Estimation and Inference in Functional Time Series Analysis

函数时间序列分析中的估计和推理

• 批准号: RGPIN-2016-03723
• 负责人: Rice, Gregory
• 金额: $ 1.97万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=663304
• 关键词: Estimation; Inference; Functional; Time; Series

This proposal aims to extend the theory and applications of functional time series analysis (FTSA). Functional data analysis (FDA) came into prominence in the 1990's, and the early developments focused on simple random samples of observations that can be framed as curves or functions. Functional data are, however, often obtained sequentially by breaking nearly continuous time records into smaller segments. For example, high frequency records of pollution levels may be segmented to form a series of daily pollution curves. Other examples include sequentially observed functions that describe physical phenomena, as in functional magnetic resonance imaging, where functions describing blood flow in the brain are computed over time. The assumption of a simple random sample is often too strong in these cases, and a central issue then becomes how to account for and utilize temporal dependence in such complex data. FTSA provides theory and methodology for addressing this issue. The research outlined in this proposal expands the knowledge of FTSA in two primary directions:******(1) Estimation of the long run covariance operator:******A covariance object used in the study of functional time series is the long run covariance operator, which describes the second order behavior of the sample mean function and incorporates information about the dependence within the series. To date, the theoretical and empirical properties of estimators of the long run covariance operator have been only lightly investigated. A data driven bandwidth selection procedure for nonparametric estimators of the long run covariance is proposed below that bridges a significant methodological gap in FTSA. This addresses a difficulty in the analysis of sequentially observed summary functions that describe how biological agents interact with each other, as frequently arise in the study of agent based models, and applications along these lines are proposed.******(2) Differentiating between structural breaks and integration with functional time series:******Many methods used to forecast time series data rely on the assumption of stationarity. In case of traditional time series, testing this assumption has been thoroughly studied in the statistics and econometrics literature, with the most widely used tests belonging to the Dickey-Fuller and KPSS families. When trend stationarity is rejected, it is often because the trend changes within the sample (structural break), or the error process is itself non-stationary (integration), and a wealth of literature exists on differentiating between the two possible sources of non-stationarity. Recently, tests for stationarity with functional time series have been developed, however methods for identifying specific sources of non-stationarity remain unstudied. The proposed research culminates in methodology for differentiating between structural breaks and "unit roots" with functional time series data.**

本文旨在扩展功能时间序列分析（FTSA）的理论与应用。功能数据分析（FDA）在20世纪90年代开始崭露头角，早期的发展集中在简单的随机观察样本上，这些样本可以被框定为曲线或函数。然而，功能数据通常是通过将几乎连续的时间记录分解成更小的片段来顺序获得的。例如，污染水平的高频记录可以分割成一系列的日污染曲线。其他例子包括描述物理现象的顺序观察功能，如在功能磁共振成像中，描述大脑血液流动的功能随着时间的推移而计算。在这些情况下，简单随机样本的假设往往过于强烈，然后一个中心问题就变成了如何在如此复杂的数据中解释和利用时间依赖性。FTSA为解决这一问题提供了理论和方法。本文所概述的研究在两个主要方向上扩展了FTSA的知识：******(1)长期协方差算子的估计：******函数时间序列研究中使用的协方差对象是长期协方差算子，它描述了样本均值函数的二阶行为，并包含了序列内相关性的信息。迄今为止，长期协方差算子的估计量的理论和经验性质只进行了很少的研究。下面提出了一个数据驱动的带宽选择程序，用于长期协方差的非参数估计，它弥补了FTSA中重要的方法差距。这解决了在描述生物制剂如何相互作用的顺序观察总结函数的分析中的一个困难，正如在基于agent的模型的研究中经常出现的那样，并提出了沿着这些路线的应用。******(2)区分结构断裂与函数时间序列的整合：******用于预测时间序列数据的许多方法依赖于平稳性假设。在传统时间序列的情况下，检验这一假设已经在统计学和计量经济学文献中进行了深入研究，最广泛使用的检验属于Dickey-Fuller和KPSS家族。当趋势平稳性被拒绝时，通常是因为样本内的趋势发生了变化（结构断裂），或者误差过程本身是非平稳的（积分），并且存在大量关于区分两种可能的非平稳性来源的文献。最近，已经开发了功能时间序列的平稳性测试，但是识别非平稳性的具体来源的方法仍然没有研究。提出的研究最终在区分结构断裂和“单位根”与功能时间序列数据的方法。**