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I have a cloud of points, and I want to compute its 'diversity'. Variance is not appropriate, because a cloud clustering around few points can still have a large variance.

To that end, I see the cloud of points as the realizations of a random variable.

I want to compute the topological entropy of a random variable, with values in a metric space $(X,d)$.

I only know $N$ (large) realisations of this variable, and I can compute the relative distances between them.

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  • $\begingroup$ see definition 2.5 in this thesis $\endgroup$
    – Carlo Beenakker
    Feb 2, 2018 at 15:53
  • $\begingroup$ Very uninformative comment. It’s the Bowen-Dinaburg definition of entropy.I can't use it here because I don't know the whole map, only N realisations. $\endgroup$
    – Mostafa
    Feb 2, 2018 at 21:25
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    $\begingroup$ Perhaps you should give a bit more of context. Your question is a bit vague. $\endgroup$
    – Rafael Alcaraz Barrera
    Feb 3, 2018 at 5:06
  • $\begingroup$ the context is that I have a cloud of points, and I want to measure its 'diversity' without using variance $\endgroup$
    – Mostafa
    Feb 3, 2018 at 12:59

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  1. Estimate the probability density, using Kernel density estimation

  2. Compute the differential entropy.

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  • $\begingroup$ Differential entropy with respect to what? $\endgroup$
    – R W
    Feb 6, 2018 at 13:14

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