The investigator plans to conduct research on identifying structure in multivariate data without imposing rigid structural assumptions and on the development of structural adaptation techniques for dimensionality reduction in nonparametric regression and density estimation. Building on recent advances in nonparametric and semiparametric estimation, it is proposed to develop iterative algorithms which alternate between identification of the lower dimensional structure, using estimates of average derivative functionals, and model estimation improved by using the current structural information. Identification and estimation of linear and nonlinear components in partially linear regression models and of independent component analysis model with unspecified component densities will be considered.
With the dramatic increase in large, complex data bases and in computer power, it has become increasingly more desirable and possible to develop nonparametric models, concepts, and procedures that can be used to study relationships between variables and to construct models without relying on rigid parametric assumptions on the structure of mean responses and error distributions. Algorithms developed in this research will provide new statistical learning tools for reducing dimensionality of multivariate data in order to identify and visualize its structure. After the algorithms are investigated on simulated data and their parameters are fine-tuned, they will be applied to statistical learning problems from bioinformatics, risk management, and other areas.
