Resumen de New estimation methods for high dimensional inverse covariance matrices
Vahe Avagyan
The estimation of inverse covariance matrix (also known as precision matrix) is an important problem in various research fields and methodologies, especially in the current age of high-dimensional data abundance. In addition, the classical estimation methods are no longer stable and applicable in high dimensional settings, i.e., when the dimensionality has the same order as the sample size or is much larger. This thesis focuses on the estimation of the precision matrices as well as their applications. In particular, the goal of this thesis is to develop and analyse accurate precision matrix estimators for problems in high-dimensional settings. Moreover, the proposed precision matrix estimators should emulate the existing prominent estimators in terms of different statistical measures without being computationally more extensive. This thesis is comprised of two articles on estimation of precision matrices in high dimensional settings. In what follows, we summarize the main contributions of this thesis. First, we propose a simple improvement of the popular Graphical LASSO (GLASSO) framework that is able to attain better statistical performance without increasing signi cantly the computational cost. The proposed improvement is based on computing a root of the sample covariance matrix to reduce the spread of the associated eigenvalues. Through extensive numerical results, using both simulated and real datasets, we show that the proposed modiffication improves the GLASSO procedure. Our results reveal that the square-root improvement can be a reasonable choice in practice. Second, we introduce two adaptive extensions of the recently proposed l1 norm penalized D-trace loss minimization method. It is well known that the l1 norm penalization often fails to control the bias of the obtained estimator because of its overestimation behavior. Our proposed extensions are based on the adaptive and weighted adaptive thresholding operators and intend to diminish the bias produced by the l1 penalty term. We present the algorithm for solving our proposed approaches, which is based on the alternating direction method. Extensive numerical results, using both simulated and real datasets, show the advantage of our proposed estimators.
