Models of random sparse eigenmatrices matrices and Bayesian analysis of multivariate structure

TL;DR

This paper introduces probabilistic models for sparse eigenmatrices in covariance structures, utilizing Givens rotations, and applies Bayesian methods to analyze multivariate data, with implications for graphical models and genomics.

Contribution

It proposes a novel framework for modeling sparse eigenmatrices using Givens rotations and develops Bayesian analysis techniques for multivariate structures.

Findings

Effective identification of sparse multivariate structures

Enhanced interpretability of covariance models

Application to genomics data demonstrates practical utility

Abstract

We discuss probabilistic models of random covariance structures defined by distributions over sparse eigenmatrices. The decomposition of orthogonal matrices in terms of Givens rotations defines a natural, interpretable framework for defining distributions on sparsity structure of random eigenmatrices. We explore theoretical aspects and implications for conditional independence structures arising in multivariate Gaussian models, and discuss connections with sparse PCA, factor analysis and Gaussian graphical models. Methodology includes model-based exploratory data analysis and Bayesian analysis via reversible jump Markov chain Monte Carlo. A simulation study examines the ability to identify sparse multivariate structures compared to the benchmark graphical modelling approach. Extensions to multivariate normal mixture models with additional measurement errors move into the framework of latent structure analysis of broad practical interest. We explore the implications and utility of the new models with summaries of a detailed applied study of a 20-dimensional breast cancer genomics data set.