On Robust Gaussian Graphical Modeling

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.

[1]  V. Yohai,et al.  Robust Statistics: Theory and Methods , 2006 .

[2]  T. W. Anderson.  An Introduction to Multivariate Statistical Analysis , 1959 .

[3]  Victor J. Yohai,et al.  The Behavior of the Stahel-Donoho Robust Multivariate Estimator , 1995 .

[4]  C. Croux,et al.  Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator , 1999 .

[5]  John Law,et al.  Robust Statistics—The Approach Based on Influence Functions , 1986 .

[6]  Gesellschaft für Klassifikation. Jahrestagung,et al.  Classification - the Ubiquitous Challenge, Proceedings of the 28th Annual Conference of the Gesellschaft für Klassifikation e.V., University of Dortmund, March 9-11, 2004 , 2005, GfKl.

[7]  Ruben H. Zamar,et al.  Robust Estimates of Location and Dispersion for High-Dimensional Datasets , 2002, Technometrics.

[8]  Y. Zuo.  Robust location and scatter estimators in multivariate analysis , 2006 .

[9]  Martin Schader,et al.  Between Data Science and Applied Data Analysis , 2003 .

[10]  R. Maronna.  Robust $M$-Estimators of Multivariate Location and Scatter , 1976 .

[11]  A. Roverato.  Cholesky decomposition of a hyper inverse Wishart matrix , 2000 .

[12]  David E. Tyler.  A Distribution-Free $M$-Estimator of Multivariate Scatter , 1987 .

[13]  Anna Gottard,et al.  Robust concentration graph model selection , 2010, Comput. Stat. Data Anal..

[14]  P. L. Davies,et al.  Breakdown and groups , 2005, math/0508497.

[15]  M. Bilodeau,et al.  Theory of multivariate statistics , 1999 .

[16]  Daniel Vogel.  On generalizing Gaussian graphical models , 2009 .

[17]  J. N. R. Jeffers,et al.  Graphical Models in Applied Multivariate Statistics. , 1990 .

[18]  Daniel Gervini,et al.  A robust and efficient adaptive reweighted estimator of multivariate location and scatter , 2003 .

[19]  D. Edwards,et al.  A fast model selection procedure for large families of models , 1987 .

[20]  R. Christensen.  Introduction to Graphical Modeling , 2001 .

[21]  Georg Ch. Pflug,et al.  Mathematical statistics and applications , 1985 .

[22]  W. Deming,et al.  On a Least Squares Adjustment of a Sampled Frequency Table When the Expected Marginal Totals are Known , 1940 .

[23]  Susan A. Murphy,et al.  Monographs on statistics and applied probability , 1990 .

[24]  Daniel Gervini.  The influence function of the Stahel-Donoho estimator of multivariate location and scatter , 2002 .

[25]  Anna Gottard,et al.  On the impact of contaminations in graphical Gaussian models , 2007, Stat. Methods Appl..

[26]  Y. Kano,et al.  Robust Gaussian graphical modeling , 2006 .

[27]  David M. Rocke.  Robustness properties of S-estimators of multivariate location and shape in high dimension , 1996 .

[28]  D. Edwards,et al.  A fast procedure for model search in multidimensional contingency tables , 1985 .

[29]  Kuldeep Kumar,et al.  Robust Statistics, 2nd edn , 2011 .

[30]  H. P. Lopuhaä.  ASYMPTOTICS OF REWEIGHTED ESTIMATORS OF MULTIVARIATE LOCATION AND SCATTER , 1999 .

[31]  Laurie Davies,et al.  The identification of multiple outliers , 1993 .

[32]  P. Rousseeuw,et al.  Wiley Series in Probability and Mathematical Statistics , 2005 .

[33]  J. Magnus,et al.  Matrix Differential Calculus with Applications in Statistics and Econometrics , 1991 .

[34]  F. Hampel.  A General Qualitative Definition of Robustness , 1971 .

[35]  Robert Castelo,et al.  A Robust Procedure For Gaussian Graphical Model Search From Microarray Data With p Larger Than n , 2006, J. Mach. Learn. Res..

[36]  P. Rousseeuw.  Multivariate estimation with high breakdown point , 1985 .

[37]  P. L. Davies,et al.  The asymptotics of Rousseeuw's minimum volume ellipsoid estimator , 1992 .

[38]  Davy Paindaveine,et al.  A canonical definition of shape , 2008 .

[39]  T. Speed,et al.  Gaussian Markov Distributions over Finite Graphs , 1986 .

[40]  Roland Fried,et al.  Partial correlation estimates based on signs , 2008 .

[41]  Werner A. Stahel,et al.  Robust Statistics: The Approach Based on Influence Functions , 1987 .

[42]  M. Jhun,et al.  Asymptotics for the minimum covariance determinant estimator , 1993 .

[43]  Hengjian Cui,et al.  Depth weighted scatter estimators , 2005 .

[44]  Poul Svante Eriksen.  Tests in covariance selection models , 1996 .

[45]  Søren Ladegaard Buhl.  On the Existence of Maximum Likelihood Estimators for Graphical Gaussian Models , 1993 .

[46]  D. Edwards.  Introduction to graphical modelling , 1995 .

[47]  A. Dawid,et al.  Hyper Markov Laws in the Statistical Analysis of Decomposable Graphical Models , 1993 .

[48]  H. Oja,et al.  Sign and rank covariance matrices , 2000 .

[49]  P. Rousseeuw,et al.  A fast algorithm for the minimum covariance determinant estimator , 1999 .

[50]  Peter J. Smith.  Assessing the Power of Model Selection Procedures used when Graphical Modelling , 1992 .

[51]  Claudia Becker.  Iterative Proportional Scaling Based on a Robust Start Estimator , 2004, GfKl.

[52]  N. Meinshausen,et al.  High-dimensional graphs and variable selection with the Lasso , 2006, math/0608017.

[53]  Mathias Drton,et al.  A SINful approach to Gaussian graphical model selection , 2005 .

[54]  David E. Tyler.  Robustness and efficiency properties of scatter matrices , 1983 .

[55]  M. Drton,et al.  Model selection for Gaussian concentration graphs , 2004 .

[56]  P. L. Davies,et al.  Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices , 1987 .

[57]  D L Streiner,et al.  An Introduction to Multivariate Statistics , 1993, Canadian journal of psychiatry. Revue canadienne de psychiatrie.

[58]  Sonja Kuhnt,et al.  Sensitivity of graphical modeling against contamination , 2002 .

[59]  H. P. Lopuhaä.  Multivariate τ‐estimators for location and scatter , 1991 .

[60]  Hendrik P. Lopuhaä.  Multivariate τ-Estimators for Location and Scatter@@@Multivariate t-Estimators for Location and Scatter , 1991 .

[61]  Bell Telephone,et al.  ROBUST ESTIMATES, RESIDUALS, AND OUTLIER DETECTION WITH MULTIRESPONSE DATA , 1972 .

[62]  Jianqing Fan,et al.  Frontiers in Statistics , 2006 .

[63]  Nanny Wermuth,et al.  Multivariate Dependencies: Models, Analysis and Interpretation , 1996 .

[64]  R. Shibata,et al.  PARTIAL CORRELATION AND CONDITIONAL CORRELATION AS MEASURES OF CONDITIONAL INDEPENDENCE , 2004 .

[65]  H. P. Lopuhaä.  On the relation between S-estimators and M-estimators of multivariate location and covariance , 1989 .

[66]  David E. Tyler,et al.  Constrained M-estimation for multivariate location and scatter , 1996 .

[67]  B. T. Porteous.  Stochastic Inequalities Relating a Class of Log-Likelihood Ratio Statistics to their Asymptotic $\chi^2$ Distribution , 1989 .

[68]  Steffen L. Lauritzen,et al.  Graphical models in R , 1996 .