Statistical Properties of the Entropy from Ordinal Patterns

TL;DR

This paper analyzes the statistical distribution of Shannon's Entropy from ordinal patterns in time series, providing asymptotic results and a new bilateral test for comparing entropy across signals, with practical applications to temperature data.

Contribution

It characterizes the asymptotic distribution of empirical Shannon's Entropy for a broad class of models and introduces a bilateral test for comparing entropy between signals.

Findings

Asymptotic distribution derived using CLT and Delta Method.

Bilateral test effectively distinguishes signals with different entropy.

Application to temperature data yields meaningful results.

Abstract

The ultimate purpose of the statistical analysis of ordinal patterns is to characterize the distribution of the features they induce. In particular, knowing the joint distribution of the pair Entropy-Statistical Complexity for a large class of time series models would allow statistical tests that are unavailable to date. Working in this direction, we characterize the asymptotic distribution of the empirical Shannon's Entropy for any model under which the true normalized Entropy is neither zero nor one. We obtain the asymptotic distribution from the Central Limit Theorem (assuming large time series), the Multivariate Delta Method, and a third-order correction of its mean value. We discuss the applicability of other results (exact, first-, and second-order corrections) regarding their accuracy and numerical stability. Within a general framework for building test statistics about Shannon's Entropy, we present a bilateral test that verifies if there is enough evidence to reject the hypothesis that two signals produce ordinal patterns with the same Shannon's Entropy. We applied this bilateral test to the daily maximum temperature time series from three cities (Dublin, Edinburgh, and Miami) and obtained sensible results.