Theory and Methods for Dependent Tensor Data

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Modern scientific studies often gather data under combinations of multiple factors. For example, neuro imaging experiments record brain activity at multiple spatial locations, at multiple time points, and under a variety of experimental stimuli. Studies of social networks record social links of a variety of types from multiple initiators of social activity to multiple receivers of the activity. Data such as these are naturally represented not as lists or tables of numbers, but as multi-indexed arrays, or tensors. However, few tools are available for the statistical analysis of such data, and as a result, scientists frequently analyze tensor data using methods that ignore the tensor structure of the data. This can lead to inefficient use of data, and important patterns in the data being overlooked. This project will remedy this situation by developing usable, practical statistical tools for the analysis of tensor data. Currently available tools for statistical inference and parameter estimation are generally based on least-squares criteria and the assumption of residual independence. Such limitations can lead to highly sub-optimal inference: In general, great improvements in estimator performance can be obtained by appropriately accounting for residual dependence. Additionally, estimation of high-dimensional datasets can often be greatly improved by employing shrinkage techniques, such as empirical Bayes methods, that are based on variance models for the parameters. In this project, we will first develop theory and methods for covariance modeling of tensor-valued data. These methods will lead to the development of Bayes and empirical Bayes methods, from which we will develop a general data analysis framework for tensor data of a variety of types.