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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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closed as unclear what you're asking by R W, Ben McKay, Mikhail Katz, Pace Nielsen, Stefan Kohl Feb 6 '18 at 20:27

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ see definition 2.5 in this thesis $\endgroup$ – Carlo Beenakker Feb 2 '18 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 '18 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 '18 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 '18 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 '18 at 13:14

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