Dimension reduction for multivariate response data

This article concerns the analysis of multivariate response data with multivariate regressors. Methods for reducing the dimensionality of response variables are developed, with the goal of preserving as much regression information as possible. We note parallels between this goal and the goal of sliced inverse regression, which intends to reduce the regressor dimension in a univariate regression while preserving as much regression information as possible. A detailed discussion is given for the case where the response is a curve measured at fixed points. The problem in this setting is to select basis functions for fitting an aggregate of curves. We propose that instead of focusing on goodness of fit, attention should be shifted to the problem of explaining the variation of the curves in terms of the regressor variables. A data-adaptive basis searching method based on dimension reduction theory is proposed. Simulation results and an application to a climatology problem are given.