Group-wise sufficient dimension reduction with principal fitted components

Sufficient dimension reduction methodologies in regressions of Y on a p-variate X aim at obtaining a reduction \(R(X) \in {\mathbb R}^{d}, d \le p\), that retains all the regression information of Y in X. When the predictors fall naturally into a number of known groups or domains, it has been established that exploiting the grouping information often leads to more effective sufficient dimension reduction of the predictors. In this article, we consider group-wise sufficient dimension reduction based on principal fitted components, when the grouping information is unknown. Principal fitted components methodology is coupled with an agglomerative clustering procedure to identify a suitable grouping structure. Simulations and real data analysis demonstrate that the group-wise principal fitted components sufficient dimension reduction is superior to the standard principal fitted components and to general sufficient dimension reduction methods.