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

Juan Adrián Ramírez Belman, Juan Bory Reyes, José Oscar González Cervantes, Gamaliel Yafte Tellez Sanchez

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 $(ρ_{1}, \ldots,ρ_{N})$ can be established by the action of the hyperbolic derivative on the generating function $\sum_{s=1}^{N}ρ_{s}^{-ξ}$ with respect to the hyperbolic variable $ξ$ and then we tend $ξ$ 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 Rényi entropy and hyperbolic extropy.

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

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 with the notion of valued hyperbolic probability. It is shown that the Shannon strong hyperbolic entropy over a discrete hyperbolic probability distribution can be established by the action of the hyperbolic derivative on the generating function with respect to the hyperbolic variable and then we tend to . 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 Rényi entropy and hyperbolic extropy.
Paper Structure (13 sections, 8 theorems, 66 equations)

This paper contains 13 sections, 8 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Shannon entropy
  4. The Rényi Entropy and Lesche Stability
  5. Hyperbolic numbers
  6. Hyperbolic derivative
  7. Hyperbolic Probability and Entropy
  8. Hyperbolic probability
  9. Hyperbolic entropy
  10. Rewriting the Shannon strong hyperbolic entropy
  11. Lesche stability for the Shannon strong hyperbolic entropy
  12. The hyperbolic extension of Rényi entropy
  13. Final comments

Key Result

Proposition 2.2

Let $X$ be a discrete random variable with support $\{X_1,\ldots, X_N\}$ and with probability mass function (distribution) $\{p_1,\ldots, p_N\}$. If $N = 2$, so $X$ contains only two states, then $S((p_1,\ldots, p_N)) = J((p_1,\ldots, p_N))$, but while $X$ contains three or more states; $N \geq 3$ t

Theorems & Definitions (33)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5: Lesche Stability
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 23 more