On the Statistical Properties of Multiscale Permutation Entropy and its Refinements, with Applications on Surface Electromyographic Signals

Permutation entropy (PE) and multiscale permutation entropy (MPE) are extensively used to measure regularity in the analysis of time series, particularly in the context of biomedical signals. As accuracy is crucial for researchers to obtain optimal interpretations, it becomes increasingly important to take into account the statistical properties of MPE. Therefore, in the present work we begin by expanding on the statistical theory behind MPE, with an emphasis on the characterization of its first two moments in the context of multiscaling. Secondly, we explore the composite versions of MPE in order to understand the underlying properties behind their improved performance; we also created an entropy benchmark through the calculation of MPE expected values for widely used Gaussian stochastic processes, since that gives us a reference point to use with real biomedical signals. Finally, we differentiate between muscle activity dynamics in isometric contractions through the application of the classical and composite MPE methods on surface electromyographic (sEMG) data. As a result of our project, we found MPE to be a biased statistic that decreases with respect to the multiscaling factor, regardless of the signal’s probability distribution. We also noticed that the variance of the MPE statistic is highly dependent on the value of MPE itself, and almost equal to its Cram´er-Rao lower bound - in other words, confirming it is an efficient estimator. Despite showing improved results, we realized that the composite versions also modify the MPE estimation due to the measuring of redundant information. In light of our findings, we decided to replace the multiscaling coarse-graining procedure with one of our own, with the intention of improving our estimations. Since our team observed the MPE statistic to be completely characterized by the model parameters when applied to correlated Gaussian models, we developed a general formulation for expected MPE with low-embedding dimensions. When applied to real sEMG signals, we were able to distinguish between fatigue and non-fatigue states with all methods, especially for high-embedding dimensions. Moreover, we found that our proposed MPE method makes an even clearer difference between the two aforementioned activity states.