Fuzzy permutation time irreversibility for nonequilibrium analysis of complex system

Permutation time irreversibility is an important method to quantify the nonequilibrium characteristics; however, ordinal pattern is a coarse-graining alternative of temporal structure and cannot accurately represent detailed structural information. In this paper, a fuzzy permutation time irreversibility (fpTIR) is proposed by measuring the difference between vector elements based on a negative exponential function. Amplitude permutation of vector is constructed and its membership degree is calculated, then the difference in probability distribution between the forward sequence and backward sequence is measured for fpTIR. For comparison, Shannon entropy is calculated as the average amount of information in the fuzzy permutation probability distribution, i.e. fuzzy permutation entropy (fPEn), to measure the complexity of the system. According to the surrogate theory, mode series are generated by logistic, Henon, and first-order autoregressive systems to verify the fpTIR, which is then used to analyze heart rates of congestive heart failure, healthy elderly and healthy young subjects from PhysioNet database. The results suggest that fpTIR effectively measures the nonequilibrium characteristic of system and improves the accuracy of heart rate analysis. Since fpTIR and fPEn are different in analyzing probability distributions, they have discrepancies in chaotic series and even opposite results in the heart rate signals, where the results of fpTIR are consistent with theory of complexity loss in aging and disease. In conclusion, the fpTIR not only accurately characterizes the structure of sequences and enhances the effect of the nonequilibrium analysis of complex systems, but also provides a new perspective and theoretical basis for exploring complex systems from the perspectives of nonequilibrium dynamics and entropy complexity.