Financial Signal Processing and Machine Learning

Chapter 6Approaches to High-Dimensional Covariance and Precision Matrix Estimations

Jianqing Fan¹, Yuan Liao² and Han Liu³

¹Bendheim Center for Finance, Princeton University, USA

²Department of Mathematics, University of Maryland, USA

³Department of Operations Research and Financial Engineering, Princeton University, USA

6.1 Introduction

Large covariance and precision (inverse covariance) matrix estimations have become fundamental problems in multivariate analysis that find applications in many fields, ranging from economics and finance to biology, social networks, and health sciences. When the dimension of the covariance matrix is large, the estimation problem is generally challenging. It is well-known that the sample covariance based on the observed data is singular when the dimension is larger than the sample size. In addition, the aggregation of a huge amount of estimation errors can make considerable adverse impacts on the estimation's accuracy. Therefore, estimating large covariance and precision matrices has attracted rapidly growing research attention in the past decade. Many regularized methods have been developed: see Bickel and Levina (2008), El Karoui (2008), Friedman et al. (2008), Fryzlewicz (2013), Han et al. (2012), Lam and Fan (2009), Ledoit and Wolf (2003), Pourahmadi (2013), Ravikumar et al., (2011b), Xue and Zou (2012), among others.

One of the commonly used approaches to estimating large matrices is to assume the covariance matrix to be sparse, that is, ...