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.
Highlights
To characterize a physical structure, confirming the underlying dynamics of the observation is either deterministic or stochastic has been found a vital part
We introduce a new fuzzy entropy (FE), Fuzzy Permutation Entropy (FPE), which can be used for detecting determinism in time series
There are still a few mis-matched results, less than 5%, for signals with slight or moderate faults. This shows that the quadratic Support Vector Machine (SVM) using FPE, which is extracted from the vibration scitation.org/journal/adv signal as a feature, has the ability to identify faults in rolling bearings
Summary
To characterize a physical structure, confirming the underlying dynamics of the observation is either deterministic or stochastic has been found a vital part. Based on the Lempel–Ziv complexity method, the surrogate verification scheme provides a more reliable distinction between the chaotic time series and non-Gaussian noise.. Ye et al distinguished chaos from noise based on the distribution of eigenvalues of the corresponding Wishart or Wigner matrix, Carroll and Byers proposed a data-partitioning method to distinguish Sprott signals, Xiong et al constructed an entropy–complexity plane under the Bandt–Pompe framework to characterize chaos and noise.. Ye et al distinguished chaos from noise based on the distribution of eigenvalues of the corresponding Wishart or Wigner matrix, Carroll and Byers proposed a data-partitioning method to distinguish Sprott signals, Xiong et al constructed an entropy–complexity plane under the Bandt–Pompe framework to characterize chaos and noise.19 From their achievements, researchers realize that the information entropy is considered as a prior statistical measure.
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