Multivariate Nonparametric Methodology Studies

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9971797This research project focuses on the development of nonparametric density and regression methodology in mid-range dimensions. Closely related applications such as clustering, mixture estimation, and dimension reduction are examined with a new point of view, relating locally adaptive and spatial estimation and recent extensions of nonparametric criteria to parametric problems. The new data-based parametric estimation algorithm, based upon integrated squared error, is investigated for its flexibility and robustness. An investigation of the dual solutions to the bandwidth choice problem for locally adaptive curve estimates continues, with the surprising finding that one solution is asymptotically a large constant. Three algorithms for finding interesting subspaces will be investigated. One measures the number of modes; a second finds maximal bias subspaces; and a third is a new least-normal criterion. Visualization work continues with the acquisition of an ImmersaDesk, which will allow improved implementations of algorithms such as the density grand tour. A somewhat new methodology is called Variable Clustering Analysis, which assists in semiparametric density estimation, data analysis, and interpretable dimension reduction. Also in the area of clustering, an algorithm for simplifying complex mixture models fitted by EM is developed, as is a new estimation and testing algorithms for the number of components. The project continues innovative work on spatial modeling and combining many data surveys into useful data modeling and conditional estimation of factors and their covariates in collaboration with researchers in the Department of Agriculture.Nonparametric methodology is widely used in one and two dimensions, but less so in higher dimensions. This research focuses on the mid-range dimensions and provides a deeper understanding of the implications to data modeling of the curse of dimensionality and problems associated with massive data sets. Particular emphasis is given to multivariate regression and density estimation problems, and closely related applications such as clustering, mixture estimation, and dimension reduction. Visualization is especially important when dealing with medium-dimensional data and the growing body of massive data sets. Of special interest is discovering and displaying data in visual clustering and visual discrimination applications. Rice University has acquired an ImmersaDesk, which will allow the implementation of recently developed algorithms in a virtual reality environment. At a recent National Research Council workshop, numerous scientists identified critical statistical needs in their work with massive data sets: alternatives to principal components, specialized visualization tools for exploring massive data, better clustering algorithms, and techniques for handling nonstationary data. Results from this research directly impact three of these four critical opportunities. This program represents a comprehensive and long-term attack on a host of important data analytic problems in multivariate estimation. Graduate training is significant component of this project. The results will be of long-term theoretical interest and will provide short-term solutions to real-world problems.