Information theory for non-stationary processes with stationary increments

TL;DR

This paper introduces a practical information-theoretic framework for analyzing non-stationary processes with stationary increments, enabling estimation from single realizations and providing insights into self-similarity and multi-fractality.

Contribution

It develops ersatz entropy and mutual information measures from time-averaged distributions, applicable to single realizations of complex processes, and demonstrates their effectiveness on various signals.

Findings

Ersatz entropy rate is robust and matches analytical expectations.

The approach can quantify self-similarity and multi-fractality.

Method works for Gaussian, non-Gaussian, and multi-fractal signals.

Abstract

We describe how to analyze the wide class of non stationary processes with stationary centered increments using Shannon information theory. To do so, we use a practical viewpoint and define ersatz quantities from time-averaged probability distributions. These ersatz versions of entropy, mutual information and entropy rate can be estimated when only a single realization of the process is available. We abundantly illustrate our approach by analyzing Gaussian and non-Gaussian self-similar signals, as well as multi-fractal signals. Using Gaussian signals allow us to check that our approach is robust in the sense that all quantities behave as expected from analytical derivations. Using the stationarity (independence on the integration time) of the ersatz entropy rate, we show that this quantity is not only able to fine probe the self-similarity of the process but also offers a new way to quantify the multi-fractality.