Lesche stability of the Shannon strong hyperbolic entropy and some hyperbolic extensions

TL;DR

This paper extends Shannon entropy into the hyperbolic number plane, proving its Lesche stability and exploring hyperbolic versions of Rényi entropy and extropy, thus broadening entropy's mathematical framework.

Contribution

It introduces a hyperbolic extension of Shannon entropy with Lesche stability and explores related hyperbolic entropy measures, expanding the theoretical foundation of entropy in hyperbolic spaces.

Findings

Hyperbolic Shannon entropy is Lesche stable.

Hyperbolic entropy can be derived via hyperbolic derivatives.

Extensions to Rényi entropy and extropy are established.

Abstract

In recent decades, several definitions of new entropy measures have been proposed, which expands the range of applications for this important tool. The present work focuses on the extension of the classical Shannon entropy to the hyperbolic number plane $\mathbb{D}$ with the notion of valued hyperbolic probability. It is shown that the Shannon strong hyperbolic entropy over a discrete hyperbolic probability distribution $(\rho_{1}, \ldots,\rho_{N})$ can be established by the action of the hyperbolic derivative on the generating function $\sum_{s=1}^{N}{\rho}_{s}^{-\xi}$ with respect to the hyperbolic variable $\xi$ and then we tend $\xi$ to $-1_{\mathbb{D}}$. Furthermore, we prove that this hyperbolic extension possesses the Lesche stability property, also known as experimental robustness. Finally, we present some results on the hyperbolic extension of the R\'enyi entropy and hyperbolic extropy.