  A novel general framework is proposed in this paper for dimension reduction
in regression to fill the gap between linear and fully nonlinear dimension
reduction. The main idea is to transform first each of the raw predictors
monotonically, and then search for a low-dimensional projection in the space
defined by the transformed variables. Both user-specified and data-driven
transformations are suggested. In each case, the methodology is discussed first
in a general manner, and a representative method, as an example, is then
proposed and evaluated by simulation. The proposed methods are applied to a
real data set for illustration.
