Multiscale Permutation Entropy for Time Series Analysis

Multiscale Permutation Entropy (MPE) is a nonlinear dynamic method for analyzing time series complexity. By integrating multiscale analysis with permutation entropy computation, MPE reveals dynamic characteristic variations across different temporal scales. The core concept of permutation entropy involves converting time series data into ordinal patterns and quantifying sequence complexity by statistically analyzing the distribution of these patterns. Multiscale permutation entropy extends this approach by decomposing the original sequence into subsequences at different scales through a coarse-graining process, then computing permutation entropy for each scale to obtain multi-scale complexity information. In code implementation, the MPE algorithm typically follows these steps: 1. Coarse-graining procedure: Divide the original time series into non-overlapping segments of increasing length (scale factor τ) 2. For each scale: Calculate the mean of each segment to create a new coarse-grained time series 3. Ordinal pattern generation: For each coarse-grained series, create ordinal patterns by comparing neighboring values using embedding dimension m and time delay τ 4. Probability distribution: Count frequency of each ordinal pattern type 5. Entropy calculation: Apply Shannon entropy formula to the pattern distribution This method is particularly suitable for analyzing financial time series and physiological signals. In finance, MPE can identify changes in market volatility patterns, detect market regime shifts, or identify anomalous events. In physiological signal research (such as EEG, ECG), MPE assesses signal complexity and health status, enabling detection of epileptic seizures or sleep stage changes in EEG analysis. MPE's advantages include computational simplicity, strong noise resistance, and effective capture of nonlinear dynamic characteristics. Compared to traditional entropy methods (like sample entropy and approximate entropy), MPE shows lower dependency on data length and parameter selection, demonstrating greater robustness in short time series analysis. Key MATLAB functions for implementation would involve creating coarse-grained series, generating ordinal patterns using sort operations, and calculating probability distributions with histogram functions.