On the Entropy of Fractionally Integrated Gauss–Markov Processes

Mario Abundo,Enrica Pirozzi

doi:10.3390/math8112031

Mario Abundo, Enrica Pirozzi

Open Access

https://doi.org/10.3390/math8112031

Copy DOI

Abstract

This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order α∈(0,1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα(t) is a decreasing function of α∈(0,1).

Highlights

In the study of a biological system, whose time evolution is modeled by a stochastic process that depends on a certain parameter α, often there is a need to find how a change in the value of α affects the qualitative behavior of the system, as well as its complexity degree, or entropy
GM processes and their fractional integrals over time are very relevant in various application fields, especially in Biology—e.g., in stochastic models for neuronal activity
The fractional integral of order α ∈ (0, 1) of a Gauss–Markov process Y (t), say Xα (t), is suitable to describe stochastic phenomena with long range memory dynamics, involving correlated input processes, which are very relevant in Biology

Summary

Introduction

In the study of a biological system, whose time evolution is modeled by a stochastic process that depends on a certain parameter α, often there is a need to find how a change in the value of α affects the qualitative behavior of the system, as well as its complexity degree, or entropy Another useful information is the knowledge of a stochastic ordering, with respect to expectation of functionals of the process (e.g., its mean and variance), when varying α. We previously found that, in all the considered cases of GM processes, for large t the variance σα (t) of their fractional integral Xα (t) is an increasing function of α, while for small t it decreases with α; instead, the covariance function has more diversified behaviors (see [2]). The array x provides the simulated path—i.e., a realization ( x1 , x2 , . . . , x N ), of ( Xα (t1 ), . . . , Xα (t N )), whose components have the assigned covariance

The Approximate Entropy

Conclusions and Final Remarks