Numerous data are nowadays collected over time and across multiple, often large number of sources, for example, the BOLD time signals across multiple brain regions arising from fMRI, the ocean wave height series across multiple spatial locations collected from buoys or satellites, or the multiple economic indicators (GPD, unemployment, and so on) gathered over time by the government agencies or private entities. Under the project, novel statistical modeling tools will be developed that can capture adequately both the temporal features of such data and also their dependencies across multiple sources. Available techniques often either neglect temporal dependencies for such high-dimensional data arising from multiple sources, or do not apply to the situations when the number of sources is large. With the fMRI data, for example, proper accounting for temporal dependence and large number of brain regions will lead to better distinction among various clinical categories (ADHD, autism or other). Understanding the temporal and spatial dependencies in wave height data will lead to better predictions of storm activity across the oceans. Further insight into economic activity is expected from the analysis of multiple economic indicators.
The project aims at developing an integrated approach to analyzing large multidimensional time series data, including their statistical models, estimation, computation (algorithms), and practice. The proposed research covers both short-range and long-range dependent multidimensional time series. For short-range dependent series, the focus is on sparse vector autoregressive and related models, dimension reduction, change point detection and some nonlinear models. The problems to be addressed concern regularization techniques, statistical significance, models exhibiting cyclical variations and other issues. Multidimensional long-range dependence is suggested as the important class complementing vector autoregressive and related short-range dependent series, thus gathering the two general classes of models employed in modern time series analysis. The goal is to develop a new methodology for multidimensional long-range dependent series with the so-called general phase, which controls the (a)symmetry properties of multidimensional time series, in both linear and nonlinear settings. The developed methods should be useful across a wide range of areas, including Neuroscience, Oceanography and Environmental Sciences, Geophysics, Economics and Finance, and others.