An overview of large-dimensional covariance and precision matrix estimators with applications in chemometrics

The covariance matrix (or its inverse, the precision matrix) is central to many chemometric techniques. Traditional sample estimators perform poorly for high-dimensional data such as metabolomics data. Because of this, many traditional inference techniques break down or produce unreliable results. In this paper, we selectively review several modern estimators of the covariance and precision matrix that improve upon the traditional sample estimator. We focus on 3 general techniques: eigenvalue-shrinkage estimation, ridge-type estimation, and structured estimation. These methods rely on different assumptions regarding the structure of the covariance or precision matrix. Various examples, in particular using metabolomics data, are used to compare these techniques and to demonstrate that in concert with, eg, principal component analysis, multivariate analysis of variance, and Gaussian graphical models, better results are obtained. We selectively review modern estimators of the covariance and precision matrix focusing on 3 general techniques, namely, eigenvalue-shrinkage estimation, ridge-type estimation, and structured estimation. These methods rely on different structural assumptions of the covariance or precision matrix. Various examples, in particular using metabolomics data, are used to compare these techniques and to demonstrate that in concert with, eg, principal component analysis, multivariate analysis of variance, and Gaussian graphical models, better results are obtained.