Han Liu
Abstract
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.
- Felix Abramovich, Yoav Benjamini, David L. Donoho, and Iain M. Johnstone. Adapting to unknown sparsity by controlling the false discovery rate. The Annals of Statistics, 34(2):584-653, 2006.Google Scholar
Cross Ref
- Onureena Banerjee, Laurent El Ghaoui, and Alexandre d'Aspremont. Model selection through sparse maximum likelihood estimation. Journal of Machine Learning Research, 9:485-516, March 2008. Google ScholarDigital Library
- Tony Cai, Cun-Hui Zhang, and Harrison H. Zhou. Optimal rates of convergence for covariance matrix estimation. Technical report, Wharton School, Statistics Department, University of Pennsylvania, 2008.Google Scholar
- Mathias Drton and Michael D. Perlman. Multiple testing and error control in Gaussian graphical model selection. Statistical Science, 22(3):430-449, 2007.Google ScholarCross Ref
- Mathias Drton and Michael D. Perlman. A SINful approach to Gaussian graphical model selection. Journal of Statistical Planning and Inference, 138(4):1179-1200, 2008.Google ScholarCross Ref
- Jerome H. Friedman, Trevor Hastie, and Robert Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3):432-441, 2007.Google Scholar
- Trevor Hastie and Robert Tibshirani. Generalized additive models. Chapman & Hall Ltd., 1999.Google Scholar
- Chris A. J. Klaassen and Jon A. Wellner. Efficient estimation in the bivariate normal copula model: Normal margins are least-favorable. Bernoulli, 3(1):55-77, 1997.Google ScholarCross Ref
- Colin L. Mallows, editor. The collected works of John W. Tukey. Volume VI: More mathematical, 1938-1984. Wadsworth & Brooks/Cole, 1990.Google ScholarCross Ref
- Nicolai Meinshausen and Peter Bühlmann. High dimensional graphs and variable selection with the Lasso. The Annals of Statistics, 34:1436-1462, 2006.Google ScholarCross Ref
- Pradeep Ravikumar, Han Liu, John Lafferty, and Larry Wasserman. SpAM: Sparse additive models. In Advances in Neural Information Processing Systems 20, pages 1201-1208. MIT Press, Cambridge, MA, 2008.Google Scholar
- Pradeep Ravikumar, John Lafferty, Han Liu, and Larry Wasserman. Sparse additive models. Journal of the Royal Statistical Society, Series B, Methodological, 2009a. To appear.Google ScholarCross Ref
- Pradeep Ravikumar, Martin Wainwright, Garvesh Raskutti, and Bin Yu. Model selection in Gaussian graphical models: High-dimensional consistency of l
1-regularized MLE. In Advances in Neural Information Processing Systems 22, Cambridge, MA, 2009b. MIT Press.Google ScholarDigital Library
- Adam J. Rothman, Peter J. Bickel, Elizaveta Levina, and Ji Zhu. Sparse permutation invariant covariance estimation. Electronic Journal of Statistics, 2:494-515, 2008.Google ScholarCross Ref
- Abe Sklar. Fonctions de répartition à n dimensions et leurs marges. Publications de l'Institut de Statistique de L'Université de Paris 8, pages 229-231, 1959.Google Scholar
- Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, Methodological, 58:267-288, 1996.Google Scholar
- Hideatsu Tsukahara. Semiparametric estimation in copula models. Canadian Journal of Statistics, 33:357-375, 2005.Google ScholarCross Ref
- Aad W. van der Vaart. Asymptotic Statistics. Cambridge University Press, 1998.Google Scholar
- Aad W. van der Vaart and Jon A. Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics. Springer-Verlag, 1996.Google Scholar
- AnjaWille et al. Sparse Gaussian graphical modelling of the isoprenoid gene network in Arabidopsis thaliana. Genome Biology, 5:R92, 2004.Google ScholarCross Ref
- Ming Yuan and Yi Lin. Model selection and estimation in the Gaussian graphical model. Biometrika, 94(1):19-35, 2007.Google Scholar
Index Terms
- The Nonparanormal: Semiparametric Estimation of High Dimensional Undirected Graphs
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