On Variable Ordination of Modified Cholesky Decomposition for Sparse Covariance Matrix Estimation

TL;DR

This paper introduces a variable order ensemble approach for sparse covariance matrix estimation using the modified Cholesky decomposition, addressing the order dependency issue and ensuring positive definiteness.

Contribution

It proposes a novel ensemble method that combines multiple variable order estimates to improve sparse covariance estimation, with proven convergence properties.

Findings

The method effectively captures sparse structures in covariance matrices.

It guarantees positive definiteness of the estimated matrices.

Simulation and real data demonstrate improved performance.

Abstract

Estimation of large sparse covariance matrices is of great importance for statistical analysis, especially in the high-dimensional settings. The traditional approach such as the sample covariance matrix performs poorly due to the high dimensionality. The modified Cholesky decomposition (MCD) is a commonly used method for sparse covariance matrix estimation. However, the MCD method relies on the order of variables, which is often not available or cannot be pre-determined in practice. In this work, we solve this order issue by obtaining a set of covariance matrix estimates under different orders of variables used in the MCD. Then we consider an ensemble estimator as the "center" of such a set of covariance matrix estimates with respect to the Frobenius norm. The proposed method not only ensures the estimator to be positive definite, but also can capture the underlying sparse structure of the covariance matrix. Under some weak regularity conditions, we establish both algorithmic convergence and asymptotical convergence of the proposed method. The merits of the proposed method are illustrated through simulation studies and one real data example.