Entropic Analysis of non-Stationary Sequences

TL;DR

This paper investigates the analysis of non-stationary time series using diffusion entropy, providing a method to understand scaling behaviors and entropy evolution from single sequences.

Contribution

It introduces a probabilistic approach to analyze non-stationary sequences, linking ensemble and single-sequence analysis, and explains entropy dynamics with two uncertainty sources.

Findings

Nonstationary sequences can show anomalous or ordinary scaling.

A recipe accurately fits numerical results for entropy evolution.

Entropy is influenced by initial condition uncertainty and trajectory randomness.

Abstract

The aim of this paper is to shed light on the analysis of non-stationary time series by means of the method of diffusion entropy. For this purpose, we first study the case when infinitely many time series, as different realizations of the same dynamic process, are available, so as to adopt the Gibbs ensemble perspective. We solve the problem of establishing under which conditions scaling emerges from within this perspective. Then, we study the more challenging problem of creating a diffusion process from only one single (non-stationary) time series. The conversion of this single sequence into many diffusional trajectories is equivalent to creating a non-Gibbsian ensemble. However, adopting a probabilistic approach to evaluate the contribution of any system of this non-Gibbsian ensemble, and using for it the theoretical Gibbsian prescription of the earlier case, we find a recipe that fits accurately the numerical results. With the help of this recipe we show that nonstationary time series produce either anomalous scaling with ordinary statistics or ordinary scaling with anomalous statistics. From this recipe we also derive an attractive way to explain the entropy time evolution, as resulting from two distinct uncertainty sources, the lack of information on the trajectory initial condition, and the lack of control on random trajectories.