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.


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

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  • $\begingroup$ 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
  1. Estimate the probability density, using Kernel density estimation

  2. Compute the differential entropy.

  • $\begingroup$ Differential entropy with respect to what? $\endgroup$ – R W Feb 6 '18 at 13:14

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