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Robust Statistics Part 3: Regression Analysis
Robust Statistics Part 3: Regression analysis Peter Rousseeuw LARS-IASC School, May 2019 Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 1 Linear regression Linear regression: Outline 1 Classical regression estimators 2 Classical outlier diagnostics 3 Regression M-estimators 4 The LTS estimator 5 Outlier detection 6 Regression S-estimators and MM-estimators 7 Regression with categorical predictors 8 Software Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 2 Linear regression Classical estimators The linear regression model The linear regression model says: yi = β0 + β1xi1 + ... + βpxip + εi ′ = xiβ + εi 2 ′ ′ with i.i.d. errors εi ∼ N(0,σ ), xi = (1,xi1,...,xip) and β =(β0,β1,...,βp) . ′ Denote the n × (p + 1) matrix containing the predictors xi as X =(x1,..., xn) , ′ ′ the vector of responses y =(y1,...,yn) and the error vector ε =(ε1,...,εn) . Then: y = Xβ + ε Any regression estimate βˆ yields fitted values yˆ = Xβˆ and residuals ri = ri(βˆ)= yi − yˆi . Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 3 Linear regression Classical estimators The least squares estimator Least squares estimator n ˆ 2 βLS = argmin ri (β) β i=1 X If X has full rank, then the solution is unique and given by ˆ ′ −1 ′ βLS =(X X) X y The usual unbiased estimator of the error variance is n 1 σˆ2 = r2(βˆ ) LS n − p − 1 i LS i=1 X Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 4 Linear regression Classical estimators Outliers in regression Different types of outliers: vertical outlier good leverage point • • y • • • regular data • ••• • •• ••• • • • • • • • • • bad leverage point • • •• • x Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. -
Robustbase: Basic Robust Statistics
Package ‘robustbase’ June 2, 2021 Version 0.93-8 VersionNote Released 0.93-7 on 2021-01-04 to CRAN Date 2021-06-01 Title Basic Robust Statistics URL http://robustbase.r-forge.r-project.org/ Description ``Essential'' Robust Statistics. Tools allowing to analyze data with robust methods. This includes regression methodology including model selections and multivariate statistics where we strive to cover the book ``Robust Statistics, Theory and Methods'' by 'Maronna, Martin and Yohai'; Wiley 2006. Depends R (>= 3.5.0) Imports stats, graphics, utils, methods, DEoptimR Suggests grid, MASS, lattice, boot, cluster, Matrix, robust, fit.models, MPV, xtable, ggplot2, GGally, RColorBrewer, reshape2, sfsmisc, catdata, doParallel, foreach, skewt SuggestsNote mostly only because of vignette graphics and simulation Enhances robustX, rrcov, matrixStats, quantreg, Hmisc EnhancesNote linked to in man/*.Rd LazyData yes NeedsCompilation yes License GPL (>= 2) Author Martin Maechler [aut, cre] (<https://orcid.org/0000-0002-8685-9910>), Peter Rousseeuw [ctb] (Qn and Sn), Christophe Croux [ctb] (Qn and Sn), Valentin Todorov [aut] (most robust Cov), Andreas Ruckstuhl [aut] (nlrob, anova, glmrob), Matias Salibian-Barrera [aut] (lmrob orig.), Tobias Verbeke [ctb, fnd] (mc, adjbox), Manuel Koller [aut] (mc, lmrob, psi-func.), Eduardo L. T. Conceicao [aut] (MM-, tau-, CM-, and MTL- nlrob), Maria Anna di Palma [ctb] (initial version of Comedian) 1 2 R topics documented: Maintainer Martin Maechler <[email protected]> Repository CRAN Date/Publication 2021-06-02 10:20:02 UTC R topics documented: adjbox . .4 adjboxStats . .7 adjOutlyingness . .9 aircraft . 12 airmay . 13 alcohol . 14 ambientNOxCH . 15 Animals2 . 18 anova.glmrob . 19 anova.lmrob . -
CFE-Cmstatistics 2020 Book of Abstracts
CFE-CMStatistics 2020 PROGRAMME AND ABSTRACTS 14th International Conference on Computational and Financial Econometrics (Virtual CFE 2020) http://www.cfenetwork.org/CFE2020 and 13th International Conference of the ERCIM (European Research Consortium for Informatics and Mathematics) Working Group on Computational and Methodological Statistics (Virtual CMStatistics 2020) http://www.cmstatistics.org/CMStatistics2020 19 – 21 December 2020 Computational and Methodological Statistics CMStatistics Computational and CFENetwork Financial Econometrics ⃝c ECOSTA ECONOMETRICS AND STATISTICS. All rights reserved. I CFE-CMStatistics 2020 ISBN 978-9963-2227-9-7 ⃝c 2020 - ECOSTA ECONOMETRICS AND STATISTICS All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any other form or by any means without the prior permission from the publisher. II ⃝c ECOSTA ECONOMETRICS AND STATISTICS. All rights reserved. CFE-CMStatistics 2020 International Organizing Committee: Ana Colubi, Erricos Kontoghiorghes and Manfred Deistler. CFE 2020 Co-chairs: Anurag Banerjee, Scott Brave, Peter Pedroni and Mike So. CFE 2020 Programme Committee: Knut Are Aastveit, Alessandra Amendola, David Ardia, Josu Arteche, Anindya Banerjee, Travis Berge, Mon- ica Billio, Raffaella Calabrese, Massimiliano Caporin, Julien Chevallier, Serge Darolles, Luca De Angelis, Filippo Ferroni, Ana-Maria Fuertes, Massimo Guidolin, Harry Haupt, Masayuki Hirukawa, Benjamin Hol- cblat, Rustam Ibragimov, Laura Jackson Young, Michel Juillard, Edward Knotek, Robinson Kruse-Becher, Svetlana Makarova, Ilia Negri, Ingmar Nolte, Jose Olmo, Yasuhiro Omori, Jesus Otero, Michael Owyang, Alessia Paccagnini, Indeewara Perera, Jean-Yves Pitarakis, Tommaso Proietti, Artem Prokhorov, Tatevik Sekhposyan, Etsuro Shioji, Michael Smith, Robert Taylor, Martin Wagner and Ralf Wilke. CMStatistics 2020 Co-chairs: Tapabrata Maiti, Sofia Olhede, Michael Pitt, Cheng Yong Tang and Tim Verdonck. -
Robust Linear Regression: Optimal Rates in Polynomial Time
Robust Linear Regression: Optimal Rates in Polynomial Time Ainesh Bakshi* Adarsh Prasad [email protected] [email protected] CMU CMU Abstract We obtain robust and computationally efficient estimators for learning several linear mod- els that achieve statistically optimal convergence rate under minimal distributional assump- tions. Concretely, we assume our data is drawn from a k-hypercontractive distribution and an ǫ-fraction is adversarially corrupted. We then describe an estimator that converges to the 2 2/k optimal least-squares minimizer for the true distribution at a rate proportional to ǫ − , when the noise is independent of the covariates. We note that no such estimator was known prior to our work, even with access to unbounded computation. The rate we achieve is information- theoretically optimal and thus we resolve the main open question in Klivans, Kothari and Meka [COLT’18]. Our key insight is to identify an analytic condition that serves as a polynomial relaxation of independence of random variables. In particular, we show that when the moments of the noise and covariates are negatively-correlated, we obtain the same rate as independent noise. 2 4/k Further, when the condition is not satisfied, we obtain a rate proportional to ǫ − , and again match the information-theoretic lower bound. Our central technical contribution is to algo- rithmically exploit independence of random variables in the ”sum-of-squares” framework by formulating it as the aforementioned polynomial inequality. arXiv:2007.01394v4 [stat.ML] 4 Dec 2020 *AB would like to thank the partial support from the Office of Naval Research (ONR) grant N00014-18-1-2562, and the National Science Foundation (NSF) under Grant No. -
Robust Statistics for Outlier Detection Peter J
Focus Article Robust statistics for outlier detection Peter J. Rousseeuw and Mia Hubert When analyzing data, outlying observations cause problems because they may strongly influence the result. Robust statistics aims at detecting the outliers by searching for the model fitted by the majority of the data. We present an overview of several robust methods and outlier detection tools. We discuss robust proce- dures for univariate, low-dimensional, and high-dimensional data such as esti- mation of location and scatter, linear regression, principal component analysis, and classification. C 2011 John Wiley & Sons, Inc. WIREs Data Mining Knowl Discov 2011 1 73–79 DOI: 10.1002/widm.2 INTRODUCTION ESTIMATING UNIVARIATE n real data sets, it often happens that some obser- LOCATION AND SCALE I vations are different from the majority. Such ob- As an example, suppose we have five measurements servations are called outliers. Outlying observations of a length: may be errors, or they could have been recorded un- der exceptional circumstances, or belong to another 6.27, 6.34, 6.25, 6.31, 6.28 (1) population. Consequently, they do not fit the model and we want to estimate its true value. For this, well. It is very important to be able to detect these one usually computes the mean x¯ = 1 n x , which outliers. n i=1 i in this case equals x¯ = (6.27 + 6.34 + 6.25 + 6.31 + In practice, one often tries to detect outliers 6.28)/5 = 6.29. Let us now suppose that the fourth using diagnostics starting from a classical fitting measurement has been recorded wrongly and the data method. -
Robust Covariance Estimation for Financial Applications
Robust covariance estimation for financial applications Tim Verdonck, Mia Hubert, Peter Rousseeuw Department of Mathematics K.U.Leuven August 30 2011 Tim Verdonck, Mia Hubert, Peter Rousseeuw Robust covariance estimation August 30 2011 1 / 44 Contents 1 Introduction Robust Statistics 2 Multivariate Location and Scatter Estimates 3 Minimum Covariance Determinant Estimator (MCD) FAST-MCD algorithm DetMCD algorithm 4 Principal Component Analysis 5 Multivariate Time Series 6 Conclusions 7 Selected references Tim Verdonck, Mia Hubert, Peter Rousseeuw Robust covariance estimation August 30 2011 2 / 44 Introduction Robust Statistics Introduction Robust Statistics Real data often contain outliers. Most classical methods are highly influenced by these outliers. What is robust statistics? Robust statistical methods try to fit the model imposed by the majority of the data. They aim to find a ‘robust’ fit, which is similar to the fit we would have found without outliers (observations deviating from robust fit). This also allows for outlier detection. Robust estimate applied on all observations is comparable with the classical estimate applied on the outlier-free data set. Robust estimator A good robust estimator combines high robustness with high efficiency. ◮ Robustness: being less influenced by outliers. ◮ Efficiency: being precise at uncontaminated data. Tim Verdonck, Mia Hubert, Peter Rousseeuw Robust covariance estimation August 30 2011 3 / 44 Introduction Robust Statistics Univariate Scale Estimation: Wages data set 6000 households with male head earning -
Robust Statistics Part 1: Introduction and Univariate Data General References
Robust Statistics Part 1: Introduction and univariate data Peter Rousseeuw LARS-IASC School, May 2019 Peter Rousseeuw Robust Statistics, Part 1: Univariate data LARS-IASC School, May 2019 p. 1 General references General references 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, John Wiley and Sons, New York, 1986. Rousseeuw, P.J., Leroy, A. Robust Regression and Outlier Detection. Wiley Series in Probability and Mathematical Statistics. John Wiley and Sons, New York, 1987. Maronna, R.A., Martin, R.D., Yohai, V.J. Robust Statistics: Theory and Methods. Wiley Series in Probability and Statistics. John Wiley and Sons, Chichester, 2006. Hubert, M., Rousseeuw, P.J., Van Aelst, S. (2008), High-breakdown robust multivariate methods, Statistical Science, 23, 92–119. wis.kuleuven.be/stat/robust Peter Rousseeuw Robust Statistics, Part 1: Univariate data LARS-IASC School, May 2019 p. 2 General references Outline of the course General notions of robustness Robustness for univariate data Multivariate location and scatter Linear regression Principal component analysis Advanced topics Peter Rousseeuw Robust Statistics, Part 1: Univariate data LARS-IASC School, May 2019 p. 3 General notions of robustness General notions of robustness: Outline 1 Introduction: outliers and their effect on classical estimators 2 Measures of robustness: breakdown value, sensitivity curve, influence function, gross-error sensitivity, maxbias curve. Peter Rousseeuw Robust Statistics, Part 1: Univariate data LARS-IASC School, May 2019 p. 4 General notions of robustness Introduction What is robust statistics? Real data often contain outliers. -
Mia Hubert, Peter Rousseeuw, & Karlien Vanden Branden
Home About Scientific Press Room Contact Us ● ScienceWatch Home ● Inside This Month... ● Interviews Featured Interviews Author Commentaries 2009 : January 2009 - Fast Moving Fronts : Mia Hubert, Peter Rousseeuw, & Karlien Vanden Branden Institutional Interviews Journal Interviews FAST MOVING FRONTS - 2009 Podcasts January 2009 ● Analyses Mia Hubert, Peter Rousseeuw, & Karlien Vanden Branden talk with ScienceWatch.com and answer Featured Analyses a few questions about this month's Fast Moving Front in the field of Mathematics. What's Hot In... Article: ROBPCA: A new approach to robust principal component Special Topics analysis Authors: Hubert, M;Rousseeuw, PJ;Vanden Branden, K Journal: TECHNOMETRICS, 47 (1): 64-79, FEB 2005 ● Data & Rankings Addresses: Katholieke Univ Leuven, Dept Math, B-3001 Louvain, Belgium. Katholieke Univ Leuven, Dept Math, B-3001 Louvain, Belgium. Sci-Bytes Univ Antwerp, Dept Math & Comp Sci, B-2020 Antwerp, Belgium. Fast Breaking Papers Katholieke Univ Leuven, Dept Math, B-3001 Heverlee, Belgium. New Hot Papers Emerging Research Fronts Fast Moving Fronts Why do you think your paper is highly cited? Research Front Maps Our paper offers a solution to an important problem in statistics and data analysis: how to perform data Current Classics reduction when the observations may be contaminated with outlying values. This problem is especially Top Topics important for the analysis of high-dimensional data sets, such as spectral data in chemometrics and Rising Stars genetic data in bio-informatics. We propose an algorithm which is highly robust and computationally New Entrants feasible, and we also provide a graphical tool for outlier detection. Country Profiles Moreover, our method serves as the cornerstone of new highly robust calibration methods (principal component regression and partial least squares regression), a robust classifier, and robust multi-way ● About Science Watch techniques. -
Fast Detection of Outliers in Data Streams with the $ Q N $ Estimator
Fast Detection of Outliers in Data Streams with the Qn Estimator Massimo Cafaroa,∗, Catiuscia Mellea, Marco Pulimenoa, Italo Epicocoa aUniversity of Salento, Lecce, Italy Abstract We present fqn (Fast Qn), a novel algorithm for fast detection of outliers in data streams. The algorithm works in the sliding window model, checking if an item is an outlier by cleverly computing the Qn scale estimator in the current window. We thoroughly compare our algorithm for online Qn with the state of the art competing algorithm by Nunkesser et al, and show that fqn (i) is faster, (ii) its computational complexity does not depend on the input distribution and (iii) it requires less space. Extensive experimental results on synthetic datasets confirm the validity of our approach. Keywords: data streams, outliers, sliding window model, Qn estimator. Declaration of interest Declarations of interest: none. 1. Introduction A data stream σ can be thought as a sequence of n items drawn from a universe U. In particular, the items need not be distinct, so that an item may appear multiple times in the stream. Data streams are ubiquitous, and, depending on the specific context, items may be IP addresses, graph edges, points, geographical coordinates, numbers etc. Since the items in the input data stream come at a very high rate, and the stream may be of potentially infinite length (in which case n refers to the number of items seen so far), it is hard for an algorithm in charge of processing its items to compute an expensive function of a large piece of the input. -
Package 'Mrfdepth'
Package ‘mrfDepth’ August 26, 2020 Type Package Version 1.0.13 Date 2020-08-24 Title Depth Measures in Multivariate, Regression and Functional Settings Description Tools to compute depth measures and implementations of related tasks such as outlier detection, data exploration and classification of multivariate, regression and functional data. Depends R (>= 3.6.0), ggplot2 Imports abind, geometry, grid, matrixStats, reshape2, Suggests robustbase LinkingTo RcppEigen (>= 0.3.2.9.0), Rcpp (>= 0.12.6), RcppArmadillo (>= 0.7.600.1.0) License GPL (>= 2) LazyLoad yes URL https://github.com/PSegaert/mrfDepth BugReports https://github.com/PSegaert/mrfDepth/issues RoxygenNote 6.1.0 NeedsCompilation yes Author Pieter Segaert [aut], Mia Hubert [aut], Peter Rousseeuw [aut], Jakob Raymaekers [aut, cre], Kaveh Vakili [ctb] Maintainer Jakob Raymaekers <[email protected]> Repository CRAN Date/Publication 2020-08-26 16:10:33 UTC 1 2 R topics documented: R topics documented: adjOutl . .3 bagdistance . .6 bagplot . .9 bloodfat . 11 cardata90 . 12 characterA . 13 characterI . 14 cmltest . 15 compBagplot . 16 depthContour . 19 dirOutl . 22 distSpace . 25 dprojdepth . 28 dprojmedian . 30 fheatmap . 31 fom ............................................. 32 fOutl . 34 geological . 37 glass . 37 hdepth . 38 hdepthmedian . 42 medcouple . 43 mfd ............................................. 45 mfdmedian . 47 mrainbowplot . 49 mri.............................................. 50 octane . 52 outlyingness . 52 plane . 56 plotContours . 58 projdepth . 59 projmedian . 61 rdepth . 63 rdepthmedian . 65 sdepth . 66 sprojdepth . 68 sprojmedian . 70 stars . 72 symtest . 73 tablets . 74 wine............................................. 75 Index 76 adjOutl 3 adjOutl Adjusted outlyingness of points relative to a dataset Description Computes the skew-adjusted outlyingness of p-dimensional points z relative to a p-dimensional dataset x. -
Review of Robustbase Software for R
Research Collection Review Article Review of ‘Robustbase’ software for R Author(s): Finger, Robert Publication Date: 2010-11 Permanent Link: https://doi.org/10.3929/ethz-b-000159379 Originally published in: Journal of Applied Econometrics 25(7), http://doi.org/10.1002/jae.1194 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Postprint This is the accepted version of a paper published in the Journal of Applied Econometrics. This paper has been peer-reviewed but does not include the final publisher proof- corrections or journal pagination. Citation for the original published paper: Finger, R. (2010). Review of ‘robustbase’ software for R. Journal of Applied Econometrics 25(7): 1205-1210 https://doi.org/10.1002/jae.1194 Access to the published version may require subscription. N.B. When citing this work, cite the original published paper Review of ‘robustbase’ software for R Robert Finger (ETH Zürich, Switzerland) 1. Introduction Robust statistical methods are powerful tools to increase the reliability and accuracy of statistical modeling and data analysis because these methods still work well when some observations deviate from (or violate) the assumed model. For many statistical models (e.g. location, scale and regression), several robust methods are available as alternative approaches to classical statistical methods (Maronna et al., 2006). However, there has been a lack of availability of robust methods in statistical software (e.g. Stromberg, 2004). The R package ‘robustbase’ (as well as other related packages) helps to fill the gap between theoretically developed robust methods and their availability in standard statistical software and makes both basic and advanced methods available to a broad range of researchers. -
Robust Regression and Outlier Detection with the ROBUSTREG Procedure
SUGI 27 Statistics and Data Analysis Paper265-27 Robust Regression and Outlier Detection with the ROBUSTREG Procedure Colin Chen, SAS Institute Inc., Cary, NC Many methods have been developed for these prob- lems. However, in statistical applications of outlier Abstract detection and robust regression, the methods most commonly used today are Huber M estimation, high Robust regression is an important tool for analyz- breakdown value estimation, and combinations of ing data that are contaminated with outliers. It these two methods. The ROBUSTREG procedure can be used to detect outliers and to provide re- provides four such methods: M estimation, LTS es- sistant (stable) results in the presence of outliers. timation, S estimation, and MM estimation. This paper introduces the ROBUSTREG procedure, which is experimental in SAS/STAT Version 9. The 1. M estimation was introduced by Huber (1973), ROBUSTREG procedure implements the most com- and it is the simplest approach both computa- monly used robust regression techniques. These tionally and theoretically. Although it is not ro- include M estimation (Huber, 1973), LTS estima- bust with respect to leverage points, it is still tion (Rousseeuw, 1984), S estimation (Rousseeuw used extensively in analyzing data for which and Yohai, 1984), and MM estimation (Yohai, 1987). it can be assumed that the contamination is The paper will provide an overview of robust re- mainly in the response direction. gression methods, describe the syntax of PROC ROBUSTREG, and illustrate the use of the procedure 2. Least Trimmed Squares (LTS) estimation is a to fit regression models and display outliers and lever- high breakdown value method introduced by age points.