Local Legendre Polynomial Fitting-Based Preprocessing for Improving the Interpretation of Permutation Entropy in Stationary Time Series

Permutation entropy (PE) and its variants are ordinal pattern-based techniques that have become widely used as complexity measures to quantify the degree of disorder or randomness in a time series. Despite their popularity, these techniques have several limitations, such as sensitivity to the embedding dimension, the sampling frequency, and their specific preprocessing strategies. The information captured by these techniques is difficult to interpret and may not fully reflect the complexity of time series. We propose an alternative PE variant to overcome these limitations. We first divide the signal into short segments of a fixed length sufficient enough to allow for a local polynomial modelling of this signal. We use a discrete orthonormal polynomial basis of a limited degree to ensure that for each segment, the model parameters obtained are uncoupled and have similar value ranges. By ranking these parameters, we construct an ordinal pattern (OP) for each segment. The proposed PE variant is then defined as the Shannon entropy applied to the probability distribution of these OPs. The proposed local polynomial fitting-based preprocessing helps improve the PE interpretation. The advantages of the proposed PE method over some existing PE variants are demonstrated using simulated signals and real data.