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in a project I want to compare 2 Histograms of similarity. My problem is to choose the right metric/ distance function.

Can someone tell me, what's the difference between the Bhattacharyya distance and the Chi-square distance?

My dataset in the best case is near of the normal distribution. In my opinion I saying that Bhattacharyya distance is the right one for me (especialy the distance metric is bounded in [0,1}). But i can't find more arguments.

Greetings

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As always, the right metric to compare too things depends on the purpose behind the comparison. –  Chris Eagle Dec 10 '12 at 16:51
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up vote 1 down vote accepted

It really depends on what you are doing. Depending on your task, you will want a metric with different properties.

The Bhattacharyya coefficient is related to the Hellinger distance that has some nice geometric properties. Both of them are symmetric:

$$ B(p,q) = B(q,p) $$ $$ H^2(p,q) = H^2(q,p)$$

Both of these are bounded - as are many others (such as the total variation).

The $\chi$ squared distance is not symmetric. (I assume you mean $\chi(p,q) = \int \frac{(p-q)^2}{p} dx$) it is the Pearson $\chi^2$ distance one way, and some else's (forgotten who's) the other:

$$\chi(p,q) \neq \chi(q,p)$$

This may or may not be a good thing depending on the context. All of them are approximately proportional to the Kullback-Leibler divergence.

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thanks for this answer :) In my task i want to compare two luminance histogram in image analysis. For this it should be symmetric –  501 - not implemented Dec 10 '12 at 19:14
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