An Entropy Formulation Based on the Generalized Liouville Fractional Derivative.

Rui A C Ferreira,J Tenreiro Machado

doi:10.3390/e21070638

Abstract

This paper presents a new formula for the entropy of a distribution, that is conceived having in mind the Liouville fractional derivative. For illustrating the new concept, the proposed definition is applied to the Dow Jones Industrial Average. Moreover, the Jensen-Shannon divergence is also generalized and its variation with the fractional order is tested for the time series.

Highlights

The quest of generalizing the Boltzmann–Gibbs entropy has become an active field of research in the past 30 years
The Dow Jones Industrial Average (DJIA) is an index based on the value of 30 large companies from the United States traded in the stock market during time
For processing the data produced by the Jensen-Shannon divergence (JSD) we adopt hierarchical clustering (HC)

Summary

Introduction

The quest of generalizing the Boltzmann–Gibbs entropy has become an active field of research in the past 30 years. Many formulations appeared in the literature extending the well-known formula (see e.g., [1,2,3,4]): S( p) = ∑ − ln( pi ) pi. The (theoretical) approaches to generalize Equation (1) may vary considerably (see e.g., [1,2,3]). We substitute the differential operator dt in Equation (2) by a suitable fractional one (see Section 2 for the details) and after some calculations, we obtain a novel (at least to the best of our knowledge) formula, which depends on a parameter 0 < α ≤ 1.

Motivation

An Example of Application

Conclusions