The identification of fractional order systems by multiscale multivariate analysis

Abstract The recognition of complex systems plays a vital role in system dynamics. To distinguish chaotic systems and stochastic processes, the complex-entropy causality plane (CECP) was proposed and received considerable attention. However, few methods have been applied to the identification of fractional order chaotic systems. In this paper, we propose multiscale multivariate fractional dispersion entropy (MMFDE) and MMFDE plane to study the structural complexity of multivariate nonlinear systems. Unlike the ordinal pattern in CECP, the dispersion pattern is used in MMFDE plane to solve the computational burden in fractional order systems. Moreover, the fractional order α in MMFDE can be used to adjust the ability of MMFDE plane to distinguish between different systems. Through simulation data, we find that the MMFDE plane can classify not only chaotic systems and stochastic processes but also different fractional order chaotic systems even if adding noise to the signals. It is then applied to financial time series and epileptic EEG recordings. The results show that the MMFDE plane can identify developed and emerging markets. For epileptic EEG recordings, both the EEG subbands and the original data without filtering can be used to distinguish different states of healthy subjects and patients with epilepsy. The coarse-graining process is also discussed to demonstrate the consequences further.