Simultaneous Statistical Modeling of Several Large Covariance Matrices

Northern Illinois University, Dekalb IL

Investigators

Abstract

Abstract PI: Mohsen Pourahmadi, DMS-0307055 TITLE: Simultaneous Statistical Modeling of Several Large Covariance Matrices This research focuses on the development of a three-stage statistical model-fitting process consisting of model formulation, estimation and diagnostics for contemporaneous covariance matrices of large multivariate responses arising, for example, in business and economics, epidemiology, environmental monitoring and global change, biotechnology and manufacturing (quality control). Correlation and covariance matrices and their spectral (eigenvalue) decomposition provide the basis for all classical multivariate techniques. For temporal correlations, however, the Cholesky decomposition lies at the core of most time series techniques. A growing body of work in step-down procedures in multivariate statistics and multivariate stochastic volatility models implicitly rely on the Cholesky decomposition or view each long random vactor as a "time series", for a given arrangement of the variables. The proposed work intends to make explicit and reveal the full potential of using Cholesky decomposition or time series techniques in modeling contemporaneous covariance matrices. It includes developing parsimonious models, their estimates and asymptotic properties, their practical implementation and uses, with an eye to guaranteeing the positive-definiteness of the covariance matrices at each stage of an iterative procedure used to compute the maximum likelihood estimates of the parameters. A major difficulty of the implementation for high-dimensional unordered vectors is the large number of possible arrangements of the variables. The methods and tools to be employed include: generalized linear models, factor analysis and random effects models, time-series model fitting process, maximum likelihood and Bayesian estimation and numerical linear algebra. The proposed research has the potential of elevating the Cholesky decomposition as a bona fide tool for modeling temporal and contemporaneous correlations and hence connecting (unifying) disparate areas like time series analysis, factor analysis and linear structural models, graphical models and Bayesian covariance modeling. Advances in technology and data collection have enabled researchers to record measurements on many characteristics of systems over finer units of time. The research proposed is motivated by statistical problems arising in settings where large amounts of multivariate data are available and the focus is on prediction, control, classification, clustering and data mining. The reliability or error rates of these tasks invariably hinge on the precise estimation of correlations among many variables and better understanding of the dynamics of large covariance matrices. The goal of the proposal is to provide a systematic and efficient method for analyzing high-dimensional data through modeling of the relevant covariance matrices. The proposal intends to use classical and recent statistical estimation and large-scale computing. Results obtained by the proposed research will have applications in diverse areas outlined above, however, particular attention will be paid to their potential use in understanding intelligence in machines and brains. A student will participate in the proposed research during summers.