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.
The research funded by the NSF award DMS 0804165 was in the field of statistics, a science which studies data. Business and industry leaders, government workers, medical and engineering researchers have large amounts of data available from which they must draw informative and useful conclusions. Data sets currently available are too large for every single entry to be studied, so there is an obvious need to develop tools that allow to make decisions based on appropriate summaries. The data studied by the Principal Investigator are called functional data. A typical example is the record of the value of the Standard and Poor's index within one trading day. The stocks forming this index are traded so frequently (dozens of trades per second), that we can view the value of the index as existing at any time, not just at the times of trades. We can thus view it as a function defined over continuous time, hence the term functional data. For decision makers, it is important to understand the behavior of such functions, to be able to draw conclusions and develop regulatory tools. A recent crash of the market which was caused by automated massive sell off over several minutes, but crashed the confidence of investors for months to come illustrates the need to understand and detect early anomalous behavior of the functions that summarize the trading activity. Other examples of functional data include time records from probes placed in the brain, records of environmental data over many locations on the globe, time records of credit card transactions, etc. The research funded by this award developed tools to detect changes in the structure of such records, evaluate trends, and to verify if a postulated model for the data is correct. In the course of the research, four PhD and two MS students have been trained in broadly applicable areas of statistics. Statistical software was developed which is now available to the research community.
