Cumulative Residual Symbolic Dispersion Entropy and its Multiscale Version: Methodology, Verification, and Application

In quantifying the complexity characteristics of neurophysiological signals, the most advanced entropy methods still have some inevitable limitations of poor accuracy, robustness, and reliability. To solve these limits, this study proposes a novel entropy estimator, termed cumulative residual symbolic dispersion entropy (CRSDE). Meanwhile, a corresponding multiscale variant, time-shift cumulative residual symbolic dispersion entropy (TCRSDE), is introduced to quantify the time series's irregularity on multiple time scales. The CRSDE starts with an improved symbolic dynamics filter (ISDF) based on the equal probability dividing criterion, mapping a raw time series to symbolic series. Next, embedding theory is applied to derive the dispersion patterns. Finally, the cumulative residual probabilities of all dispersion patterns are counted, and corresponding CRSDE results are calculated. A series of performance validations are executed using synthetic and real-world datasets. The simulation results confirm CRSDE's optimal estimation accuracy and robustness to noise and data length. The findings of EEG dataset demonstrate that CRSDE could realize the best reliability with the lowest root mean square deviation (RMSD; <0.05). In the multiscale version, TCRSDE, a time-shift coarse-graining is introduced. The verification findings confirm that the TCRSDE can avoid incorrect estimates and achieve the best-estimated stability on all scales with the smallest coefficient of variation (CV; <0.01) and shortest running time. Finally, CRSDE method is applied to neonatal sleep EEG. Compared to other methods, CRSDE shows the most significant differences, with all p-values less than 0.01 via a nonparametric Mann-Whitney U test. Meanwhile, CRSDE obtains the highest Hedges' g effect size values and least outliers at three sleep stages. Therefore, CRSDE performs best in quantifying neurodynamics of neonatal sleep stages.