Gaussian graphical model estimation with false discovery rate control

This paper studies the estimation of high dimensional Gaussian graphical model (GGM). Typically, the existing methods depend on regularization techniques. As a result, it is necessary to choose the regularized parameter. However, the precise relationship between the regularized parameter and the number of false edges in GGM estimation is unclear. Hence, it is impossible to evaluate their performance rigorously. In this paper, we propose an alternative method by a multiple testing procedure. Based on our new test statistics for conditional dependence, we propose a simultaneous testing procedure for conditional dependence in GGM. Our method can control the false discovery rate (FDR) asymptotically. The numerical performance of the proposed method shows that our method works quite well.

[1]  S. Berman.  A Law of Large Numbers for the Maximum in a Stationary Gaussian Sequence , 1962 .

[2]  T. W. Anderson.  An Introduction to Multivariate Statistical Analysis, 2nd Edition. , 1985 .

[3]  A. Zaitsev,et al.  On the gaussian approximation of convolutions under multidimensional analogues of S.N. Bernstein's inequality conditions , 1987 .

[4]  Y. Benjamini,et al.  Controlling the false discovery rate: a practical and powerful approach to multiple testing , 1995 .

[5]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[6]  Y. Benjamini,et al.  THE CONTROL OF THE FALSE DISCOVERY RATE IN MULTIPLE TESTING UNDER DEPENDENCY , 2001 .

[7]  M. Drton,et al.  Model selection for Gaussian concentration graphs , 2004 .

[8]  N. Meinshausen,et al.  High-dimensional graphs and variable selection with the Lasso , 2006, math/0608017.

[9]  M. Yuan,et al.  Model selection and estimation in the Gaussian graphical model , 2007 .

[10]  Terence Tao,et al.  The Dantzig selector: Statistical estimation when P is much larger than n , 2005, math/0506081.

[11]  Alexandre d'Aspremont,et al.  Model Selection Through Sparse Maximum Likelihood Estimation , 2007, ArXiv.

[12]  R. Tibshirani,et al.  Sparse inverse covariance estimation with the graphical lasso. , 2008, Biostatistics.

[13]  Alexandre d'Aspremont,et al.  First-Order Methods for Sparse Covariance Selection , 2006, SIAM J. Matrix Anal. Appl..

[14]  Adam J. Rothman,et al.  Sparse permutation invariant covariance estimation , 2008, 0801.4837.

[15]  M. Wainwright,et al.  HIGH-DIMENSIONAL COVARIANCE ESTIMATION BY MINIMIZING l1-PENALIZED LOG-DETERMINANT DIVERGENCE BY PRADEEP RAVIKUMAR , 2009 .

[16]  P. Bickel,et al.  SIMULTANEOUS ANALYSIS OF LASSO AND DANTZIG SELECTOR , 2008, 0801.1095.

[17]  Jianqing Fan,et al.  NETWORK EXPLORATION VIA THE ADAPTIVE LASSO AND SCAD PENALTIES. , 2009, The annals of applied statistics.

[18]  Cun-Hui Zhang,et al.  Confidence intervals for low dimensional parameters in high dimensional linear models , 2011, 1110.2563.

[19]  Cun-Hui Zhang,et al.  Scaled sparse linear regression , 2011, 1104.4595.

[20]  Cun-Hui Zhang,et al.  Confidence Intervals for Low-Dimensional Parameters With High-Dimensional Data , 2011 .

[21]  T. Cai,et al.  A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation , 2011, 1102.2233.

[22]  A. Belloni,et al.  Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming , 2011 .

[23]  H. Zou,et al.  Regularized rank-based estimation of high-dimensional nonparanormal graphical models , 2012, 1302.3082.

[24]  Peter Buhlmann.  Statistical significance in high-dimensional linear models , 2012, 1202.1377.

[25]  Larry A. Wasserman,et al.  High Dimensional Semiparametric Gaussian Copula Graphical Models. , 2012, ICML 2012.

[26]  T. Cai,et al.  Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings , 2013 .

[27]  S. Geer,et al.  On asymptotically optimal confidence regions and tests for high-dimensional models , 2013, 1303.0518.

[28]  Adel Javanmard,et al.  Hypothesis Testing in High-Dimensional Regression Under the Gaussian Random Design Model: Asymptotic Theory , 2013, IEEE Transactions on Information Theory.