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。当大量的随机过程在相对长的时间段内同时记录时，这些复杂的数据结构经常出现。数据的复杂性使得研究人员和实践者无法使用经典的时间序列方法，如平稳阿尔马理论和方法。拟议研究的长期目标是双重的。首先，从非线性系统的角度来看，一个系统的理论基础的建模和推理的一大类高维和非平稳（HDNS）的时间序列在时间和频谱域将建立。其次，基于上述理论基础，将建立一个强大的，自适应的和计算效率高的方法工具箱，用于估计，推理和预测源于各种重要应用的HDNS时态系统。在短期内，理论上，主要重点将是建立系统的高斯近似理论和自回归近似理论的HDNS时间序列;方法上，重点将是非参数统计推断的线性，双线性和非线性时频分析与信号处理的应用。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大量数据成为可能。我看到了巨大的需求，机会和挑战的统计分析HDNS时间序列出现在各个重要的实践领域。因此，统计理论和方法也应该随着这一需求而发展。然而，一个统一的统计理论HDNS时间序列分析仍然缺乏和强大的，准确的和计算效率高的方法工具箱，严格和准确的随机不确定性控制几乎不存在于许多应用中。我相信，从非线性系统的角度来看，所提出的框架将提供一个重要的理论和方法基础，在许多科学学科的HDNS时间序列分析。

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.