Cumulative Diversity Pattern Entropy (CDEn): A High-Performance, Almost-Parameter-Free Complexity Estimator for Nonstationary Time Series

Abstract

Tedious parameter settings and poor performances seriously affect the entropy estimation's effectiveness in time series analysis. To solve these limits, we propose a conceptually novel definition, cumulative diversity pattern entropy (CDEn), focusing on eliminating parameter selections and improving quantization accuracy, stability, and robustness. The CDEn algorithm consists of three steps: (1) improved phase-space reconstruction (IPSR) with constant embedding dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\bm{m\ }} = {\bm{\ }}2$</tex-math></inline-formula> and time delay <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\bm{\tau \ }} = {\bm{\ }}1$</tex-math></inline-formula> ; (2) diversity pattern partition generated by the cosine similarity between adjacent vectors; and (3) entropy calculation based on the normalized cumulative probability distribution. Numerical experiments are performed using 7 synthetic datasets and 15 baseline entropy methods for comparative validation. The results confirm CDEn's best description of chaotic/stochastic dynamics with the highest quantization accuracy and the lowest error rate of 2.04%. The coefficient of variation (CV) results also verify CDEn's excellent quantization stability with CV lower than 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> . The relative change rate (RCR) results demonstrate that CDEn achieves the best robustness to data length and noise. Finally, the entropy algorithms are applied to a real-world dataset, i.e., neonatal sleep EEG analysis. The results further confirm that suggested CDEn outperforms the state-of-the-art entropy methods, with the minimum outliers and best statistical significance (highest mean of effect size, 1.22) in characterizing the neurodynamics of different sleep stages.