MULTISCALE ANALYSIS OF COMPLEX TIME SERIES

To commemorate the first anniversary of Prof. Zheng Zhemin's death, the author first recalls the teachings that Prof. Zheng gave to himself during his postgraduate studies, then introduces their joint work in nonlinear science, which have been expanded and evolved into some general methods for analyzing nonlinear problems. Specific methods include optimal embedding of chaotic time series, direct dynamical test for chaos, multiscale analysis based on scale-dependent Lyapunov exponent (SDLE), and adaptive fractal analysis. In particular, SDLE can very well characterize all known time series models, and therefore, can unify various measures of complexity developed thus far. Adaptive fractal analysis is based on adaptive filtering, which can optimally determine trends, including various oscillation modes and nonlinear regression curves in regression analysis, and can also optimally reduce noise and decompose time series into intrinsic modes. These methods have been widely used in many fields of natural sciences, engineering, and social sciences. They are particularly useful for fault diagnosis in various fields (including operation and maintenance), analysis of biomedical data, and measurement of uncertainty. Prof. Zheng was never confined to a field of comfort, but constantly expanded into new the ones with time going by. In a period with unprecedented changes not seen in a century, we must carry forward the spirit of Prof. Zheng.