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Adjusting for High-dimensional Covariates in Sparse Precision Matrix Estimation by ℓ1-Penalization
1Center for Applied Statistics and School of Statistics, Renmin University of China, No. 59 Zhongguancun Street, Haidian District, Beijing 100872, China and Department of Biostatistics and Epidemiology, University of Pennsylvania Perelman School of Medicine, Philadelphia, PA 19104-6021, USA.
Journal of Multivariate Analysis
|May 21, 2013
Summary
This study introduces a novel two-stage method for estimating sparse precision matrices in high-dimensional genomic data, accounting for covariate effects. The approach accurately identifies relevant covariates and improves precision matrix estimation, crucial for understanding complex genetic networks.
Area of Science:
- Computational Biology and Bioinformatics
- Statistical Genetics
- High-Dimensional Statistics
Background:
- Genetical genomic data analysis often involves high-dimensional sparse precision matrices.
- Covariates can influence the mean of random vectors, complicating precision matrix estimation.
- Accurate estimation of the precision matrix is vital for understanding conditional dependencies in Gaussian graphical models.
Purpose of the Study:
- To develop a robust two-stage estimation procedure for high-dimensional sparse precision matrices.
- To adjust for a large number of covariates affecting the mean of the random vector.
- To accurately estimate the zero pattern of the true precision matrix in complex datasets.
Main Methods:
- A two-stage estimation procedure is proposed.
- Stage 1: Joint ℓ1 penalization to identify relevant covariates affecting means.
- Stage 2: Estimation of the sparse precision matrix using ℓ1-penalized log-determinant Bregman divergence on a multivariate sub-Gaussian model.
Main Results:
- Consistent estimation of regression coefficients in element-wise ℓ∞, Frobenius, and spectral norms, even when dimensions (p, q) exceed sample size (n).
- High probability of correctly identifying the zero pattern of the true precision matrix.
- Simulations demonstrate improved precision matrix estimation compared to existing methods.
Conclusions:
- The proposed two-stage method effectively estimates high-dimensional sparse precision matrices while adjusting for covariates.
- The method shows strong theoretical guarantees and practical utility, as evidenced by simulations and application to yeast genetical genomic data.
- This approach enhances the analysis of complex biological networks by providing a more accurate estimation of conditional dependencies.
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