Robust estimation of precision matrices under cellwise contamination

Tarr, Garth; Müller, Samuel; Weber, N. C.

doi:10.1016/j.csda.2015.02.005

Cited by 53 publications

(77 citation statements)

References 44 publications

(67 reference statements)

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“…Our approach is similar in spirit as Tarr et al [2015] [see also Tarr, 2014], but we emphasize the difference in Section 3. We start with pairwise robust correlation estimates from which we then estimate a covariance matrix by multiplication with robust standard deviations.…”

Section: Introductionmentioning

confidence: 99%

“…This results in a sparse, cellwise robust precision matrix estimate. Our approach differs from Tarr et al [2015] in the selection of the initial covariance estimate. We estimate robust correlations and standard deviations separately to get the robust covariances.…”

Section: Introductionmentioning

confidence: 99%

“…Afterwards the OGK estimate is applied to obtain a positive semidefinite covariance estimate. This method has been fine tuned by Tarr et al [2015] who use pairwise covariances instead of correlations [see also Tarr, 2014]. This matrix is then plugged into the graphical lasso (and similar techniques) instead of the sample covariance matrix, resulting in a sparse precision matrix estimate.…”

Section: Introductionmentioning

confidence: 99%

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Robust High-Dimensional Precision Matrix Estimation

Öllerer

Croux

2015

Modern Nonparametric, Robust and Multivariate Methods

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The dependency structure of multivariate data can be analyzed using the covariance matrix Σ. In many fields the precision matrix Σ −1 is even more informative. As the sample covariance estimator is singular in high-dimensions, it cannot be used to obtain a precision matrix estimator. A popular highdimensional estimator is the graphical lasso, but it lacks robustness. We consider the high-dimensional independent contamination model. Here, even a small percentage of contaminated cells in the data matrix may lead to a high percentage of contaminated rows. Downweighting entire observations, which is done by traditional robust procedures, would then results in a loss of information. In this paper, we formally prove that replacing the sample covariance matrix in the graphical lasso with an elementwise robust covariance matrix leads to an elementwise robust, sparse precision matrix estimator computable in high-dimensions. Examples of such elementwise robust covariance estimators are given. The final precision matrix estimator is positive definite, has a high breakdown point under elementwise contamination and can be computed fast.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Robust High-Dimensional Precision Matrix Estimation

Öllerer

Croux

2015

Modern Nonparametric, Robust and Multivariate Methods

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“…This approach works well with high levels of scattered contamination and has the advantage of being able to impose sparsity on the resulting precision matrix. The results from this chapter have been published in [7].…”

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confidence: 97%

Quantile Based Estimation of Scale and Dependence

Tarr¹

2015

Bull. Aust. Math. Soc.

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The sample quantile has a long history in statistics. The aim of this thesis is to explore some further applications of quantiles as simple, convenient and robust alternatives to classical procedures. The first application we consider is estimating confidence intervals for quantile regression coefficients, however, the core of this thesis is the development of a new, quantile based, robust scale estimator and its extension to autocovariance estimation in the time series setting and precision matrix estimation in the multivariate setting.Chapter 1 addresses the need for reliable confidence intervals for quantile regression coefficients, particularly in small samples. The existing methods for constructing confidence intervals tend to be based on complex asymptotic arguments and little is known about their finite sample performance. We consider taking xy-pair bootstrap samples and calculating the corresponding quantile regression coefficient estimates for each sample. Instead of estimating a covariance matrix based on these bootstrap samples, our approach is to take the appropriate upper and lower quantiles of the bootstrap sample estimates as the bounds of the confidence interval. The resulting confidence interval estimate is not necessarily symmetric, only covers admissible parameter values and is shown to have good coverage properties. This work demonstrates the competitive performance of our quantile based approach in a broad range of model designs with a focus on small and moderate sample sizes. These results were published in [5].A reliable estimate of the scale of the residuals from a regression model is often of interest, whether it be parametrically estimating confidence intervals, determining a goodness-of-fit measure, performing model selection, or identifying

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“…In high dimensions, or when the sample size n is close to or larger than the dimension p, we perform an additional regularization step, as in Tarr et al (2015). (4) Use S as input for the graphical lasso or GLASSO of Friedman et al (2008).…”

Section: The Ggq-estimatormentioning

confidence: 99%

Comments on: Robust estimation of multivariate location and scatter in the presence of cellwise and casewise contamination

Croux

Öllerer

2015

TEST

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We would like to congratulate the authors for their innovative paper, containing many stimulating ideas. The authors propose an estimator for multivariate location and scatter, robust to both cellwise and casewise contamination. The idea is simple: (i) check for large univariate outliers and replace these cellwise outliers by NA, and (ii) apply the S-estimator for missing values of Danilov et al. (2012). If there are no cellwise outliers detected in step (i), the proposed estimator equals the regular S-estimator and shares the affine equivariance property. The authors will agree that the main power of the estimator comes from the second step, where the casewise-or multivariate-outliers are detected using a robust version of the Mahalanobis distance. For every observation, this distance is computed in the dimension given by the number of non-missing components. Danilov et al. (2012) present a smart way to compute an S-estimator associated with Mahalanobis distances computed in different dimensions.Estimation of the scatter matrix is 'a corner stone in many applications', as the authors state. However, the applications that the authors list (principal component analysis, factor analysis, and multiple linear regression) require the precision matrix Θ = Σ −1 rather than the covariance matrix Σ. Obviously, the inverse of the proposed two-step generalized S-estimator (TSGS) yields an estimate of the precision matrix. In this discussion note we (i) investigate the performance of TSGS as precision matrix estimator by means of a modest simulation study, (ii) discuss a regularized version of This comment refers to the invited paper available at

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Robust estimation of precision matrices under cellwise contamination

Cited by 53 publications

References 44 publications

Robust High-Dimensional Precision Matrix Estimation

Robust High-Dimensional Precision Matrix Estimation

Quantile Based Estimation of Scale and Dependence

Comments on: Robust estimation of multivariate location and scatter in the presence of cellwise and casewise contamination

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