High-Dimensional Covariance Estimation

With High-Dimensional Data

Mohsen Pourahmadi

• Publisher: Wiley
• Year: 2013
• ISBN:
  9781118573662
• Edition: 1
• Topic: Mathématiques
• Subtopic: Probabilités et statistiques

Methods for estimating sparse and large covariance matrices

Covariance and correlation matrices play fundamental roles in every aspect of the analysis of multivariate data collected from a variety of fields including business and economics, health care, engineering, and environmental and physical sciences. High-Dimensional Covariance Estimation provides accessible and comprehensive coverage of the classical and modern approaches for estimating covariance matrices as well as their applications to the rapidly developing areas lying at the intersection of statistics and machine learning.

Recently, the classical sample covariance methodologies have been modified and improved upon to meet the needs of statisticians and researchers dealing with large correlated datasets. High-Dimensional Covariance Estimation focuses on the methodologies based on shrinkage, thresholding, and penalized likelihood with applications to Gaussian graphical models, prediction, and mean-variance portfolio management. The book relies heavily on regression-based ideas and interpretations to connect and unify many existing methods and algorithms for the task.

High-Dimensional Covariance Estimation features chapters on:

• Data, Sparsity, and Regularization
• Regularizing the Eigenstructure
• Banding, Tapering, and Thresholding
• Covariance Matrices
• Sparse Gaussian Graphical Models
• Multivariate Regression

The book is an ideal resource for researchers in statistics, mathematics, business and economics, computer sciences, and engineering, as well as a useful text or supplement for graduate-level courses in multivariate analysis, covariance estimation, statistical learning, and high-dimensional data analysis.

PART I

MOTIVATION AND THE BASICS

CHAPTER 1

INTRODUCTION

Is it possible to estimate a covariance matrix using the regression methodology? If so, then one may bring the vast machinery of regression analysis (regularized estimation, parametric and nonparametric methods, Bayesian analysis, …) developed in the last two centuries to the service of covariance modeling.

In this chapter, through several examples, we show that sparse estimation of high-dimensional covariance matrices can be reduced to solving a series of regularized regression problems. The examples include sparse principal component analysis (PCA), Gaussian graphical models, and the modified Cholesky decomposition of covariance matrices. The roles of sparsity, the least absolute shrinkage and smooth operator (Lasso) and particularly the soft-thresholding operator in estimating the parameters of linear regression models with a large number of predictors and large covariance matrices are reviewed briefly.

Nowadays, high-dimensional data are collected routinely in genomics, biomedical imaging, functional magnetic resonance imaging (fMRI), tomography, and finance. Let X be an n × p data matrix where n is the sample size and p is the number of variables. By the high-dimensional data usually it is meant that p is bigger than n. Analysis of high-dimensional data often poses challenges which calls for new statistical methodologies and theories (Donoho, 2000). For example, least-squares fitting of linear models and classical multivariate statistical methods cannot handle high-dimensional X since both rely on the inverse of X′X which could be singular or not well-conditioned. It should be noted that increasing n and p each has very different and opposite effects on the statistical results. In general, the focus of multivariate analysis is to make statistical inference about the dependence among variables so that increasing n has the effect of improving the precision and certainty of inference, whereas increasing p has the opposite effect of reducing the precision and certainty. Therefore the level of detail that can be inferred about correlations among variables improves with increasing n but it deteriorates with increasing p.

The dimension reduction and variable selection are of fundamental importance for high-dimensional data analysis. The sparsity principle which assumes that only a small number of predictors contribute to the response is frequently adopted as the guiding light in the analysis. Armed with the sparsity principle, a large number of estimation approaches are available to estimate sparse models and select the significant variables simultaneously. The Lasso method introduced by Tibshirani (1996) is one of the most prominent and popular estimation methods for the high-dimensional linear regression models.

Quantifying the interplay between high-dimensionality and sparsity is important in the modern data analysis environment. In the classic setup, usually p is fixed and n grows so that

However, in the modern high-dimensional setup where both p and n can grow, estimating accurately a vector β with p parameters is a real challenge. By invoking the sparsity principle one assumes that only a small number, say s(β), of the entries of β is nonzero and then proceeds in developing algorithms to estimate the nonzero parameters. Of course, it is desirable to establish the statist...

Table of contents

• Cover
• Series
• Title Page
• Copyright
• Preface
• Part I: Motivation and the Basics
• Part II: Covariance Estimation: Regularization
• Bibliography
• Index
• Wiley Series in Probability and Statistics