Nonparametric statistical inference under complex temporal dynamics

在过去的二十年里，人们对具有复杂时间动态的数据进行分析的需求急剧增加。特别是，两种类型的时间序列数据结构在理论和实践中都引起了极大的兴趣。首先，很明显，许多时间序列数据的时间相关性结构和边际分布随着时间的推移而突然而平稳地变化。其次，对于许多长时间、密集观测的时间序列数据，将时间记录划分为自然连续的区间，并将数据视为函数时间序列，在数学上是优雅的，也是有益的。 拟议的研究旨在为在非线性系统表示的统一框架下对上述两种类型的复杂结构时间序列进行建模、估计、推断和预测提供一套系统的统计方法和理论。在方法上，我们将开发高效和自适应的非参数程序，用于分析一般类型的非平稳和/或函数值时间序列，并具有严格的数学证明。这些方法包括但不限于：非平稳(泛函)时间序列的稳健、多尺度和自适应突变检测；非平稳时间序列非参数回归的有效且稳健的同时置信带；稀疏和密集观测的函数时间序列的一致推断；以及时变谱密度的高效同时非参数推断。从理论上讲，将发展出具有发散维度的非平稳时间序列的系统统计理论，这将为上述大多数研究课题提供数学基础。特别是，对于一大类具有发散维度的非平稳时间序列，本研究将建立系统的经验过程理论和高斯近似理论。 如今，技术创新使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。我看到了对非平稳时间序列进行统计分析的巨大需求、机遇和挑战，无论是有没有函数形式。事实上，统计理论和方法应该随着数据的趋势而进步。然而，由于缺乏合适的统计和概率工具，目前还缺乏用于非平稳时间序列分析的统一的统计理论。我相信，从非线性系统角度提出的框架将为许多新兴科学学科的非平稳(函数)时间序列分析提供重要的理论和方法论基础。

The last two decades have witnessed an enormous increase in the need for analyzing data with complex temporal dynamics. In particular, two types of time series data structures are of drastically growing interests in both theory and practice. First, it is evident that the temporal dependence structures and marginal distributions of many time series data change both abruptly and smoothly over time. Second, for many long and densely observed time series data, it is mathematically elegant and beneficial to separate the time record into natural consecutive intervals and treat the data as functional time series. The proposed research is aimed at providing a systematic package of statistical methodologies and theory for the modelling, estimation, inference and prediction of the aforementioned two types of complex structured time series in a unified framework of nonlinear system representation. Methodologically, we will develop efficient and adaptive nonparametric procedures for the analysis of general classes of non-stationary and/or functional valued time series with rigorous mathematical justifications. These include but are not limited to robust, multiscale and adaptive abrupt change detection in non-stationary (functional) time series; efficient and robust simultaneous confidence bands for nonparametric regression of non-stationary time series; uniform inference of both sparsely and densely observed functional time series and efficient  simultaneous nonparametric inference of time-varying spectral densities. Theoretically, a systematic statistical theory for non-stationary time series with diverging dimensionality will be developed, which provides a mathematical foundation for most of the research topics mentioned above. In particular, systematic empirical process theory and Gaussian approximation theory will be established for a wide class of non-stationary time series with diverging dimensionalities in the proposed research. Nowadays, technological innovations have made it possible to collect 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 non-stationary time series with or without functional forms. Indeed, statistical theory and methodologies should progress with the trend of the data. However, a unified statistical theory for non-stationary time series analysis is still lacking due to the lack of appropriate statistical and probabilistic tools. I believe that the proposed framework from the nonlinear system point of view will provide an important theoretical and methodological basis for non-stationary (functional) time series analysis in many scientific disciplines.