Estimation and Inference in Functional Time Series Analysis

本文旨在拓展泛函时间序列分析(FTSA)的理论和应用。函数数据分析在20世纪90年代的S时期开始出现，早期的发展集中在简单的随机样本观察，可以框架为曲线或函数。然而，功能数据通常是通过将几乎连续的时间记录分成更小的段来顺序获得的。例如，可以对污染水平的高频记录进行分段，以形成一系列每日污染曲线。其他例子包括顺序观察的描述物理现象的功能，如在功能磁共振成像中，描述大脑中血流的功能是随着时间的推移而计算出来的。在这些情况下，简单随机样本的假设往往太过强烈，于是一个中心问题变成了如何在这样复杂的数据中解释和利用时间相关性。FTSA为解决这一问题提供了理论和方法。这项建议中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，并包含了关于序列内相关性的信息。到目前为止，关于长期协方差算子估计的理论和经验性质的研究还很少。下面提出了一种用于长期协方差的非参数估计的数据驱动的带宽选择过程，该过程弥合了FTSA中显著的方法学鸿沟。这解决了在基于主体的模型研究中经常出现的描述生物制剂如何相互作用的连续观察的汇总函数的分析中的困难，并提出了沿着这些路线的应用。*(2)区分结构突变和与功能时间序列的整合：*许多用于预测时间序列数据的方法依赖于平稳假设。在传统时间序列的情况下，对这一假设的检验在统计学和计量经济学文献中已经得到了彻底的研究，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，往往是因为趋势在样本内发生变化(结构突变)，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非平稳性的两个可能来源的文献。最近，函数时间序列的平稳性检验已经发展起来，但识别非平稳性的具体来源的方法还没有研究。拟议的研究最终形成了用功能时间序列数据区分结构突变和“单位根”的方法。**

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.**