Multiscale Permutation Entropy: Algorithm and Implementation

This section introduces the concept and applications of the Multiscale Permutation Entropy (MPE) algorithm. MPE is a methodological approach for analyzing complex systems by first permuting time series data and subsequently calculating permutation entropy across multiple scales to quantify system complexity. The algorithm involves three key computational stages: time series coarse-graining at different scales, symbolic permutation encoding of embedded vectors, and entropy calculation based on pattern distribution probabilities. Implementation typically requires defining critical parameters including embedding dimension (m), time delay (τ), and scale factor (s), where the coarse-graining process divides the original series into non-overlapping windows of length s. The algorithm finds extensive applications across domains such as financial analytics, biological signal processing, and climate science, enabling researchers to characterize system dynamics and extract meaningful patterns from complex temporal data. Consequently, MPE holds significant value in both theoretical research and practical engineering applications.