Statistical Inference for Complex Temporal Systems: Non-stationarity, High Dimensionality And Beyond.

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Non-stationarity, high-dimensionality and nonlinearity are widely recognized as the three major challenges for time series analysis in the big data era. These complicated data structures arise frequently when a large number of stochastic processes are simultaneously recorded over a relatively long period of time. The complexity of the data prevents researchers and practitioners from using the classical time series approaches, such as the stationary ARMA theory and methodology. The long-term objective of the proposed research is two-fold. First, a systematic theoretical foundation for the modelling and inference of a large class of high-dimensional and non-stationary (HDNS) time series in both time and spectral domains will be established from a nonlinear system point of view. Second, based on the aforementioned theoretical foundation, a robust, adaptive and computationally efficient methodological toolbox for the estimation, inference and prediction of HDNS temporal systems stemmed from various important applications will be built. In the short term, theoretically, the main focus will be on establishing systematic Gaussian approximation theory and auto-regressive approximation theory for HDNS time series; methodologically, the focus will be on nonparametric statistical inference in linear, bilinear and nonlinear time-frequency analysis with applications to signal processing. Nowadays, technological innovations have made it possible to collect a massive amount of data with complex structures over a relatively long period of time. I see a great demand, opportunity and challenge for statistical analysis of HDNS time series emerging from various important fields of practice. Therefore, statistical theory and methodologies should progress with this demand. However, a unified statistical theory for HDNS time series analysis is still lacking and robust, accurate and computationally efficient methodological toolboxes with rigorous and accurate stochastic uncertainty control barely exist in many applications . I believe that the proposed framework from the nonlinear system point of view will provide an important theoretical and methodological basis for HDNS time series analysis in many scientific disciplines.