This is a three year research project which has as its purpose the study of computationally intensive methods in classification (pattern recognition) and regression (noisy prediction) where the data is high-dimensional and nonlinear. The research involves new approaches to binary tree structured methods, with the goal of significantly improving on the accuracy and flexiblity of the present, widely used tree structured methods initiated by the book and software presented by Breiman et. al. (1984). The advances in methodology will be useful in the many and diverse areas that make use of classification and prediction. Current areas of interest including speech recognition, image processing, medical diagnosis, and handwritten and printed charactor recognition. There will be five areas of concentration. Two of these are based on recent work by Breiman (1991) for efficiently fitting very high dimensional noisy data by continuously joined hyperplane segments. Similar algorithms will be used to construct trees by fitting hyperplanes in the nodes, and by using multivariate "ramp" functions to construct continuous approximations to prediction surfaces. The other three areas include a promising method for optimizing trees, making them multi-step optimal instead in one- step at present, hyperplane splitting for multiple response trees and a new method for doing penalized linear prediction. This latter method, called Bridge Regression, offers hope of providing uniformly better prediction than ordinary least squares.
