High-Breakdown Robust Multivariate Methods

By Mia Hubert, Peter J. Rousseeuw, S. Van Aelst. 2008

354 citations recorded by the source corpus.

When applying a statistical method in practice it often occurs that some observations deviate from the usual assumptions. However, many classical methods are sensitive to outliers. The goal of robust statistics is to develop methods that are robust against the possibility that one or several unannounced outliers may occur anywhere in the data. These methods then allow to detect outlying observations by their residuals from a robust fit. We focus on high-breakdown methods, which can deal with a substantial fraction of outliers in the data. We give an overview of recent high-breakdown robust methods for multivariate settings such as covariance estimation, multiple and multivariate regression, discriminant analysis, principal components and multivariate calibration.

Subfields: Statistics and Probability, Water Science and Technology

View Paper: https://www.semanticscholar.org/paper/958263fcefe54c066c2294b0bb1200f1b737174b

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Citations in the corpus

An Object-Oriented Framework for Robust Multivariate Analysis (2009, background): “In the context of PCA Hubert et al. (2005) defined a diagnostic plot or outlier map which helps to distinguish between the regular observations and the different types of outliers.”
Geometric median and robust estimation in Banach spaces (2013, background): “For an overview of classical and modern results in robust statistics, see [19,20], and references therein.”
Central limit theorem and influence function for the MCD estimators at general multivariate distributions (2009, background): “These estimators, in particular the covariance estimator, also serve as robust plug-ins in other multivariate statistical techniques, such as principal component analysis [6, 21], multivariate linear regression [1, 20], discriminant analysis [11], factor analysis [16], canonical correlations [22, 25”
Least quantile regression via modern optimization (2013, methodology): “We refer the interested reader to Hubert et al. (2008) for a nice review of various robust statistical methods and their applications (Meer et al., 1991; Stewart, 1999; Rousseeuw et al., 2006).”
Principal Component Analysis (2018, methodology): “Because the PCA projection can be sensitive to outliers [Rousseeuw and Hubert 2011], some studies replace the covariance matrix by an estimator referred to as robust covariance [Hubert et al. 2008; Rousseeuw and Hubert 2011].”
Anomaly detection by robust statistics (2017, background): “The MAD of (2) is the same as that of (1), namely 0.”
Robust Estimation in Signal Processing: A Tutorial-Style Treatment of Fundamental Concepts (2012, methodology): “This means intuitively that the MCD is computed from the " closest " h samples.”
Robust Correlation Analyses: False Positive and Power Validation Using a New Open Source Matlab Toolbox (2012, background): “…outlier can result in the bivariate mean giving a poor reflection of the typical response, one relies here on the minimum covariance determinant (MCD) estimator, which is a robust estimator of multivariate location and scatter (Rousseeuw, 1984; Rousseeuw and Van Drissen, 1999; Hubert et al., 2008).”
Multivariate Outlier Detection With High-Breakdown Estimators (2010, background): “Usually 0.01 ≤ α ≤ 0.05, with α = 0.025 being perhaps the most popular choice (e.g., Hubert, Rousseeuw, and Van Aelst 2008; Willems, Joe, and Zamar 2009).”
Learning from History for Byzantine Robust Optimization (2020, methodology): “However, classically the focus of robust statistics has been to design methods which can withstand a large fraction of Byzantine workers (high break down point δ max ) and not result in inﬁnities (Hubert et al., 2008).”
Efficient and Robust Median-of-Means Algorithms for Location and Regression (2016, background): “The important property to preserve is the positive deﬁniteness of the matrices (see e.g. [10]).”
Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries (2016, neutral): “Some popular robust estimators of scatter are discussed in [HRVA08], including the Minimum Covariance Determinant (MCD) estimator and the Minimum Volume Ellipsoid estimator (MVE).”