Varying coefficient tensor models for brain imaging

We revisit a multidimensional varying-coefficient model (VCM), by allowing regressor coefficients to vary smoothly in more than one dimension, thereby extending the VCM of Hastie and Tibshirani. The motivating example is 3-dimensional, involving a special type of nuclear magnetic resonance measurement technique that is being used to estimate the diffusion tensor at each point in the human brain. We aim to improve the current state of the art, which is to apply a multiple regression model for each voxel separately using information from six or more volume images. We present a model, based on P-spline tensor products, to introduce spatial smoothness of the estimated diffusion tensor. Since the regression design matrix is space-invariant, a 4-dimensional tensor product model results, allowing more efficient computation with penalized array regression.