Ensemble sparse estimation of covariance structure for exploring genetic disease data

A novel robust estimation for high-dimensional precision matrices

In this paper we propose a new robust estimation of precision matrices for high-dimensional data when the number of variables is larger than the sample size. Different from the existing methods in literature, the proposed model combines the technique of modified Cholesky decomposition (MCD) with the robust generalized M-estimators. The MCD reparameterizes a precision matrix and transforms its estimation into solving a series of linear regressions, in which the commonly used robust techniques can be conveniently incorporated. Additionally, the proposed method adopts the model averaging idea to address the ordering issue in the MCD approach, resulting in an accurate estimation for precision matrices. Simulations and real data analysis are conducted to illustrate the merits of the proposed estimator.

Thresholding Approach for Low-Rank Correlation Matrix Based on MM Algorithm

Background: Low-rank approximation is used to interpret the features of a correlation matrix using visualization tools; however, a low-rank approximation may result in an estimation that is far from zero, even if the corresponding original value is zero. In such a case, the results lead to a misinterpretation. Methods: To overcome this, we propose a novel approach to estimate a sparse low-rank correlation matrix based on threshold values. We introduce a new cross-validation function to tune the corresponding threshold values. To calculate the value of a function, the MM algorithm is used to estimate the sparse low-rank correlation matrix, and a grid search was performed to select the threshold values. Results: Through numerical simulation, we found that the false positive rate (FPR), interpretability, and average relative error of the proposed method were superior to those of the tandem approach. For the application of microarray gene expression, the FPRs of the proposed approach with d=2,3 and 5 were 0.128, 0.139, and 0.197, respectively, while the FPR of the tandem approach was 0.285. Conclusions: We propose a novel approach to estimate sparse low-rank correlation matrices. The advantage of the proposed method is that it provides results that are interpretable using a heatmap, thereby avoiding result misinterpretations. We demonstrated the superiority of the proposed method through both numerical simulations and real examples.

A Cholesky-based sparse covariance estimation with an application to genes data

The modified Cholesky decomposition (MCD) is a powerful tool for estimating a covariance matrix. The regularization can be conveniently imposed on the linear regressions to encourage the sparsity in the estimated covariance matrix to accommodate the high-dimensional data. In this paper, we propose a Cholesky-based sparse ensemble estimate for covariance matrix by averaging a set of Cholesky factor estimates obtained from multiple variable orderings used in the MCD. The sparse estimation is enabled by encouraging the sparsity in the Cholesky factor. The theoretical consistent property is established under some regular conditions. The merits of the proposed method are illustrated through simulation and a maize genes data set.