Omnibus and change point tests for functional time series

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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.