Groupwise dimension reduction

In many regression applications, the predictors fall naturally into a number of groups or domains, and it is often desirable to establish a domain-specific relation between the predictors and the response. In this article, we consider dimension reduction that incorporates such domain knowledge. The proposed method is based on the derivative of the conditional mean, where the differential operator is constrained to the form of a direct sum. This formulation also accommodates the situations where dimension reduction is focused only on part of the predictors; as such it extends Partial Dimension Reduction to cases where the blocked predictors are continuous. Through simulation and real data analyses, we show that the proposed method achieves greater accuracy and interpretability than the dimension reduction methods that ignore group information. Furthermore, the new method does not require the stringent conditions on the predictor distribution that are required by existing methods.