Defining subjects distance in hierarchical cluster analysis by copula approach
We propose a new measure to evaluate the distance between subjects expressing their preferences by rankings in order to segment them by hierarchical cluster analysis. The proposed index builds upon the Spearman’s grade correlation coefficient on a transformation, operated by the copula function, of...
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| Published in | Quality & quantity Vol. 51; no. 2; pp. 859 - 872 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.03.2017
Springer Springer Nature B.V |
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| Online Access | Get full text |
| ISSN | 0033-5177 1573-7845 |
| DOI | 10.1007/s11135-016-0444-9 |
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| Abstract | We propose a new measure to evaluate the distance between subjects expressing their preferences by rankings in order to segment them by hierarchical cluster analysis. The proposed index builds upon the Spearman’s grade correlation coefficient on a transformation, operated by the copula function, of the position/rank denoting the level of the importance assigned by subjects under classification to
k
objects. In particular, by using the copula functions with tail dependence we obtain an index suitable for emphasizing the agreement on top ranks, when the top ranks are considered more important than the lower ones. We evaluate the performance of our proposal by an example on simulated data, showing that the resulting groups contain subjects whose preferences are more similar on the most important ranks. A further application with real data confirms the pertinence and the importance of our proposal. |
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| AbstractList | We propose a new measure to evaluate the distance between subjects expressing their preferences by rankings in order to segment them by hierarchical cluster analysis. The proposed index builds upon the Spearman’s grade correlation coefficient on a transformation, operated by the copula function, of the position/rank denoting the level of the importance assigned by subjects under classification to
k
objects. In particular, by using the copula functions with tail dependence we obtain an index suitable for emphasizing the agreement on top ranks, when the top ranks are considered more important than the lower ones. We evaluate the performance of our proposal by an example on simulated data, showing that the resulting groups contain subjects whose preferences are more similar on the most important ranks. A further application with real data confirms the pertinence and the importance of our proposal. We propose a new measure to evaluate the distance between subjects expressing their preferences by rankings in order to segment them by hierarchical cluster analysis. The proposed index builds upon the Spearman's grade correlation coefficient on a transformation, operated by the copula function, of the position/rank denoting the level of the importance assigned by subjects under classification to k objects. In particular, by using the copula functions with tail dependence we obtain an index suitable for emphasizing the agreement on top ranks, when the top ranks are considered more important than the lower ones. We evaluate the performance of our proposal by an example on simulated data, showing that the resulting groups contain subjects whose preferences are more similar on the most important ranks. A further application with real data confirms the pertinence and the importance of our proposal. |
| Audience | Academic |
| Author | Nai Ruscone, Marta Bonanomi, Andrea Osmetti, Silvia Angela |
| Author_xml | – sequence: 1 givenname: Andrea surname: Bonanomi fullname: Bonanomi, Andrea organization: Department of Statistical Sciences, Università Cattolica del Sacro Cuore di Milano – sequence: 2 givenname: Marta surname: Nai Ruscone fullname: Nai Ruscone, Marta email: mnairuscone@liuc.it organization: Scuola di Economia e Management, Università Cattaneo LIUC – sequence: 3 givenname: Silvia Angela surname: Osmetti fullname: Osmetti, Silvia Angela organization: Department of Statistical Sciences, Università Cattolica del Sacro Cuore di Milano |
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| CitedBy_id | crossref_primary_10_1002_sam_11402 crossref_primary_10_1016_j_ijar_2024_109185 crossref_primary_10_1007_s00357_022_09420_0 crossref_primary_10_1142_S0219477522500584 crossref_primary_10_1007_s00357_024_09495_x crossref_primary_10_1016_j_compeleceng_2019_06_011 crossref_primary_10_1016_j_ecolind_2019_105516 |
| Cites_doi | 10.1111/j.1467-842X.2005.00413.x 10.2307/1412159 10.1016/j.csda.2012.08.008 10.1016/j.csda.2012.02.002 10.1214/lnms/1215467407 10.1016/0022-2496(91)90050-4 10.1080/02664763.2015.1125864 10.1016/j.jspi.2014.02.011 10.1093/biomet/44.1-2.114 10.1007/s11135-014-0114-8 10.1007/978-1-4939-1471-5 10.1016/S0167-7152(98)00006-6 10.2307/2347613 10.1201/b13150 10.1016/j.csda.2009.07.014 10.1007/978-3-319-00032-9_13 10.1002/mcda.313 10.1007/978-1-4612-2738-0_5 |
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| Keywords | Copula function Hierarchical cluster analysis Ordinal data |
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| References | DiaconisPGroup Representations in Probability and Statistics. Lecture Notes-Monograph Series1988HaywardInstitute of Mathematical Statistics1192 AlvoMYuPLHStatistical Methods for Ranking Data2014New YorkSpringer10.1007/978-1-4939-1471-5 FeiginPDFlignerMAVerducciJSModelling and analysing paired ranking dataProbability Models and Statistical Analyses for Ranking Data1993New YorkSpringer759110.1007/978-1-4612-2738-0_5 ZaniSCerioliAAnalisi dei dati e data mining per le decisioni aziendali2007MilanoGiuffrè Editore CritchlowDEFlignerMAVerducciJSProbability models on rankingsJ. Math. Psychol.19913529431810.1016/0022-2496(91)90050-4 NelsenRBAn Introduction to Copulas2013New YorkSpringer Quade, D., Salama, I.: A survey of weighted rank correlation. In: Order Statistics and Nonparametrics: Theory and Applications, pp. 213–224. Elsevier, Amsterdam (1992) SklarAWFonctions de répartition à n dimension et leurs margesPubl. Inst. Stat. Univ. Paris19598229231 BrentariEDancelliLManiseraMClustering ranking data in market segmentation: a case study on the Italian McDonald’s customers’ preferencesJ. Appl. Stat.2016431111810.1080/02664763.2015.1125864 ShiehGSA weighted Kendall’s tau statisticStat. Probab. Lett.198839172410.1016/S0167-7152(98)00006-6 JoeHMultivariate Models and Dependence Concepts1997Boca RatonChapman & Hall10.1201/b13150 Tarsitano, A.: Weighted rank correlation and hierarchical clustering. In: Book of Short Papers CLADAG. Parma, pp. 517–521 (2005) Dancelli, L., Manisera, M., Vezzoli, M.: On two classes of Weighted Rank Correlation measures deriving from the Spearman’s ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho$$\end{document}. In: Statistical Model and Data Analysis, pp. 107–114. Springer, New York (2013) BiernackiCJacquesJA generative model for rank data based on insertion sort algorithmComput. Stat. Data Anal.20135816217610.1016/j.csda.2012.08.008 da CostaPJSolaresCA weighted rank measure of correlationAust. N Z. J. Stat.20054751552910.1111/j.1467-842X.2005.00413.x Dancelli, L., Manisera, M., Vezzoli, M.: Weighted Rank Correlation measures in hierarchical cluster analysis. Book of Short Papers JCS-CLADAG 2012. Anacapri (2012) CritchlowDVerducciJDetecting a trend in paired rankingsAppl. Stat.199241172910.2307/2347613 SpearmanCThe proof and measurement of association between two thingsAm. J. Psychol.1904157210110.2307/1412159 EmondEMasonDA new rank correlation coefficient with application to the consensus ranking problemJ. Multi-Criteria Decis. Anal.200211172810.1002/mcda.313 BonanomiACantaluppiGNai RusconeMOsmettiSAA new estimator of Zumbo’s Ordinal Alpha: a copula approachQual. Quant.20154994195310.1007/s11135-014-0114-8 JacquesJBiernackiCModel-based clustering for multivariate partial ranking dataJ. Stat. Plan. Inference201414920121710.1016/j.jspi.2014.02.011 KojadinovicIHierarchical clustering of continuous variables based on the empirical copula process and permutation linkagesComput. Stat. Data Anal.2010549010810.1016/j.csda.2009.07.014 LeePHYuPLHMixtures of weighted distance-based models for ranking data with applications in political studiesComput. Stat. Data Anal.2012562486250010.1016/j.csda.2012.02.002 MallowsCLNon-null ranking models. IBiometrika19574411413010.1093/biomet/44.1-2.114 PJ Costa da (444_CR18) 2005; 47 P Diaconis (444_CR9) 1988 E Brentari (444_CR4) 2016; 43 I Kojadinovic (444_CR14) 2010; 54 J Jacques (444_CR12) 2014; 149 D Critchlow (444_CR6) 1992; 41 C Spearman (444_CR22) 1904; 15 E Emond (444_CR10) 2002; 11 GS Shieh (444_CR20) 1988; 39 PD Feigin (444_CR11) 1993 444_CR23 M Alvo (444_CR1) 2014 S Zani (444_CR24) 2007 C Biernacki (444_CR2) 2013; 58 CL Mallows (444_CR16) 1957; 44 444_CR7 444_CR19 AW Sklar (444_CR21) 1959; 8 DE Critchlow (444_CR5) 1991; 35 444_CR8 PH Lee (444_CR15) 2012; 56 RB Nelsen (444_CR17) 2013 A Bonanomi (444_CR3) 2015; 49 H Joe (444_CR13) 1997 |
| References_xml | – reference: DiaconisPGroup Representations in Probability and Statistics. Lecture Notes-Monograph Series1988HaywardInstitute of Mathematical Statistics1192 – reference: JoeHMultivariate Models and Dependence Concepts1997Boca RatonChapman & Hall10.1201/b13150 – reference: Tarsitano, A.: Weighted rank correlation and hierarchical clustering. In: Book of Short Papers CLADAG. Parma, pp. 517–521 (2005) – reference: Dancelli, L., Manisera, M., Vezzoli, M.: Weighted Rank Correlation measures in hierarchical cluster analysis. Book of Short Papers JCS-CLADAG 2012. Anacapri (2012) – reference: SklarAWFonctions de répartition à n dimension et leurs margesPubl. Inst. Stat. Univ. Paris19598229231 – reference: BrentariEDancelliLManiseraMClustering ranking data in market segmentation: a case study on the Italian McDonald’s customers’ preferencesJ. Appl. Stat.2016431111810.1080/02664763.2015.1125864 – reference: Quade, D., Salama, I.: A survey of weighted rank correlation. In: Order Statistics and Nonparametrics: Theory and Applications, pp. 213–224. Elsevier, Amsterdam (1992) – reference: SpearmanCThe proof and measurement of association between two thingsAm. J. Psychol.1904157210110.2307/1412159 – reference: Dancelli, L., Manisera, M., Vezzoli, M.: On two classes of Weighted Rank Correlation measures deriving from the Spearman’s ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho$$\end{document}. In: Statistical Model and Data Analysis, pp. 107–114. Springer, New York (2013) – reference: EmondEMasonDA new rank correlation coefficient with application to the consensus ranking problemJ. Multi-Criteria Decis. Anal.200211172810.1002/mcda.313 – reference: FeiginPDFlignerMAVerducciJSModelling and analysing paired ranking dataProbability Models and Statistical Analyses for Ranking Data1993New YorkSpringer759110.1007/978-1-4612-2738-0_5 – reference: NelsenRBAn Introduction to Copulas2013New YorkSpringer – reference: AlvoMYuPLHStatistical Methods for Ranking Data2014New YorkSpringer10.1007/978-1-4939-1471-5 – reference: MallowsCLNon-null ranking models. IBiometrika19574411413010.1093/biomet/44.1-2.114 – reference: ZaniSCerioliAAnalisi dei dati e data mining per le decisioni aziendali2007MilanoGiuffrè Editore – reference: BiernackiCJacquesJA generative model for rank data based on insertion sort algorithmComput. Stat. Data Anal.20135816217610.1016/j.csda.2012.08.008 – reference: JacquesJBiernackiCModel-based clustering for multivariate partial ranking dataJ. Stat. Plan. Inference201414920121710.1016/j.jspi.2014.02.011 – reference: ShiehGSA weighted Kendall’s tau statisticStat. Probab. Lett.198839172410.1016/S0167-7152(98)00006-6 – reference: CritchlowDVerducciJDetecting a trend in paired rankingsAppl. Stat.199241172910.2307/2347613 – reference: KojadinovicIHierarchical clustering of continuous variables based on the empirical copula process and permutation linkagesComput. Stat. Data Anal.2010549010810.1016/j.csda.2009.07.014 – reference: BonanomiACantaluppiGNai RusconeMOsmettiSAA new estimator of Zumbo’s Ordinal Alpha: a copula approachQual. Quant.20154994195310.1007/s11135-014-0114-8 – reference: CritchlowDEFlignerMAVerducciJSProbability models on rankingsJ. Math. Psychol.19913529431810.1016/0022-2496(91)90050-4 – reference: da CostaPJSolaresCA weighted rank measure of correlationAust. N Z. J. Stat.20054751552910.1111/j.1467-842X.2005.00413.x – reference: LeePHYuPLHMixtures of weighted distance-based models for ranking data with applications in political studiesComput. Stat. Data Anal.2012562486250010.1016/j.csda.2012.02.002 – volume: 47 start-page: 515 year: 2005 ident: 444_CR18 publication-title: Aust. N Z. J. Stat. doi: 10.1111/j.1467-842X.2005.00413.x – volume-title: An Introduction to Copulas year: 2013 ident: 444_CR17 – volume: 15 start-page: 72 year: 1904 ident: 444_CR22 publication-title: Am. J. Psychol. doi: 10.2307/1412159 – volume: 58 start-page: 162 year: 2013 ident: 444_CR2 publication-title: Comput. Stat. Data Anal. doi: 10.1016/j.csda.2012.08.008 – volume: 56 start-page: 2486 year: 2012 ident: 444_CR15 publication-title: Comput. Stat. Data Anal. doi: 10.1016/j.csda.2012.02.002 – start-page: 1 volume-title: Group Representations in Probability and Statistics. Lecture Notes-Monograph Series year: 1988 ident: 444_CR9 doi: 10.1214/lnms/1215467407 – ident: 444_CR7 – volume-title: Analisi dei dati e data mining per le decisioni aziendali year: 2007 ident: 444_CR24 – volume: 35 start-page: 294 year: 1991 ident: 444_CR5 publication-title: J. Math. Psychol. doi: 10.1016/0022-2496(91)90050-4 – ident: 444_CR19 – volume: 43 start-page: 1 issue: 11 year: 2016 ident: 444_CR4 publication-title: J. Appl. Stat. doi: 10.1080/02664763.2015.1125864 – volume: 149 start-page: 201 year: 2014 ident: 444_CR12 publication-title: J. Stat. Plan. Inference doi: 10.1016/j.jspi.2014.02.011 – volume: 44 start-page: 114 year: 1957 ident: 444_CR16 publication-title: Biometrika doi: 10.1093/biomet/44.1-2.114 – volume: 49 start-page: 941 year: 2015 ident: 444_CR3 publication-title: Qual. Quant. doi: 10.1007/s11135-014-0114-8 – volume-title: Statistical Methods for Ranking Data year: 2014 ident: 444_CR1 doi: 10.1007/978-1-4939-1471-5 – volume: 39 start-page: 17 year: 1988 ident: 444_CR20 publication-title: Stat. Probab. Lett. doi: 10.1016/S0167-7152(98)00006-6 – ident: 444_CR23 – volume: 41 start-page: 17 year: 1992 ident: 444_CR6 publication-title: Appl. Stat. doi: 10.2307/2347613 – volume: 8 start-page: 229 year: 1959 ident: 444_CR21 publication-title: Publ. Inst. Stat. Univ. Paris – volume-title: Multivariate Models and Dependence Concepts year: 1997 ident: 444_CR13 doi: 10.1201/b13150 – volume: 54 start-page: 90 year: 2010 ident: 444_CR14 publication-title: Comput. Stat. Data Anal. doi: 10.1016/j.csda.2009.07.014 – ident: 444_CR8 doi: 10.1007/978-3-319-00032-9_13 – volume: 11 start-page: 17 year: 2002 ident: 444_CR10 publication-title: J. Multi-Criteria Decis. Anal. doi: 10.1002/mcda.313 – start-page: 75 volume-title: Probability Models and Statistical Analyses for Ranking Data year: 1993 ident: 444_CR11 doi: 10.1007/978-1-4612-2738-0_5 |
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| SubjectTerms | Agreements Algorithms Classification Cluster analysis Clustering Consumers Data analysis Methodology of the Social Sciences Preferences Random variables Ratings & rankings Social Sciences Studies Transformation |
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| Title | Defining subjects distance in hierarchical cluster analysis by copula approach |
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