Foundations of Dimension Reduction and Graphics

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Regression analysis is the general area of study of how a response variable changes as one or more predictors are varied over their possible values. Regression is one of the most widely applied areas in statistical analysis, and is used for monitoring the performance of assembly lines, for determining the success or failure of social innovations, to predict the future outcomes based on passed data. Regression analysis has a long history, dating back at least 200 years. A myriad of methods for specific types of problems (e. g., problems in which the response is a survival time, or a binary variable) have been developed. The work proposed in this project looks at regression in a very general way. It is founded on asking two questions. First, how much can be learned about dependence through using graphs? And the second question: how far can one push regression methodology without making any limiting assumptions about the nature of the problem at hand? Over the last decade, substantial progress has been made on the first of these questions, summarized in two books, a theoretical summary of the area in Cook (1998a) and an applied approach to regression through graphics in Cook and Weisberg (1999a). Both theoretical and applied issues must be understood to develop methodology for regression based on graphics. The second question is important because all the existing methodology for regression through graphics is based on a few assumptions, generally concerning the distribution of the predictors. The methodology to be developed in this project will overcome the limitations that are imposed by making assumptions at the outset. In particular, the assumption that predictors must be at least approximately linearly related is not required. In addition, the method can be extended to qualitative predictors like factors.