Rajarshi Guhaniyogi and Daniel Spencer
06/21/2020 10:40 PM
Statistics
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 applications, we propose to regress responses from all voxels together as a tensor response on scalar predictors, accounting for the structural information inherent in the tensor response. To estimate model parameters with proper voxel specific shrinkage, we propose a novel multiway stick breaking shrinkage prior distribution on tensor structured regression coefficients, enabling identification of voxels 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 voxels 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. Simulation studies and analysis of brain activation data empirically verify desirable performance of the proposed model in terms of estimation and inference on voxel-level parameters.
