Parameter interpretation, regularization and source localization in multivariate linear models

Neuroimaging data are frequently analyzed with multivariate methods. Models expressing the data as a function of underlying factors related to the brain processes under study (signals) are called forward models, while models reversing this functional relationship are called backward models. Weigth vectors of backward models (called extraction filters) indicate the measurement channels informative with respect to isolating the signals. However, being a function of both signal and noise, significant weights may be observed at channels containing pure noise, while a proportion of signal-related channels may be given zero or insignificant weight. In contrast, forward model parameters (activation patterns) may exhibit significant weights only at signal-related channels, and are therefore interpretable with respect to the origin of the brain processes under study. It is sometimes incorrectly assumed that regularization (e.g., sparsification) of backward models makes extraction filters interpretable in the same sense. However, by transforming filters into patterns of corresponding forward models, as outlined here for the linear case, this can be indeed achieved. While these considerations hold for all types of data, the distinction between filters and patterns is particularly crucial for EEG and MEG data: only activation patterns can be localized to brain anatomy using customary inverse methods. We illustrate our theoretical results using a real EEG data example.