Probing High-Order Dependencies With Information Theory

Information theoretic measures (entropies, entropy rates, mutual information) are nowadays commonly used in statistical signal processing for real-world data analysis. This paper 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 statistics 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. Furthermore, 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 for these processes, indicating that one of the non-Gaussian actually has temporal dynamics that resembles that of a Gaussian process with the same covariance while the other does not.