The proposed research provides innovative methodology for the statistical inference for functional data with a focus on functional regressions and functional time series. The investigators will thereby extend the range of functional data analysis beyond the currently available models. This is relevant because the proposed models arise in a natural way in a number of applications such as engineering, geophysics, economics and finance. A characteristic common to many functional data applications is the presence of dependence. The theory of functional data, however, is as of today mainly focused on independent processes with notable exceptions being given by the functional autoregressive and linear processes. This proposal represents a comprehensive research plan for developing new estimation and model selection methods for different classes of functional linear and (novel) functional time series models. It contains the following parts: (1) develop practical algorithms for partitioning a sequence of functional data into subsequences of homogeneous curves, (2) develop methods for parameter estimation for general functional autoregressive processes of known orders, (3) develop automatic methods for selecting the order of a functional autoregressive process, (4) apply model selection techniques to select best fitting models in the fully functional linear model setting, (5) construct estimation and model selection methods for the novel functional autoregressive model with exogenous covariates, and (6) develop online monitoring procedure for both functional linear and autoregressive models. This will require the development of sophisticated new statistical methodology, requiring the refinement and extension of the theory of (vector-valued) Hilbert space-valued observations. The research will also include a significant innovative computational component. To aid the dissemination of results, the investigators plan to make the relevant software freely available via the Internet. Completion of the proposal will give statisticians and practitioners new tools for analyzing different forms of functional data.
The research in this proposal is interdisciplinary in nature, with applications in diverse fields such as engineering (floodplain management), geophysics (magnetic field readings of magnetometers), finance and economics (tick-by-tick transaction data). The research is therefore of immediate interest for practitioners and will further connect statistics and fields of science with a significant statistical component. It will also advance mathematical and computational statistics. The proposed research will produce doctoral students, among them female and minority students, theoretically and practically versed in both statistics and an area of application. The training and involvement of undergraduate students in this research is also included through regular coursework, independent study and projects.
