Nonparametric statistical inference under complex temporal dynamics

复杂时间动态下的非参数统计推断

• 批准号: RGPIN-2015-04927
• 负责人: Zhou, Zhou
• 金额: $ 1.46万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677240
• 关键词: Nonparametric; statistical; inference; under; complex

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.**

在过去的二十年里，对分析具有复杂时间动态的数据的需求大幅增加。特别是，两种类型的时间序列数据结构在理论和实践中的兴趣急剧增长。首先，很明显，许多时间序列数据的时间相关结构和边缘分布随着时间的推移既突然又平稳地变化。其次，对于许多长时间和密集观测的时间序列数据，将时间记录分离为自然连续的间隔并将数据视为函数时间序列在数学上是优雅和有益的。* 拟议的研究旨在提供一套系统的统计方法和理论，以便在非线性系统表示的统一框架内，对上述两类复杂结构的时间序列进行建模、估计、推断和预测。在方法上，我们将开发有效的和自适应的非参数程序，用于分析具有严格数学理由的一般类别的非平稳和/或函数值时间序列。这些包括但不限于非平稳（功能）时间序列中的鲁棒、多尺度和自适应突变检测;非平稳时间序列的非参数回归的高效和鲁棒的同步置信带;稀疏和密集观察的功能时间序列的均匀推断以及时变谱密度的高效同步非参数推断。在理论上，将发展一个系统的具有发散维数的非平稳时间序列的统计理论，这为上述大多数研究课题提供了数学基础。特别是，系统的经验过程理论和高斯近似理论将建立广泛的一类非平稳时间序列与发散维度的拟议研究。* 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。我看到了一个巨大的需求，机会和挑战的非平稳时间序列的统计分析，或没有功能的形式。事实上，统计理论和方法应该随着数据的趋势而发展。然而，由于缺乏合适的统计和概率工具，非平稳时间序列分析仍然缺乏统一的统计理论。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非平稳（函数）时间序列分析提供重要的理论和方法基础。