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

非平稳性、高维性和非线性被广泛认为是大数据时代时间序列分析面临的三大挑战。当在相对较长的时间段内同时记录大量随机过程时，这些复杂的数据结构经常出现。数据的复杂性阻碍了研究人员和实践者使用经典的时间序列方法，如平稳ARMA理论和方法论。这项拟议研究的长期目标有两个。首先，从非线性系统的角度为一大类高维非平稳(HDNS)时间序列在时间域和谱域的建模和推断建立了系统的理论基础。其次，基于上述理论基础，将建立一个稳健、自适应和计算高效的方法工具箱，用于各种重要应用中产生的HDNS时间系统的估计、推理和预测。在短期内，理论上将主要集中在建立HDNS时间序列的系统的高斯逼近理论和自回归逼近理论上；在方法上，重点将集中在线性、双线性和非线性时频分析中的非参数统计推断及其在信号处理中的应用。如今，技术创新使得在相对较长的时间内收集具有复杂结构的海量数据成为可能。我看到各个重要实践领域对HDNS时间序列的统计分析产生了巨大的需求、机遇和挑战。因此，统计理论和方法应该随着这一需求而进步。然而，HDNS时间序列分析仍然缺乏一个统一的统计理论，在许多应用中几乎没有稳健、准确和计算高效的方法工具箱，并且具有严格和准确的随机不确定性控制。我相信，从非线性系统的角度提出的框架将为许多科学学科中的HDNS时间序列分析提供重要的理论和方法基础。

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.