Distinguishing chaotic from stochastic dynamics via the complexity of ordinal patterns

Abstract

Distinguishing between chaotic and stochastic dynamics given an input series is a widely studied topic within the time series analysis due to high demand from the practitioners in various fields. Due to one of the fundamental properties of chaotic systems, namely, being sensitive to parameters and initial conditions, chaotic time series exhibit features also observed in randomly generated signals. In this paper, we introduce distance as a measure of similarity between segments based on the ordinal structure. Furthermore, we introduce a new fuzzy entropy, Fuzzy Permutation Entropy (FPE), which can be used to detect determinism in time series. FPE immunes from repeated equal values in signals to some extent, especially for chaotic series. With specific embedding dimensions, it can be employed to distinguish chaotic signals from noise. We show an example for white Gaussian noise, autoregressive moving-average, continuous or discrete chaotic time series, and test FPE’s performance with additive observational noise. We show an application of FPE on rolling bearings’ fault diagnosis.