Elliptical graphical modelling

By Daniel Vogel, Roland Fried. 2011

53 citations recorded by the source corpus.

We propose elliptical graphical models based on conditional uncorrelatedness as a gen- eralization of Gaussian graphical models by letting the population distribution be elliptical instead of normal, allowing the tting of data with arbitrarily heavy tails. We study the class of propor- tionally ane equivariant scatter estimators and show how they can be used to perform elliptical graphical modelling, leading to a new class of partial correlation estimators and analogues of the classical deviance test. General expressions for the asymptotic variance of partial correlation esti- mators, unconstrained and under decomposable models, are given, and the asymptotic chi square approximation of the pseudo-deviance test statistic is proved. The feasibility of our approach is demonstrated by a simulation study, using, among others, Tyler's scatter estimator, which is distribution-free within the elliptical model. Our approach provides a robustication of Gauss- ian graphical modelling. The latter is likelihood-based and known to be very sensitive to model misspecication and outlying observations. 1. Introduction and notation The statistical theory of undirected graphical models for continuous variables is usually based on the assumption of multivariate normality. In practice, data may deviate from the normal model in various ways. Outliers and heavy tails pose a problem of particular gravity: they frequently occur, and the normal likelihood methods, such as the sample covariance matrix, are very susceptible to them. Our objective is to deal with heavy-tailed data and to safeguard graphical modelling against the impact of faulty outliers. We restrict our attention to the case where we have only continuous variables and only undirected edges. Joint multivariate normality is often assumed in this situation, and the statistical methodology is called Gaussian graphical modelling. We propose the class of elliptical distributions as a more general model and call our approach elliptical graphical modelling. The lack of robustness of Gaussian graphical modelling has been noted by several authors. Four proposals of robust approaches to Gaussian graphical modelling are known to us: Becker (2005) and Gottard and Pacillo (2010) suggest replacing the sample covariance matrix by the reweighted minimum covariance determinant estimator. Miyamura and Kano (2006) propose an alternative M-type estimation, and Finegold & Drton (arXiv:1009.3669) consider robustied versions of the graphical lasso by Friedman et al. (2008). This article delivers a systematic treatment of the plug-in approach used in the rst two ref- erences. We show that the sample covariance matrix may be replaced by any ane equivariant,

Subfields: Analytical Chemistry, Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6ff5eb8379542f6b59c0552d0a8c0f6af266e2d8

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In Analytical Chemistry

  1. Marloes H. Maathuis
  2. Mathias Drton
  3. Sacha Epskamp
  4. D. Hevey

In Environmental Engineering

  1. Mathias Drton

In Statistics and Probability

  1. Marloes H. Maathuis
  2. Mathias Drton
  3. Peter Buhlmann
  4. Sacha Epskamp
  5. Nicolai Meinshausen
  6. Diego Colombo
  7. Preetam Nandy
  8. Martyn Plummer

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