Advances in high-dimensional covariance matrix estimation

Abstract

Many applications require precise estimates of high-dimensional covariance matrices. The standard estimator is the sample covariance matrix, which is conceptually simple, fast to compute and has favorable properties in the limit of infinitely many observations. The picture changes when the dimensionality is of the same order as the number of observations. In such cases, the eigenvalues of the sample covariance matrix are highly biased, the condition number becomes large and the inversion of the matrix gets numerically unstable. A number of alternative estimators are superior in the high-dimensional setting, which include as subcategories structured estimators, regularized estimators and spectrum correction methods. In this thesis I contribute to all three areas. In the area of structured estimation, I focus on models with low intrinsic dimensionality. I analyze the bias in Factor Analysis, the state-of-the-art factor model and propose Directional Variance Adjustment (DVA) Factor Analysis, which reduces bias and yields improved estimates of the covariance matrix. Analytical shrinkage of Ledoit and Wolf (LW-Shrinkage) is the most popular regularized estimator. I contribute in three aspects: first, I provide a theoretical analysis of the behavior of LW-Shrinkage in the presence of pronounced eigendirections, a case of great practical relevance. I show that LW-Shrinkage does not perform well in this setting and propose aoc-Shrinkage which yields significant improvements. Second, I discuss the effect of autocorrelation on LW-Shrinkage and review the Sancetta-Estimator, an extension of LW-Shrinkage to autocorrelated data. I show that the Sancetta-Estimator is biased and propose a theoretically and empirically superior estimator with reduced bias. Third, I propose an extension of shrinkage to multiple shrinkage targets. Multi-Target Shrinkage is not restricted to covariance estimation and allows for many interesting applications which go beyond regularization, including transfer learning. I provide a detailed theoretical and empirical analysis. Spectrum correction approaches the problem of covariance estimation by improving the estimates of the eigenvalues of the sample covariance matrix. I discuss the state-of-the-art approach, Nonlinear Shrinkage, and propose a cross-validation based covariance (CVC) estimator which yields competitive performance at increased numerical stability and greatly reduced complexity and computational cost. On all data sets considered, CVC is on par or superior in comparison to the regularized and structured estimators. In the last chapter, I conclude with a discussion of the advantages and disadvantages of all covariance estimators presented in this thesis and give situation-specific recommendations. In addition, the appendix contains a systematic analysis of Linear Discriminant Analysis as a model application, which sheds light on the interdependency between the generative model of the data and various covariance estimators.