Generalized Linear Models for Large Correlation Matrices Via Partial Autocorrelations

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This research will focus on developing statistical models for large correlation matrices in the spirit of the generalized linear models using the partial correlation as the new unconstrained parameters. In particular, computationally efficient procedures will be developed for simulating random correlation matrices which are of great interest in simulation testing of new statistical methods and data mining algorithms, digital signal processing, and working correlation matrices in the analysis of longitudinal data. Large correlation matrices arise quite often in business and economics, epidemiology, environmental monitoring, biotechnology and spectroscopy where modern technological innovations have made it possible to collect massive amount of data with relatively low cost. The three major difficulties in modeling and simulating correlation matrices are (i) the positive-definiteness constraint, (ii) the high-dimensionality and (iii) the additional constraint that its diagonal entries must equal to one. While the Cholesky decomposition and other techniques can handle (i) and (ii), they are unable to handle (iii). The proposed research intends to reparameterize a correlation matrix in an unconstrained and statistically interpretable manner using the basic concept of partial correlation. Consequently, sparse and flexible statistical models, data analytic and graphical tools for correlation matrices will be developed in analogy with those commonly used in regression and time series analysis. The methods and tools to be employed include: the theory of generalized linear models, time series analysis, numerical linear algebra, theory of orthogonal polynomials and the Monte Carlo methods. The proposed work has the potential of elevating the basic concept of partial autocorrelation as a bona fide tool for modeling standard multivariate data in a manner similar to its well-established role in time series analysis, signal processing, the theory of orthogonal polynomials and graphical models. It has the added feature of connecting these apparently disparate areas, which brings out the interdisciplinary nature of the work. The focus on high-dimensional data analysis has immediate impacts on settings where large amounts of multivariate data are collected. Important examples of such settings are financial markets, environmental monitoring and global change, biotechnology and manufacturing. Graduate students will be involved in various phases of the project, the results will be incorporated in courses and presented in seminars and workshops accessible to researchers outside the field of statistics.