Cross-fuzzy entropy: A new method to test pattern synchrony of bivariate time series

A new method, namely cross-fuzzy entropy (C-FuzzyEn) analysis, that can enable the measurement of the synchrony or similarity of patterns between two distinct signals, is presented in this study. With the inclusion of fuzzy sets, the similarity of vectors is fuzzily defined in C-FuzzyEn based on the exponential function and their shapes, rather than on the Heaviside function used in the conventional cross sample entropy (C-SampEn). Tests on simulated data sets and real EEG signals showed that C-FuzzyEn was superior to C-SampEn in several aspects, including giving the entropy definition in the case of small parameters, better relative consistency, and less dependence on record length. The proposed C-FuzzyEn was then applied for the analysis of simultaneously recorded electromyography (EMG) and mechanomyography (MMG) signals during sustained isometric contraction for monitoring local muscle fatigue. The results showed that the C-FuzzyEn of EMG–MMG signals decreased significantly during the development of muscle fatigue. The C-FuzzyEn showed a similar trend with the mean frequency (MNF) of EMG, the commonly used muscle fatigue indicator. However, C-FuzzyEn of EMG–MMG demonstrated a better robustness to the length of the analysis window in comparison with the MNF of EMG. The results suggested that the proposed C-FuzzyEn of EMG–MMG may potentially become a new reliable method for muscle fatigue assessment. It can also be applied to other bivariate signals extracted from complex systems with short data lengths in noisy backgrounds.