Partial inverse regression

Abstract

In regression with a vector of quantitative predictors, sufficient dimension reduction methods can effectively reduce the predictor dimension, while preserving full regression information and assuming no parametric model. However, all current reduction methods require the sample size <it>n</it> to be greater than the number of predictors <it>p</it>. It is well known that partial least squares can deal with problems with <it>n</it> < <it>p</it>. We first establish a link between partial least squares and sufficient dimension reduction. Motivated by this link, we then propose a new dimension reduction method, entitled partial inverse regression. We show that its sample estimator is consistent, and that its performance is similar to or superior to partial least squares when <it>n</it> < <it>p</it>, especially when the regression model is nonlinear or heteroscedastic. An example involving the spectroscopy analysis of biscuit dough is also given.