Real systems are often complex, nonlinear, and noisy in various fields, including mathematics, natural science, and social science. We present the symplectic entropy (SymEn) measure as well as an analysis method based on SymEn to estimate the nonlinearity of a complex system by analyzing the given time series. The SymEn estimation is a kind of entropy based on symplectic principal component analysis (SPCA), which represents organized but unpredictable behaviors of systems. The key to SPCA is to preserve the global submanifold geometrical properties of the systems through a symplectic transform in the phase space, which is a kind of measure-preserving transform. The ability to preserve the global geometrical characteristics makes SymEn a test statistic for the detection of the nonlinear characteristics in several typical chaotic time series, and the stochastic characteristic in Gaussian white noise. The results are in agreement with findings in the approximate entropy (ApEn), the sample entropy (SampEn), and the fuzzy entropy (FuzzyEn). Moreover, the SymEn method is also used to analyze the nonlinearities of real signals (including the electroencephalogram (EEG) signals for Autism Spectrum Disorder (ASD) and healthy subjects, and the sound and vibration signals for mechanical systems). The results indicate that the SymEn estimation can be taken as a measure for the description of the nonlinear characteristics in the data collected from natural complex systems.
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