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，而不是标量或向量，在时间或空间上顺序记录。功能观察是从高分辨率测量（物理，工程，金融）或从平滑不规则间隔的观察（生物学，心理测量学，环境科学）。在过去的十年里，出现了许多这种数据的模型。 与标量或向量时间序列不同，没有系统的方法可用于验证给定的模型是否适合数据，或者通过使用拟合度量在几个竞争模型中进行选择。 也没有方法来检查是否可以为整个功能记录假设一个单一的随机结构。 拟议的研究集中在两种类型的测试：1）综合测试，旨在检测偏离任何方向从一个指定的模型，和2）变点测试，旨在检测模型的变化，在一些未知的时间。 变点检验对于时间序列尤其重要，因为“条件”可能会随着时间而变化，并且假设一个模型用于整个实现可能会导致非常误导的推断。 实际的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中没有遇到的挑战。 虽然许多方法在直觉上看起来很吸引人，但那些可行的和最佳的方法是重点，并制定出重要的细节。 一个工具箱的测试和全面的方法验证的理论，模拟和一些applications.Recent进展，测量和数据存储技术的出现，导致功能的时间序列在许多科学和工程领域。函数时间序列由曲线或曲面的集合组成，而不是数字。 例如，在不确定性和高波动性的时期，监管机构和市场参与者关注的不是道琼斯或纳斯达克等经济指标指数的收盘日价值，而是曲线的日内演变，显示指数如何从一分钟到另一分钟的变化。了解典型的每日指数曲线是什么样子，有多少可以用规律的变化来解释，什么是不寻常的，需要采取行动是重要的实际问题。函数时间序列出现在许多其他领域，最显着的是在物理学，工程学，生物学，医学和环境科学。在后者中，显示每15分钟污染物浓度的每日曲线比每日最高值或平均值更能提供信息，后者在评估对公众的实际风险方面可能没有用处。该研究开发了用于检测偏离通常曲线模式的统计程序。特别强调的是放在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.