Multidimensional Penalized Signal Regression

We propose a general approach to regression on digitized multidimensional signals that can pose severe challenges to standard statistical methods. The main contribution of this work is to build a two-dimensional coefficient surface that allows for interaction across the indexing plane of the regressor array. We aim to use the estimated coefficient surface for reliable (scalar) prediction. We assume that the coefficients are smooth along both indices. We present a rather straightforward and rich extension of penalized signal regression using penalized B-spline tensor products, where appropriate difference penalties are placed on the rows and columns of the tensor product coefficients. Our methods are grounded in standard penalized regression, and thus cross-validation, effective dimension, and other diagnostics are accessible. Further, the model is easily transplanted into the generalized linear model framework. An illustrative example motivates our proposed methodology, and performance comparisons are made to other popular methods.