Estimating the Structural Dimension of Regressions

George Washington University, Washington DC

Investigators

Abstract

Abstract (DMS-0204563) PI: E. Bura Title: ESTIMATING THE STRUCTURAL DIMENSION OF REGRESSIONS Traditional approaches to reducing dimension in a regression context focus on modeling: a sequence of models are fitted and associated diagnostic tools that may lead to a reduction of the dimension of the predictors a posteriori are used in order to select the most reliable yet simplest model. A different approach is taken in this proposal where the goal is to investigate and develop methods of reducing the dimension of the regressor vector at the outset of the analysis without applying any a priori fitting or model selection procedures. The methods proposed in this project use inverse regression as a tool to estimate the structural dimension of a regression, which is defined to be the dimension of the subspace generated by a sufficient number of linear projections of the regressor vector. Furthermore, the case of intrinsically nonlinear relationship between response and regressors will be considered. A "translation" of the nonlinear manifold of the regressors to an equivalent linear subspace (equivalent in the sense that all information on the response is retained) is proposed. The research should make a significant contribution to the available inferential tools and software for analyzing high-dimensional data. The goals of the proposed research are (a) to extend and generalize existing parametric-based dimension reduction methodology to nonparametric techniques, (b) to apply the developed methods and devise new ones in order to fully estimate the structural dimension of a regression and not only a lower bound on the full dimension, (c) to generalize the methods so that they apply to nonlinear manifolds of the regressor vector, (d) to develop state-of-the-art implementation computer code, which will be freely available to the research community, and (e) to compare the proposed dimension reduction techniques to the existing ones using simulated and real world data sets. High-dimensional data have become increasingly common in practically all scientific and business related fields. Modeling such data is challenging, mostly because of our inability to visualize them. Consequently, it is practically imperative to reduce the dimension of the input data prior to any attempts at modeling. Dimension reduction is especially relevant and appealing in the era of cheap and easily available computing power and technology, where huge multivariate data sets are collected or continuously generated and accumulated. The proposed research will extend existing and develop new technology for reducing the complexity of the input data at the outset of the modeling process.