Omnibus and change point tests for functional time series

函数时间序列由曲线或曲面的集合组成，而不是标量或矢量，按时间或空间顺序记录。功能观测是通过高分辨率测量（物理、工程、金融）或平滑不规则间隔观测（生物学、心理测量学、环境科学）获得的。在过去的十年中，出现了许多这类数据的模型。与标量或矢量时间序列不同，没有系统的方法可用于验证给定模型是否适合数据，或通过使用拟合度量在几个相互竞争的模型中进行选择。也没有方法来检查是否可以假设整个功能记录具有单一的随机结构。提出的研究重点是两种类型的测试：1)综合测试，旨在检测偏离指定模型的任何方向，2)变化点测试，旨在检测模型在某个未知时间的变化。变更点测试对于时间序列尤其重要，因为“条件”可能随时间而变化，并且假设整个实现的一个模型可能会导致非常误导的推断。实际实现是基于坚实的理论理解，这需要克服在建模标量时间序列时没有遇到的挑战。虽然许多方法在直觉上看起来很吸引人，但那些可行和最优的方法是重点关注的，并制定了重要的细节。一个工具箱的测试和综合方法验证了理论，模拟和一些应用开发。测量和数据存储技术的最新进展导致了函数时间序列在许多科学和工程领域的出现。函数时间序列由曲线或曲面的集合组成，而不是数字。例如，在不确定和高度波动的时期，监管机构和市场参与者关注的不是像道琼斯或纳斯达克这样的经济指标指数的收盘日价值，而是显示指数每分钟变化的曲线的日内演变。了解典型的每日指数曲线是什么样的，有多少可以用常规变化来解释，以及什么是不寻常的，需要采取行动，这些都是重要的实际问题。函数时间序列出现在许多其他领域，最显著的是在物理，工程，生物，医学和环境科学。在后一种情况下，每15分钟显示一种污染物浓度的日曲线比最大或平均日浓度提供的信息要多，这在评估对公众的实际风险时可能没有用处。该研究开发了统计程序来检测偏离通常的曲线模式。特别强调的是检测这些模式的突然变化。这些方法在很大程度上是自动化的，便于决策。

A functional time series consists of a collection of curves or surfaces, rather than scalars or vectors, recorded sequentially over time or space. Functional observations are obtained from high resolution measurements (physics, engineering, finance) or from smoothing irregularly spaced observations (biology, psychometrics, environmental science). The last decade has seen the emergence of a number of models for such data. Unlike for scalar or vector time series, no systematic methodology is available to verify if a given model is appropriate for the data, or for choosing among several competing models by using a measures of fit. No methodology to check if a single stochastic structure can be assumed for the whole functional record is available either. The proposed research focuses on two types of tests: 1) omnibus tests intended to detect departures in any direction from a specified model, and 2) change point tests designed to detect a model change at some unknown time. Change point tests are particularly important for time series, as "conditions" may change with time, and assuming one model for the whole realization may lead to very misleading inference. Practical implementations is based on solid theoretical understanding which requires overcoming challenges not encountered in modeling scalar time series. While many approaches seem intuitively appealing, those that are feasible and optimal are focused on, and nontrivial details are worked out. A tool box of tests and comprehensive methodology validated by theory, simulations and a number of applications is developed.Recent advances in measurement and data storage technology have led to the emergence of functional time series in many fields of science and engineering. A functional time series consists of a collection of curves or surfaces, rather than numbers. For example, rather than looking at a closing daily value of an economic indicator index like Dow Jones or NASDAQ, in times of uncertainty and high volatility, regulators and market participants focus on the intra-day evolution of the curve which shows how an index changes from minute to minute. Understanding how typical daily index curves look like, how much can be explained by regular variability, and what is unusual and requires action are important practical questions. Functional time series appear in many other fields, most notably in physics, engineering, biology, medicine and environmental science. In the latter, daily curves showing the concentration of a pollutant every 15 minutes are much more informative than a maximum or average daily values, which may not be useful in assessing the actual risk to the public. The research develops statistical procedures for detecting departures from a ususal pattern of curves. Special emphasis is placed on detecting a sudden change in these patterns. The methods are automated to a large degree and facilitate decision making.