied to texture segmentation.^[9]

See also[]

• Bhattacharyya angle
• Kullback–Leibler divergence
• Hellinger distance
• Mahalanobis distance
• Chernoff bound
• Rényi entropy
• F-divergence
• Fidelity of quantum states

References[]

• Nielsen, F.; Boltz, S. (2010). "The Burbea–Rao and Bhattacharyya centroids". IEEE Transactions on Information Theory. 57 (8): 5455–5466. arXiv:1004.5049. doi:10.1109/TIT.2011.2159046.

• Kailath, T. (1967). "The Divergence and Bhattacharyya Distance Measures in Signal Selection". IEEE Transactions on Communication Technology. 15 (1): 52–60. doi:10.1109/TCOM.1967.1089532.

• Djouadi, A.; Snorrason, O.; Garber, F. (1990). "The quality of Training-Sample estimates of the Bhattacharyya coefficient". IEEE Transactions on Pattern Analysis and Machine Intelligence. 12 (1): 92–97. doi:10.1109/34.41388.

• For a short list of properties, see: http://www.mtm.ufsc.br/~taneja/book/node20.html

External links[]

• "Bhattacharyya distance", Encyclopedia of Mathematics, EMS Press, 2001 [1994]