Voxelwise encoding models with non-spherical multivariate normal priors

Predictive models for neural or fMRI data are often fit using regression methods that employ priors on the model parameters. One widely used method is ridge regression, which employs a spherical Gaussian prior that assumes equal and independent variance for all parameters. However, a spherical prior is not always optimal or appropriate. There are many cases where expert knowledge or hypotheses about the structure of the model parameters could be used to construct a better prior. In these cases, non-spherical Gaussian priors can be employed using a generalized form of ridge known as Tikhonov regression. Yet Tikhonov regression is only rarely used in neuroscience. In this paper we discuss the theoretical basis for Tikhonov regression, demonstrate a computationally efficient method for its application, and show several examples of how Tikhonov regression can improve predictive models for fMRI data. We also show that many earlier studies have implicitly used Tikhonov regression by linearly transforming the regressors before performing ridge regression.