# Probing high order dependencies with information theory

## TL;DR

This paper introduces the use of Auto Mutual Information and entropy rate to detect high-order dependencies in signals, revealing details missed by traditional correlation analysis and distinguishing between complex non-Gaussian processes.

## Contribution

It demonstrates how Auto Mutual Information and entropy rate can assess refined, higher-order temporal dependencies in signals, surpassing traditional correlation-based methods.

## Key findings

- Auto Mutual Information captures higher-order dependencies.
- Entropy rate distinguishes non-Gaussian processes with similar covariance.
- Estimators' performance analyzed via Monte Carlo simulations.

## Abstract

Information theoretic measures (entropies, entropy rates, mutual information) are nowadays commonly used in statistical signal processing for real-world data analysis. The present work proposes the use of Auto Mutual Information (Mutual Information between subsets of the same signal) and entropy rate as powerful tools to assess refined dependencies of any order in signal temporal dynamics. Notably, it is shown how two-point Auto Mutual Information and entropy rate unveil information conveyed by higher order statistic and thus capture details of temporal dynamics that are overlooked by the (two-point) correlation function. Statistical performance of relevant estimators for Auto Mutual Information and entropy rate are studied numerically, by means of Monte Carlo simulations, as functions of sample size, dependence structures and hyper parameters that enter their definition. Further, it is shown how Auto Mutual Information permits to discriminate between several different non Gaussian processes, having exactly the same marginal distribution and covariance function. Assessing higher order statistics via multipoint Auto Mutual Information is also shown to unveil the global dependence structure fo these processes, indicating that one of the non Gaussian actually has temporal dynamics that ressembles that of a Gaussian process with same covariance while the other does not.