Cosine Similarity Entropy: Self-Correlation-Based Complexity Analysis of Dynamical Systems

The nonparametric Sample Entropy (SE) estimator has become a standard for the quantification of structural complexity of nonstationary time series, even in critical cases of unfavorable noise levels. The SE has proven very successful for signals that exhibit a certain degree of the underlying structure, but do not obey standard probability distributions, a typical case in real-world scenarios such as with physiological signals. However, the SE estimates structural complexity based on uncertainty rather than on (self) correlation, so that, for reliable estimation, the SE requires long data segments, is sensitive to spikes and erratic peaks in data, and owing to its amplitude dependence it exhibits lack of precision for signals with long-term correlations. To this end, we propose a class of new entropy estimators based on the similarity of embedding vectors, evaluated through the angular distance, the Shannon entropy and the coarse-grained scale. Analysis of the effects of embedding dimension, sample size and tolerance shows that the so introduced Cosine Similarity Entropy (CSE) and the enhanced Multiscale Cosine Similarity Entropy (MCSE) are amplitude-independent and therefore superior to the SE when applied to short time series. Unlike the SE, the CSE is shown to yield valid entropy values over a broad range of embedding dimensions. By evaluating the CSE and the MCSE over a variety of benchmark synthetic signals as well as for real-world data (heart rate variability of three different cardiovascular pathologies), the proposed algorithms are demonstrated to be able to quantify degrees of structural complexity in the context of self-correlation over small to large temporal scales, thus offering physically meaningful interpretations and rigor in the understanding the intrinsic properties of the structural complexity of a system, such as the number of its degrees of freedom.