On Robust Gaussian Graphical Modeling

  • Daniel Vogel
  • Roland Fried


The objective of this exposition is to give an overview of the existing approaches to robust Gaussian graphical modeling. We start by thoroughly introducing Gaussian graphical models (also known as covariance selection models or concentration graph models) and then review the established, likelihood-based statistical theory (estimation, testing and model selection). Afterwards we describe robust methods and compare them to the classical approaches.


Partial Correlation Graphical Model Conditional Independence Robust Estimator Sample Covariance Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Baba, K., Shibata, R., Sibuya, M.: Partial correlation and conditional correlation as measures of conditional independence. Aust. N. Z. J. Stat. 46(4), 657–664 (2004) CrossRefMathSciNetMATHGoogle Scholar
  2. Becker, C.: Iterative proportional scaling based on a robust start estimator. In: Weihs, C., Gaul, W. (eds.) Classification—The Ubiquitous Challenge, pp. 248–255. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  3. Bilodeau, M., Brenner, D.: Theory of Multivariate Statistics. Springer Texts in Statistics. Springer, New York (1999) MATHGoogle Scholar
  4. Buhl, S.L.: On the existence of maximum likelihood estimators for graphical Gaussian models. Scand. J. Stat. 20(3), 263–270 (1993) MathSciNetMATHGoogle Scholar
  5. Butler, R.W., Davies, P.L., Jhun, M.: Asymptotics for the minimum covariance determinant estimator. Ann. Stat. 21(3), 1385–1400 (1993) CrossRefMathSciNetMATHGoogle Scholar
  6. Castelo, R., Roverato, A.: A robust procedure for Gaussian graphical model search from microarray data with p larger than n. J. Mach. Learn. Res. 7, 2621–2650 (2006) MathSciNetGoogle Scholar
  7. Cox, D.R., Wermuth, N.: Multivariate Dependencies: Models, Analysis and Interpretation. Monographs on Statistics and Applied Probability, vol. 67. Chapman and Hall, London (1996) MATHGoogle Scholar
  8. Croux, C., Haesbroeck, G.: Influence function and efficiency of the minimum covariance determinant scatter matrix estimator. J. Multivar. Anal. 71(2), 161–190 (1999) CrossRefMathSciNetMATHGoogle Scholar
  9. Davies, P.L.: Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices. Ann. Stat. 15, 1269–1292 (1987) CrossRefMATHGoogle Scholar
  10. Davies, P.L.: The asymptotics of Rousseeuw’s minimum volume ellipsoid estimator. Ann. Stat. 20(4), 1828–1843 (1992) CrossRefGoogle Scholar
  11. Davies, P.L., Gather, U.: The identification of multiple outliers. J. Am. Stat. Assoc. 88(423), 782–801 (1993) CrossRefMathSciNetMATHGoogle Scholar
  12. Davies, P.L., Gather, U.: Breakdown and groups. Ann. Stat. 33(3), 977–1035 (2005) CrossRefMathSciNetMATHGoogle Scholar
  13. Dawid, A.P., Lauritzen, S.L.: Hyper Markov laws in the statistical analysis of decomposable graphical models. Ann. Stat. 21(3), 1272–1317 (1993) CrossRefMathSciNetMATHGoogle Scholar
  14. Deming, W.E., Stephan, F.F.: On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Stat. 11, 427–444 (1940) CrossRefMathSciNetMATHGoogle Scholar
  15. Dempster, A.P.: Covariance selection. Biometrics 28, 157–175 (1972) CrossRefGoogle Scholar
  16. Donoho, D.L.: Breakdown properties of multivariate location estimators. PhD thesis, Harvard University (1982) Google Scholar
  17. Donoho, D.L., Huber, P.J.: The notion of breakdown point. In: Bickel, P.J., Doksum, K.A., Hodges, J.L. (eds.): Festschrift for Erich L. Lehmann, pp. 157–183. Wadsworth, Belmont (1983) Google Scholar
  18. Drton, M., Perlman, M.D.: Model selection for Gaussian concentration graphs. Biometrika 91(3), 591–602 (2004) CrossRefMathSciNetMATHGoogle Scholar
  19. Drton, M., Perlman, M.D.: A SINful approach to Gaussian graphical model selection. J. Stat. Plan. Inference 138(4), 1179–1200 (2008) CrossRefMathSciNetMATHGoogle Scholar
  20. Edwards, D.: Introduction to graphical modeling. Springer Texts in Statistics. Springer, New York (2000) Google Scholar
  21. Edwards, D., Havránek, T.: A fast procedure for model search in multidimensional contingency tables. Biometrika 72, 339–351 (1985) CrossRefMathSciNetMATHGoogle Scholar
  22. Edwards, D., Havránek, T.: A fast model selection procedure for large families of models. J. Am. Stat. Assoc. 82, 205–213 (1987) CrossRefMATHGoogle Scholar
  23. Eriksen, P.S.: Tests in covariance selection models. Scand. J. Stat. 23(3), 275–284 (1996) MathSciNetMATHGoogle Scholar
  24. Gervini, D.: The influence function of the Stahel–Donoho estimator of multivariate location and scatter. Stat. Probab. Lett. 60(4), 425–435 (2002) CrossRefMathSciNetMATHGoogle Scholar
  25. Gervini, D.: A robust and efficient adaptive reweighted estimator of multivariate location and scatter. J. Multivar. Anal. 84(1), 116–144 (2003) CrossRefMathSciNetMATHGoogle Scholar
  26. Gnanadesikan, R., Kettenring, J.R.: Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics 28(1), 81–124 (1972) CrossRefGoogle Scholar
  27. Gottard, A., Pacillo, S.: On the impact of contaminations in graphical Gaussian models. Stat. Methods Appl. 15(3), 343–354 (2007) CrossRefMathSciNetGoogle Scholar
  28. Gottard, A., Pacillo, S.: Robust concentration graph model selection. Comput. Stat. Data Anal. (2008). doi: 10.1016/j.csda.2008.11.021 Google Scholar
  29. Hampel, F.R.: A general qualitative definition of robustness. Ann. Math. Stat. 42, 1887–1896 (1971) CrossRefMathSciNetGoogle Scholar
  30. Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics. The Approach Based on Influence Functions. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1986) MATHGoogle Scholar
  31. Huber, P.J., Ronchetti, E.M.: Robust Statistics, 2nd edn. Wiley Series in Probability and Statistics. Wiley, Hoboken (2009) MATHGoogle Scholar
  32. Kent, J.T., Tyler, D.E.: Constrained M-estimation for multivariate location and scatter. Ann. Stat. 24(3), 1346–1370 (1996) CrossRefMathSciNetMATHGoogle Scholar
  33. Kuhnt, S., Becker, C.: Sensitivity of graphical modeling against contamination. In: Schader, Martin, et al. (eds.) Between Data Science and Applied Data Analysis, Proceedings of the 26th Annual Conference of the Gesellschaft für Klassifikation e. V., Mannheim, Germany, July 22–24, 2002, pp. 279–287. Springer, Berlin (2003) Google Scholar
  34. Lauritzen, S.L.: Graphical Models. Oxford Statistical Science Series, vol. 17. Oxford University Press, Oxford (1996) Google Scholar
  35. Lopuhaä, H.P.: On the relation between S-estimators and M-estimators of multivariate location and covariance. Ann. Stat. 17(4), 1662–1683 (1989) CrossRefMATHGoogle Scholar
  36. Lopuhaä, H.P.: Multivariate τ-estimators for location and scatter. Can. J. Stat. 19(3), 307–321 (1991) CrossRefMATHGoogle Scholar
  37. Lopuhaä, H.P.: Asymptotics of reweighted estimators of multivariate location and scatter. Ann. Stat. 27(5), 1638–1665 (1999) CrossRefMATHGoogle Scholar
  38. Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd edn. Wiley Series in Probability and Statistics. Wiley, Chichester (1999) MATHGoogle Scholar
  39. Maronna, R.A.: Robust M-estimators of multivariate location and scatter. Ann. Stat. 4, 51–67 (1976) CrossRefMathSciNetMATHGoogle Scholar
  40. Maronna, R.A., Yohai, V.J.: The behavior of the Stahel-Donoho robust multivariate estimator. J. Am. Stat. Assoc. 90(429), 330–341 (1995) CrossRefMathSciNetMATHGoogle Scholar
  41. Maronna, R.A., Zamar, R.H.: Robust estimates of location and dispersion for high-dimen sional datasets. Technometrics 44, 307–317 (2002) CrossRefMathSciNetGoogle Scholar
  42. Maronna, R.A., Martin, D.R., Yohai, V.J.: Robust Statistics: Theory and Methods. Wiley Series in Probability and Statistics. Wiley, Chichester (2006) CrossRefMATHGoogle Scholar
  43. Meinshausen, N., Bühlmann, P.: High-dimensional graphs and variable selection with the Lasso. Ann. Stat. 34(3), 1436–1462 (2006) CrossRefMATHGoogle Scholar
  44. Miyamura, M., Kano, Y.: Robust Gaussian graphical modeling. J. Multivar. Anal. 97(7), 1525–1550 (2006) CrossRefMathSciNetMATHGoogle Scholar
  45. Paindaveine, D.: A canonical definition of shape. Stat. Probab. Lett. 78(14), 2240–2247 (2008) CrossRefMathSciNetMATHGoogle Scholar
  46. Porteous, B.T.: Stochastic inequalities relating a class of log-likelihood ratio statistics to their asymptotic χ 2 distribution. Ann. Stat. 17(4), 1723–1734 (1989) CrossRefMathSciNetMATHGoogle Scholar
  47. Rocke, D.M.: Robustness properties of S-estimators of multivariate location and shape in high dimension. Ann. Stat. 24(3), 1327–1345 (1996) CrossRefMathSciNetMATHGoogle Scholar
  48. Rousseeuw, P.J.: Multivariate estimation with high breakdown point. In: Grossmann, W., Pflug, G.C., Vincze, I., Wertz, W. (eds.) Mathematical Statistics and Applications, Vol. B, Proc. 4th Pannonian Symp. Math. Stat., Bad Tatzmannsdorf, Austria, September 4–10, 1983, pp. 283–297. Reidel, Dordrecht (1985) Google Scholar
  49. Rousseeuw, P.J., Van Driessen, K.: A fast algorithm for the minimum covariance determinant estimator. Technometrics 41, 212–233 (1999) CrossRefGoogle Scholar
  50. Roverato, A.: Cholesky decomposition of a hyper inverse Wishart matrix. Biometrika 87(1), 99–112 (2000) CrossRefMathSciNetMATHGoogle Scholar
  51. Smith, P.W.F.: Assessing the power of model selection procedures used when graphical modeling. In: Dodge, Y., Whittaker, J. (eds.) Computational Statistics, vol. I, pp. 275–280. Physica, Heidelberg (1992) Google Scholar
  52. Speed, T.P., Kiiveri, H.T.: Gaussian Markov distributions over finite graphs. Ann. Stat. 14, 138–150 (1986) CrossRefMathSciNetMATHGoogle Scholar
  53. Srivastava, M., Khatri, C.: An Introduction to Multivariate Statistics. North Holland, New York (1979) MATHGoogle Scholar
  54. Stahel, W.: Robust estimation: Infinitesimal optimality and covariance matrix estimation. PhD thesis, ETH Zürich (1981) Google Scholar
  55. Tyler, D.E.: Robustness and efficiency properties of scatter matrices. Biometrika 70, 411–420 (1983) CrossRefMathSciNetMATHGoogle Scholar
  56. Tyler, D.E.: A distribution-free M-estimator of multivariate scatter. Ann. Stat. 15, 234–251 (1987) CrossRefMathSciNetMATHGoogle Scholar
  57. Visuri, S., Koivunen, V., Oja, H.: Sign and rank covariance matrices. J. Stat. Plan. Inference 91(2), 557–575 (2000) CrossRefMathSciNetMATHGoogle Scholar
  58. Vogel, D., Köllmann, C., Fried, R.: Partial correlation estimates based on signs. In Heikkonen, J. (ed.) Proceedings of the 1st Workshop on Information Theoretic Methods in Science and Engineering. TICSP series # 43 (2008) Google Scholar
  59. Vogel, D.: On generalizing Gaussian graphical models. In: Ciumara, R., Bădin, L. (eds.) Proceedings of the 16th European Young Statisticians Meeting, University of Bucharest, pp. 149–153 (2009) Google Scholar
  60. Whittaker, J.: Graphical Models in Applied Multivariate Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley, Chichester (1990) Google Scholar
  61. Zuo, Y.: Robust location and scatter estimators in multivariate analysis. In: Fan, J., Koul, H. (eds.) Frontiers in Statistics. Dedicated to Peter John Bickel on Honor of his 65th Birthday, pp. 467–490. Imperial College Press, London (2006) Google Scholar
  62. Zuo, Y., Cui, H.: Depth weighted scatter estimators. Ann. Stat. 33(1), 381–413 (2005) CrossRefMathSciNetMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Fakultät StatistikTechnische Universität DortmundDortmundGermany

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