The Nonparanormal: Semiparametric Estimation of High Dimensional Undirected Graphs

By Han Liu, John D. Lafferty, Larry A. Wasserman. 2009

848 citations recorded by the source corpus.

Recent methods for estimating sparse undirected graphs for real-valued data in high dimensional problems rely heavily on the assumption of normality. We show how to use a semiparametric Gaussian copula---or "nonparanormal"---for high dimensional inference. Just as additive models extend linear models by replacing linear functions with a set of one-dimensional smooth functions, the nonparanormal extends the normal by transforming the variables by smooth functions. We derive a method for estimating the nonparanormal, study the method's theoretical properties, and show that it works well in many examples.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes (2019, methodology): “HYPERPARAMETER VALUE OR RANGE SEARCHED learning rate [1e-4, 1e-4, 1e-2] LSTM cells [10, 20, 40] LSTM layers 2 rank 10 num eval samples 400 conditioning length m 100 sampling dimension B 20 dropout 0.”
Analysis of elliptical copula correlation factor model with Kendall's tau (2013, background): “copulas, sometimes referred to in recent literature as the nonparanormal model [30], by choosing the particular characteristic generator φ(t) = exp(−t/2).”
The huge Package for High-dimensional Undirected Graph Estimation in R (2012, methodology): “…methods have been implemented, including the nonparanormal [Liu et al., 2009] method for estimating a high dimensional Gaussian copula graph, the StARS [Liu et al., 2010] approach for stability-based graphical model selection, and correlation screening [Fan and Lv, 2008] for graph estimation.”
Multi-task Sparse Structure Learning with Gaussian Copula Models (2016, background): “A more general formulation of the Gaussian copula is the semiparametric Gaussian copulas (Liu et al., 2009; Xue and Zou, 2012), which allows the marginals to follow any non-parametric distribution.”
High Dimensional Semiparametric Gaussian Copula Graphical Models. (2012, methodology): “…like the graphical lasso, graphical Dantzig selector, CLIME, and the Meinshausen-B¨uhlmann estimator, the only extra cost of the non-paranormal skeptic is the computation of S , which requires us to calculate the d ( d − 1) / 2 pairs of Spearman’s rho or Kendall’s tau statistics. b A naive…”
Estimating the effect of joint interventions from observational data in sparse high-dimensional settings (2014, positive): “In this section, we show that a simple modification of the joint-IDA estimators can be applied to nonparanormal distributions [12, 17], which form an interesting nonlinear and non-Gaussian generalization of linear Gaussian SEMs.”
Structure Learning in Graphical Modeling (2016, methodology): “In a diﬀerent vein, Liu et al. (2009) propose semiparametric methods based on Gaussian copula models.”
Transformation Autoregressive Networks (2018, methodology): “Another common semiparametric approach is to use a nonparanormal type model (Liu et al., 2009).”
Learning structured densities via infinite dimensional exponential families (2015, methodology): “To further show the strength of our algorithm, we test it on a nonparanormal (NPN) distribution ([18]).”
PC algorithm for nonparanormal graphical models (2013, methodology): “Phrased in the terminology of Liu et al. (2009), we considernonparanormaldistributions.”
High Dimensional Semiparametric Gaussian Copula Graphical Models (2012, methodology): “Under certain conditions, Liu et al. (2009) show that ‖Ω̂− Ω‖(2)2 = OP ( s · log d · n−1/2 ) .”
Learning Sparse Nonparametric DAGs (2019, background): “Ideally, the problem could be formulated as a conventional optimization problem that can be tackled with general purpose solvers, much like the current state-of-the-art for undirected graphical models (e.g. Suggala et al., 2017; Yang et al., 2015; Liu et al., 2009; Hsieh et al., 2013; Banerjee et al”
Kernel Embeddings of Latent Tree Graphical Models (2011, methodology): “The second existing approach we compare to is the Nonparanormal (NPN) [13] which assumes that there exist marginal transformations f1, .”
Sparse and Compositionally Robust Inference of Microbial Ecological Networks (2014, methodology): “In addition, nonparametric approaches, such as sparse additive models, can be used to “gaussianize” the data prior to network inference [45].”
Copulas in Machine Learning (2013, background): “Nonparanormal [26] 100-1000s MN Gaussian high-dimensional estimation Section 3.”