Abstract
This article proposes a novel Bayesian implementation of regression with multi-dimensional array (tensor) response on scalar covariates. The recent emergence of complex datasets in various disciplines presents a pressing need to devise regression models with a tensor valued response. This article considers one such application of detecting neuronal activation in fMRI experiments in presence of tensor valued brain images and scalar predictors. The overarching goal in this application is to identify spatial regions (voxels) of a brain activated by an external stimulus. In such and related applications, we propose to regress responses from all cells (or voxels in brain activation studies) together as a tensor response on scalar predictors, accounting for the structural information inherent in the tensor response. To estimate model parameters with proper cell specific shrinkage, we propose a novel multiway stick breaking shrinkage prior distribution on tensor structured regression coefficients, enabling identification of cells which are related to the predictors. The major novelty of this article lies in the theoretical study of the contraction properties for the proposed shrinkage prior in the tensor response regression when the number of cells grows faster than the sample size. Specifically, estimates of tensor regression coefficients are shown to be asymptotically concentrated around the true sparse tensor in L2-sense under mild assumptions. Various simulation studies and analysis of a brain activation data empirically verify desirable performance of the proposed model in terms of estimation and inference on cell-level parameters.
Highlights
Of late, neuroscience or related applications routinely encounter regression scenarios involving a multidimensional array or tensor structured response and scalar predictors
The BOLD measure tensor response at each time point is presumed to be associated with the task related predictors and it is of scientific interest to delineate the nature and regions of activation using a regression framework involving the tensor response and task related predictors
Since the major motivation of model development is drawn from the Functional MRI (fMRI) based brain activation study, the simulation study is performed on simulated datasets which closely mimic the real world fMRI data
Summary
Neuroscience or related applications routinely encounter regression scenarios involving a multidimensional array or tensor structured response and scalar predictors. The resulting data is a two-dimensional matrix where the readings are both spatially and temporally correlated These matrix responses are often regressed on a set of scalar predictors (e.g., if a subject is alcoholic or not) to identify their variation with the predictors. All these applications involve a response tensor Y t ∈ Rp1×···×pD and a vector of predictors xt ∈ Rm at time t, respectively, with an objective to understand the cells in Y t influenced by the changes in xt. The tensor response regression framework is motivated by the aforementioned neuroimaging studies, the proposed methodology applies to a variety of scientific applications, including chemometrics (Bro, 2006), psychometrics (Kiers and Mechelen, 2001) and relational data (Gerard and Hoff, 2015), among others, where tensor valued responses are collected routinely
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