Statistical Properties of the Entropy from Ordinal Patterns
This work addresses a statistical gap for researchers analyzing time series by providing tools to test entropy equality, but it is incremental as it builds on existing ordinal pattern methods.
The authors tackled the problem of characterizing the distribution of Shannon's Entropy from ordinal patterns in time series, enabling statistical tests; they derived its asymptotic distribution and applied a bilateral test to temperature data from three cities, obtaining sensible results.
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.