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I'm looking for some good terminology to describe what I'm trying to do, to make it easier to look for resources.

So, say I have two clusters of points A and B, each associated to two values, X and Y, and I want to measure the "distance" between A and B - i.e. how likely is it that they were sampled from the same distribution (I can assume that the distributions are normal). For example, if X and Y are correlated in A but not in B, the distributions are different.

Intuitively, I would get the covariance matrix of A, and then look at how likely each point in B is to fit in there, and vice-versa (probably using someting like Mahalanobis distance).

But that is a bit "ad-hoc", and there is probably a more rigorous way of describing this (of course, in practice I have more than two datasets with more than two variables - I'm trying to identify which of my datasets are outliers).

Thanks!

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  • $\begingroup$ Dunno why, but a Mantel test flashed in front of my eyes when I read your post. $\endgroup$ – Roman Luštrik Nov 6 '10 at 15:43
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There is also the Kullback-Leibler divergence, which is related to the Hellinger Distance you mention above.

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    $\begingroup$ can one calculate the Kullback-Leibler divergence of points without making an assumption of the underlying probability density the points came from ? $\endgroup$ – Andre Holzner Nov 6 '10 at 16:22
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Hmm, the Bhattacharyya distance seems to be what I'm looking for, though the Hellinger distance works too.

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  • $\begingroup$ you mention Bhattacharyya and Helling then accept an answer speaking about KL... At the end what was your choice and why? $\endgroup$ – Simon C. Jan 23 at 16:20
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    $\begingroup$ I believe it was KL divergence, but ... that was in 2010 and my memory is far from perfect. $\endgroup$ – Emile Jan 23 at 17:21
  • $\begingroup$ ahah yes I guessed that, but thank you anyway! $\endgroup$ – Simon C. Jan 23 at 17:40
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Heuristic

  • Minkowski-form
  • Weighted-Mean-Variance (WMV)

Nonparametric test statistics

  • 2 (Chi Square)
  • Kolmogorov-Smirnov (KS)
  • Cramer/von Mises (CvM)

Information-theory divergences

  • Kullback-Liebler (KL)
  • Jensen–Shannon divergence (metric)
  • Jeffrey-divergence (numerically stable and symmetric)

Ground distance measures

  • Histogram intersection
  • Quadratic form (QF)
  • Earth Movers Distance (EMD)
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The most complete survey is provided in Statistical Inference Based on Divergence Measures by Leandro Pardo, Complutense University, Chapman Hall 2006.

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Few more measures of "Statistical Difference"

  • Permutation test (by Fisher)
  • Central Limit Theorem & Slutsky’s theorem
  • Mann-Whitney-Wilcoxin test
  • Anderson–Darling test
  • Shapiro–Wilk test
  • Hosmer–Lemeshow test
  • Kuiper's test
  • kernelized Stein discrepancy
  • Jaccard similarity
  • Also, hierarchical clustering deals with similarity measures between groups. The most popular measures of group similarity are perhaps the single linkage, complete linkage, and average linkage.
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