On the breakdown behavior of the TCLUST clustering procedure
Clustering procedures allowing for general covariance structures of the obtained clusters need some constraints on the solutions. With this in mind, several proposals have been introduced in the literature. The TCLUST procedure works with a restriction on the “eigenvalues-ratio” of the clusters scat...
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Published in | Test (Madrid, Spain) Vol. 22; no. 3; pp. 466 - 487 |
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Main Authors | , , , |
Format | Journal Article Web Resource |
Language | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.09.2013
Springer Nature B.V Springer |
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Abstract | Clustering procedures allowing for general covariance structures of the obtained clusters need some constraints on the solutions. With this in mind, several proposals have been introduced in the literature. The TCLUST procedure works with a restriction on the “eigenvalues-ratio” of the clusters scatter matrices. In order to try to achieve robustness with respect to outliers, the procedure allows to trim off a proportion
α
of the most outlying observations. The resistance to infinitesimal contamination of the TCLUST has already been studied. This paper aims to look at its resistance to a higher amount of contamination by means of the study of its breakdown behavior. The rather new concept of restricted breakdown point will demonstrate that the TCLUST procedure resists to a proportion
α
of contamination as soon as the data set is sufficiently “well clustered”. |
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AbstractList | Clustering procedures allowing for general covariance structures of the obtained clusters need some constraints on the solutions. With this in mind, several proposals have been introduced in the literature. The TCLUST procedure works with a restriction on the “eigenvalues-ratio” of the clusters scatter matrices. In order to try to achieve robustness with respect to outliers, the procedure allows to trim off a proportion
α
of the most outlying observations. The resistance to infinitesimal contamination of the TCLUST has already been studied. This paper aims to look at its resistance to a higher amount of contamination by means of the study of its breakdown behavior. The rather new concept of restricted breakdown point will demonstrate that the TCLUST procedure resists to a proportion
α
of contamination as soon as the data set is sufficiently “well clustered”. Clustering procedures allowing for general covariance structures of the obtained clusters need some constraints on the solutions. With this in mind, several proposals have been introduced in the literature. The TCLUST procedure works with a restriction on the "eigenvalues-ratio" of the clusters scatter matrices. In order to try to achieve robustness with respect to outliers, the procedure allows to trim off a proportion [alpha] of the most outlying observations. The resistance to infinitesimal contamination of the TCLUST has already been studied. This paper aims to look at its resistance to a higher amount of contamination by means of the study of its breakdown behavior. The rather new concept of restricted breakdown point will demonstrate that the TCLUST procedure resists to a proportion [alpha] of contamination as soon as the data set is sufficiently "well clustered".[PUBLICATION ABSTRACT] Clustering procedures allowing for general covariance structures of the obtained clusters need some constraints on the solutions. With this in mind, several proposals have been introduced in the literature. The TCLUST procedure works with a restriction on the "eigenvalues-ratio" of the clusters scatter matrices. In order to try to achieve robustness with respect to outliers, the procedure allows to trim off a proportion of the most outlying observations. The resistance to infinitesimal contamination of the TCLUST has already been studied. This paper aims to look at its resistance to a higher amount of contamination by means of the study of its breakdown behavior. The rather new concept of restricted breakdown point will demonstrate that the TCLUST procedure resists to a proportion of contamination equal to the trimming rate as soon as the data set is sufficiently "well clustered". Clustering procedures allowing for general covariance structures of the obtained clusters need some constraints on the solutions. With this in mind, several proposals have been introduced in the literature. The TCLUST procedure works with a restriction on the "eigenvalues-ratio" of the clusters scatter matrices. In order to try to achieve robustness with respect to outliers, the procedure allows to trim off a proportion alpha of the most outlying observations. The resistance to infinitesimal contamination of the TCLUST has already been studied. This paper aims to look at its resistance to a higher amount of contamination by means of the study of its breakdown behavior. The rather new concept of restricted breakdown point will demonstrate that the TCLUST procedure resists to a proportion alpha of contamination as soon as the data set is sufficiently "well clustered". |
Author | Mayo-Iscar, A. Ruwet, C. García-Escudero, L. A. Gordaliza, A. |
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Cites_doi | 10.1111/1467-9868.00373 10.1016/j.jmva.2007.07.002 10.1214/aos/1031833664 10.1002/9780470316801 10.1214/aos/1176349557 10.1007/s11634-009-0044-9 10.1007/s11634-012-0107-1 10.1016/j.csda.2006.12.024 10.1198/016214502760047131 10.1214/07-AOS515 10.1002/0471721182 10.1007/s11634-010-0064-5 10.1214/009053604000000940 10.1007/s11222-010-9194-z |
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Keywords | TCLUST 62H30 Robustness Clustering 62F35 Breakdown point 62G35 Trimming |
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References | Gallegos, Ritter (CR7) 2009; 3 Zhong, Ghosh (CR19) 2004; 4 Dennis (CR2) 1982 García-Escudero, Gordaliza, Matrán, Mayo-Iscar (CR10) 2010; 4 Hennig (CR14) 2008; 99 Ruwet, García-Escudero, Gordaliza, Mayo-Iscar (CR18) 2012; 6 McLachlan, Peel (CR16) 2000 Cuesta-Albertos, Gordaliza, Matrán (CR1) 1997; 25 Fraley, Raftery (CR4) 2002; 97 García-Escudero, Gordaliza, Matrán, Mayo-Iscar (CR11) 2011; 21 Genton, Lucas (CR12) 2003; 65 Donoho, Huber (CR3) 1983 Kaufman, Rousseeuw (CR15) 1990 García-Escudero, Gordaliza (CR8) 1999; 94 García-Escudero, Gordaliza, Matrán, Mayo-Iscar (CR9) 2008; 36 Neykov, Filzmoser, Dimova, Neytchev (CR17) 2007; 52 Gallegos, Ritter (CR6) 2009; 71 Gallegos, Ritter (CR5) 2005; 33 Hathaway (CR13) 1985; 13 D Donoho (312_CR3) 1983 MT Gallegos (312_CR6) 2009; 71 G McLachlan (312_CR16) 2000 JE Dennis Jr. (312_CR2) 1982 N Neykov (312_CR17) 2007; 52 MT Gallegos (312_CR5) 2005; 33 MT Gallegos (312_CR7) 2009; 3 LA García-Escudero (312_CR8) 1999; 94 LA García-Escudero (312_CR11) 2011; 21 MG Genton (312_CR12) 2003; 65 LA García-Escudero (312_CR10) 2010; 4 S Zhong (312_CR19) 2004; 4 C Ruwet (312_CR18) 2012; 6 RJ Hathaway (312_CR13) 1985; 13 JA Cuesta-Albertos (312_CR1) 1997; 25 LA García-Escudero (312_CR9) 2008; 36 C Hennig (312_CR14) 2008; 99 L Kaufman (312_CR15) 1990 C Fraley (312_CR4) 2002; 97 |
References_xml | – volume: 94 start-page: 956 year: 1999 end-page: 969 ident: CR8 article-title: Robustness properties of means and trimmed means publication-title: J Am Stat Assoc contributor: fullname: Gordaliza – volume: 65 start-page: 81 year: 2003 end-page: 94 ident: CR12 article-title: Comprehensive definitions of breakdown points for independent and dependent observations publication-title: J R Stat Soc, Ser B, Stat Methodol doi: 10.1111/1467-9868.00373 contributor: fullname: Lucas – volume: 99 start-page: 1154 year: 2008 end-page: 1176 ident: CR14 article-title: Dissolution point and isolation robustness: robustness criteria for general cluster analysis methods publication-title: J Multivar Anal doi: 10.1016/j.jmva.2007.07.002 contributor: fullname: Hennig – volume: 25 start-page: 553 year: 1997 end-page: 576 ident: CR1 article-title: Trimmed -means: an attempt to robustify quantizers publication-title: Ann Stat doi: 10.1214/aos/1031833664 contributor: fullname: Matrán – year: 1990 ident: CR15 publication-title: Finding groups in data: an introduction to cluster analysis doi: 10.1002/9780470316801 contributor: fullname: Rousseeuw – volume: 71 start-page: 164 year: 2009 end-page: 220 ident: CR6 article-title: Trimmed ML estimation of contaminated mixtures publication-title: Sankhyā contributor: fullname: Ritter – volume: 13 start-page: 795 year: 1985 end-page: 800 ident: CR13 article-title: A constrained formulation of maximum-likelihood estimation for normal mixture distributions publication-title: Ann Stat doi: 10.1214/aos/1176349557 contributor: fullname: Hathaway – volume: 3 start-page: 135 year: 2009 end-page: 167 ident: CR7 article-title: Trimming algorithms for clustering contaminated grouped data and their robustness publication-title: Adv Data Anal Classif doi: 10.1007/s11634-009-0044-9 contributor: fullname: Ritter – volume: 6 start-page: 107 year: 2012 end-page: 130 ident: CR18 article-title: The influence function of the TCLUST robust clustering procedure publication-title: Adv Data Anal Classif doi: 10.1007/s11634-012-0107-1 contributor: fullname: Mayo-Iscar – volume: 52 start-page: 299 year: 2007 end-page: 308 ident: CR17 article-title: Robust fitting of mixtures using the trimmed likelihood estimator publication-title: Comput Stat Data Anal doi: 10.1016/j.csda.2006.12.024 contributor: fullname: Neytchev – volume: 97 start-page: 611 year: 2002 end-page: 631 ident: CR4 article-title: Model-based clustering, discriminant analysis, and density estimation publication-title: J Am Stat Assoc doi: 10.1198/016214502760047131 contributor: fullname: Raftery – start-page: 67 year: 1982 end-page: 78 ident: CR2 article-title: Algorithms for nonlinear fitting publication-title: Nonlinear optimization contributor: fullname: Dennis – volume: 36 start-page: 1324 year: 2008 end-page: 1345 ident: CR9 article-title: A general trimming approach to robust cluster analysis publication-title: Ann Stat doi: 10.1214/07-AOS515 contributor: fullname: Mayo-Iscar – year: 2000 ident: CR16 publication-title: Finite mixture models doi: 10.1002/0471721182 contributor: fullname: Peel – volume: 4 start-page: 1001 year: 2004 end-page: 1037 ident: CR19 article-title: A unified framework for model-based clustering publication-title: J Mach Learn Res contributor: fullname: Ghosh – volume: 4 start-page: 89 year: 2010 end-page: 109 ident: CR10 article-title: A review of robust clustering methods publication-title: Adv Data Anal Classif doi: 10.1007/s11634-010-0064-5 contributor: fullname: Mayo-Iscar – volume: 33 start-page: 347 year: 2005 end-page: 380 ident: CR5 article-title: A robust method for cluster analysis publication-title: Ann Stat doi: 10.1214/009053604000000940 contributor: fullname: Ritter – start-page: 157 year: 1983 end-page: 184 ident: CR3 article-title: The notion of breakdown point publication-title: A festschrift for Erich L. Lehmann contributor: fullname: Huber – volume: 21 start-page: 585 year: 2011 end-page: 599 ident: CR11 article-title: Exploring the number of groups in robust model-based clustering publication-title: Stat Comput doi: 10.1007/s11222-010-9194-z contributor: fullname: Mayo-Iscar – volume: 13 start-page: 795 year: 1985 ident: 312_CR13 publication-title: Ann Stat doi: 10.1214/aos/1176349557 contributor: fullname: RJ Hathaway – volume: 97 start-page: 611 year: 2002 ident: 312_CR4 publication-title: J Am Stat Assoc doi: 10.1198/016214502760047131 contributor: fullname: C Fraley – volume: 25 start-page: 553 year: 1997 ident: 312_CR1 publication-title: Ann Stat doi: 10.1214/aos/1031833664 contributor: fullname: JA Cuesta-Albertos – volume: 71 start-page: 164 year: 2009 ident: 312_CR6 publication-title: Sankhyā contributor: fullname: MT Gallegos – volume-title: Finite mixture models year: 2000 ident: 312_CR16 doi: 10.1002/0471721182 contributor: fullname: G McLachlan – volume: 94 start-page: 956 year: 1999 ident: 312_CR8 publication-title: J Am Stat Assoc contributor: fullname: LA García-Escudero – volume: 52 start-page: 299 year: 2007 ident: 312_CR17 publication-title: Comput Stat Data Anal doi: 10.1016/j.csda.2006.12.024 contributor: fullname: N Neykov – volume: 3 start-page: 135 year: 2009 ident: 312_CR7 publication-title: Adv Data Anal Classif doi: 10.1007/s11634-009-0044-9 contributor: fullname: MT Gallegos – volume: 21 start-page: 585 year: 2011 ident: 312_CR11 publication-title: Stat Comput doi: 10.1007/s11222-010-9194-z contributor: fullname: LA García-Escudero – start-page: 67 volume-title: Nonlinear optimization year: 1982 ident: 312_CR2 contributor: fullname: JE Dennis Jr. – volume: 36 start-page: 1324 year: 2008 ident: 312_CR9 publication-title: Ann Stat doi: 10.1214/07-AOS515 contributor: fullname: LA García-Escudero – volume: 99 start-page: 1154 year: 2008 ident: 312_CR14 publication-title: J Multivar Anal doi: 10.1016/j.jmva.2007.07.002 contributor: fullname: C Hennig – volume: 4 start-page: 1001 year: 2004 ident: 312_CR19 publication-title: J Mach Learn Res contributor: fullname: S Zhong – volume: 4 start-page: 89 year: 2010 ident: 312_CR10 publication-title: Adv Data Anal Classif doi: 10.1007/s11634-010-0064-5 contributor: fullname: LA García-Escudero – start-page: 157 volume-title: A festschrift for Erich L. Lehmann year: 1983 ident: 312_CR3 contributor: fullname: D Donoho – volume: 33 start-page: 347 year: 2005 ident: 312_CR5 publication-title: Ann Stat doi: 10.1214/009053604000000940 contributor: fullname: MT Gallegos – volume-title: Finding groups in data: an introduction to cluster analysis year: 1990 ident: 312_CR15 doi: 10.1002/9780470316801 contributor: fullname: L Kaufman – volume: 6 start-page: 107 year: 2012 ident: 312_CR18 publication-title: Adv Data Anal Classif doi: 10.1007/s11634-012-0107-1 contributor: fullname: C Ruwet – volume: 65 start-page: 81 year: 2003 ident: 312_CR12 publication-title: J R Stat Soc, Ser B, Stat Methodol doi: 10.1111/1467-9868.00373 contributor: fullname: MG Genton |
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SubjectTerms | Breakdown Breakdown point Cluster analysis Clustering Datasets Economics Eigenvalues Euclidean space Finance Insurance Management Mathematics Mathematics and Statistics Mathématiques Original Paper Physical, chemical, mathematical & earth Sciences Physique, chimie, mathématiques & sciences de la terre Robustness Statistical Theory and Methods Statistics Statistics for Business Studies TCLUST Test methods Trimming |
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