NETWORK EXPLORATION VIA THE ADAPTIVE LASSO AND SCAD PENALTIES

By Jianqing Fan, Yang Feng, Yichao Wu. Annals of Applied Statistics, 2009

434 citations recorded by the source corpus.

Graphical models are frequently used to explore networks, such as genetic networks, among a set of variables. This is usually carried out via exploring the sparsity of the precision matrix of the variables under consideration. Penalized likelihood methods are often used in such explorations. Yet, positive-definiteness constraints of precision matrices make the optimization problem challenging. We introduce non-concave penalties and the adaptive LASSO penalty to attenuate the bias problem in the network estimation. Through the local linear approximation to the non-concave penalty functions, the problem of precision matrix estimation is recast as a sequence of penalized likelihood problems with a weighted L1 penalty and solved using the efficient algorithm of Friedman et al. (2008). Our estimation schemes are applied to two real datasets. Simulation experiments and asymptotic theory are used to justify our proposed methods.

Subfields: Genetics, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ef5d4979a2234857084099bf3894901c331c330f

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Citations in the corpus

Covariance Estimation: The GLM and Regularization Perspectives (2011, methodology): “Fan, Huang and Li (2007) have studied a semiparametric model for a covariance structure by estimating the marginal variances via kernel smoothing and used specific parametric models for the correlation matrix such as the ARMA(1,1).”
Estimating time-varying networks (2008, methodology): “Meinshausen and Bühlmann (2006) proposed a procedure based on neighborhood selection of each node via the `1 penalized regression. Their procedure selects the model consistently under a set of suitable conditions, of which the most important one is commonly known in the literature as the irrepresent”
Sparsistency and Rates of Convergence in Large Covariance Matrix Estimation (2007, methodology): “We give a brief summary of a breast cancer data analysis with p n > n considered in Fan, Feng and Wu (2009).”
Generalized Thresholding of Large Covariance Matrices (2009, neutral): “0 instead of log p/n !”
A Projection-based Conditional Dependence Measure with Applications to High-dimensional Undirected Graphical Models. (2015, methodology): “We follow the implementation in Fan et al. (2009) for the three penalized likelihood estimators.”
The Graphical Horseshoe Estimator for Inverse Covariance Matrices (2017, methodology): “In a penalized likelihood framework, two of the most notable methods for inverse covariance estimation are the graphical lasso (Friedman, Hastie, and Tibshirani 2008) and the graphical SCAD (Fan, Feng, and Wu 2009).”
Change-point detection in high-dimensional covariance structure (2016, result): “A rather similar result is provided in paper [3] for adaptive versions of graphical lasso suggested and studied in [44] [17] [18] [43] .”
Regularized M-estimators with nonconvexity: statistical and algorithmic theory for local optima (2013, methodology): “The same framework is considered by Fan et al. [5].”