I have exactely no idea of where to start when it come to cluster distribution and find out similar the similar one. is there a package in R that do the job ?

There are quite a few ways to measure the difference between two distributions. Take a look at this overview article on wikipedia.

One very common way that is used often in Machine and Deep Learning is the Kullback-Leibler (KL) Divergence. It is most commonly used in minimising the cross-entropy between the distribution of your training data and the expected (generalised) distribution of the problem you are analysing. In general, a value close to 0 indicates that two distributions are expected to show similar behaviour, while a large value indicates the distributions behave very differently - so knowing the first distribution doesn't help you know anything about the second.

A more general class of measures of dissimilarity between distributions is the so called $f$-divergence, "an average, weighted by the function $f$ of the odds ratio given by [probability measures] $P$ and $Q$". The Wikipedia article contains a formal definition and a table with functions $f$ that give some popular dissimilarities/distances, including the KL-divergence.

The following table lists many of the common divergences between probability distributions and the $f$ function to which they correspond (cf. Liese & Vajda (2006))

Distribution distance metrics

KL-Divergence is combined with the entropy itself, to define the cross-entropy. Take a look here, under the Information Theory View, for a bit more info.

Use the popular K-means clustering algorithm combined with Hellinger distance as a metric of distance.

Hellinger distance quantifies the similarity between two distributions / histograms, thus it can be very easily merged with K-means for your purpose :)

  • Can you say anything more about why you recommend the Hellinger distance (as opposed to any other distance metric)? – D.W. Jun 12 at 23:13
  • to be honest I am more biased towards it because of some publications I have been recently reading that explicitly used it, such as this one link.springer.com/article/10.1007/s10618-017-0538-6. I don't recommend it over other metrics, I suggested it as a starting point because OP said he has no idea where to start and it is actually a concrete starting point when combined with K-means for his problem :) – pcko1 Jun 12 at 23:21
  • actually, from the publication I cited above: "It is a proper metric, it is quick to compute and it has no problem with empty bins". Hope it helps ! – pcko1 Jun 12 at 23:23
  • 1
    Got it, thank you for the explanation! – D.W. Jun 12 at 23:28

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