High Dimensional Semiparametric Gaussian Copula Graphical Models.

By Han Liu, Fang Han, Ming Yuan, John D. Lafferty, Larry A. Wasserman. 2012

402 citations recorded by the source corpus.

In this paper, we propose a semiparametric approach, named nonparanormal skeptic, for efficiently and robustly estimating high dimensional undirected graphical models. To achieve modeling flexibility, we consider Gaussian Copula graphical models (or the nonparanormal) as proposed by Liu et al. (2009). To achieve estimation robustness, we exploit nonparametric rank-based correlation coefficient estimators, including Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This celebrating result suggests that the Gaussian copula graphical models can be used as a safe replacement of the popular Gaussian graphical models, even when the data are truly Gaussian. Besides theoretical analysis, we also conduct thorough numerical simulations to compare different estimators for their graph recovery performance under both ideal and noisy settings. The proposed methods are then applied on a large-scale genomic dataset to illustrate their empirical usefulness. The R language software package huge implementing the proposed methods is available on the Comprehensive R Archive Network: this http URL r-project.org/.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Kernel‐based tests for joint independence (2016, methodology): “correlations (Székely and Rizzo, 2009, 2014), rank-based correlations (Bergsma and Dassios, 2014; Leung and Drton, 2016; Nandy et al., 2016) or also non- and semiparametric copula-based correlations (Liu et al., 2012; Xue and Zou, 2012; Wegkamp and Zhao, 2016; Gaißer et al., 2010). One of our motiva”
Sufficient Forecasting Using Factor Models (2015, methodology): “For example, Forni et al. (2000, 2015) studied factor estimation based on projection, Liu et al. (2012), Xue and Zou (2012) and Fan et al. (2015b) employed rank-based methods for robust estimation of the covariance matrices, and Connor et al. (2012) applied a weighted additive nonparametric…”
A Unified Theory of Confidence Regions and Testing for High-Dimensional Estimating Equations (2015, methodology): “In this subsection we consider the transelliptical graphical models (TGM), proposed by Liu et al. (2012b). We recall several definitions before we proceed.”
High-Dimensional Statistics with a View Toward Applications in Biology (2014, methodology): “Extensions for non-Gaussian continuous distributions, based on copula models, are given by Liu et al. (2012) and Xue and Zou (2012); this is exemplified in Section 4.3.1 below using the so-called nonparanormal transformation.”
Challenges of Big Data Analysis (2013, neutral): “However,ithasa greaterbalancebetweendiseasedandnon-diseased subjects.”
Finding and Leveraging Structure in Learning Problems (2012, background): “While fully non-parametric learning seems hopeless in high-dimensions, variants of additive models have been shown to be a useful compromise in many problems [20, 77, 130, 157].”
Multi-task Sparse Structure Learning with Gaussian Copula Models (2016, methodology): “…than simply Gaussian; and (ii) we employ a semiparametric Gaussian copula model to capture task relationship, which does not rely of Gaussian assumption on the marginals and have shown to be more robust to outliers (Liu et al., 2012), then traditional Gaussian model used in Rothman et al. (2010).”
High‐dimensional covariance matrix estimation (2019, neutral): “See also Liu et al. (2012)”
Regularized rank-based estimation of high-dimensional nonparanormal graphical models (2012, methodology): “A referee informed us in his/her review report that Liu et al. (2012) also independently used the rank-based correlation in the context of nonparametric Gaussian graphical model estimation.”
High Dimensional Semiparametric Gaussian Copula Graphical Models (2012, background): “The proof can be found in Theorem 4.1 of the long version of this paper; see Liu et al. (2012).”
A Model of Text for Experimentation in the Social Sciences (2016, methodology): “2014), we consider more nuanced procedures based on the literature on graph recovery (Blei and Lafferty 2007; Meinshausen and Bühlmann 2006; Liu et al. 2012).”
Variation in the human immune system is largely driven by non-heritable influences (2015, methodology): “To identify meaningful relationships between measurements, we used the recently developed non-paranormal SKEPTIC approach (Liu et al., 2012), with a transformed Spearman/rank correlation matrix as input.”
Multitask Quantile Regression under the Transnormal Model (2016, background): “…or quadratic discriminant analysis (Lin and Jeon 2003; Mai and Zou 2015; Fan et al. 2013), graphical modeling (Liu, Lafferty, and Wasserman 2009; Liu et al. 2012; Xue and Zou 2012), covariance matrix estimation (Xue and Zou 2014; Wegkamp and Zhao 2016), and principal component analysis (Han and…”
PC algorithm for nonparanormal graphical models (2013, background): “In particular,Liu et al. (2012a) show that if (X,Y) are bivariate normal with Corr(X,Y) = ρ, then P (∣ ∣ ∣ 2sin (π 6 ρ̂S ) −ρ ∣ ∣ ∣ > ε ) ≤ 2exp ( − 2 9π2 nε2 ) (4) and P (∣ ∣ ∣ sin (π 2 τ̂ ) −ρ ∣ ∣ ∣ > ε ) ≤ 2exp ( − 2 π2 nε2 ) .”
An Overview on the Estimation of Large Covariance and Precision Matrices (2015, neutral): “Liu et al. (2014a) introduced a covariance estimation method named EC2 (Estimation of Covariance with Eigenvalue Constraints).”