Statistical Models, Inference, and Computation for Multidimensional Time Series Data

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It is now commonplace for data to be collected over time across multiple (often many) sources. Examples include the time signals across multiple brain regions arising from fMRI, ocean wave height series across multiple spatial locations collected from buoys or satellites, and the multiple economic indicators (GPD, unemployment, and so on) gathered over time by government agencies and other parties. Available techniques often either neglect temporal dependencies for such high-dimensional data arising from multiple sources or do not apply to situations when the number of sources is large. This research project aims to develop novel statistical modeling tools that can capture adequately both the temporal features of such data and also their dependencies across multiple sources. Such tools have the potential to greatly enhance knowledge gained from such data. With fMRI data, for example, proper accounting for temporal dependence and large number of brain regions may facilitate better distinction among various clinical categories (ADHD, autism, etc.). Understanding the temporal and spatial dependencies in wave height data can lead to better predictions of storm activity across the oceans, and further insight into economic activity is expected from improved 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 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 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.