Partial Correlation Estimation by Joint Sparse Regression Models

Abstract

In this paper, we propose a computationally efficient approach —space(Sparse PArtial Correlation Estimation)— for selecting non-zero partial correlations under the high-dimension-low-sample-size setting. This method assumes the overall sparsity of the partial correlation matrix and employs sparse regression techniques for model fitting. We illustrate the performance of space by extensive simulation studies. It is shown that space performs well in both non-zero partial correlation selection and the identification of hub variables, and also outperforms two existing methods. We then apply space to a microarray breast cancer data set and identify a set of hub genes which may provide important insights on genetic regulatory networks. Finally, we prove that, under a set of suitable assumptions, the proposed procedure is asymptotically consistent in terms of model selection and parameter estimation.

Random matrix theory in statistics: A review

“…Lam and Fan (2009) compared different penalization and thresholding methods for sparse precision matrix estimation and derived rates of convergence under the Frobenius norm, while obtained rates of convergence of the l 1 -constrained estimator under the spectral norm. Indirect l 1 -type penalization schemes for determining the zero entries in the precision matrix have been proposed by Meinshausen and Bühlmann (2006) and Peng et al (2009).…”

Recent developments of constructing adjacency matrix in network analysis

“…Yuan and Lin (2007) discussed penalized maximum likelihood with a lasso penalty on inverse of covariance matrix. Peng et al (2009) suggested an algorithm called SPACE (Sparse PArtial Correlation Estimation) for selecting nonzero partial correlations and hub identification by the lasso in high dimensional setting. All the approaches mentioned above are called conditional dependency, and its corresponding graphical model is called a Markov network.…”

“…Since then, inspired by Dempster (1972), outstanding achievements have been made. Among them, Whittaker (1990), Edward (2000), Meinshausen and Buhlmann (2006), Yuan and Lin (2007), Peng et al (2009), Rothman et al (2010), Bien and Tibshirani (2011), Cai and Yuan (2012) and Chandrasekaran et al (2012) are often referred. Also, noticeable results on the distribution of the largest eigenvalue of the sample covariance matrix under various assumptions are done by Johnstone (2001Johnstone ( , 2008, Bickel and Levina (2008), Rothman et al (2009), Cai and Liu (2011), Cai and Zhou (2012 and Birnbaum et al (2013).…”

Integrated Network Analysis of Genetic and Epigenetic Factors in Glioblastoma Multiform

“…A co-mRNA expression or co-miRNA expression network can be constructed by joint sparse regression for estimating the concentration matrix in which off-diagonal elements represents the covariance between the corresponding variables given all other variables in the network (Peng, et al, 2009 ) . Sparse regression for reconstruction of co-expression network is briefly introduced here.…”