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 inQuality & quantity Vol. 51; no. 2; pp. 859 - 872
Main Authors Bonanomi, Andrea, Nai Ruscone, Marta, Osmetti, Silvia Angela
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.03.2017
Springer
Springer Nature B.V
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ISSN0033-5177
1573-7845
DOI10.1007/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.
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
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Hierarchical cluster analysis
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– 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
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Snippet We propose a new measure to evaluate the distance between subjects expressing their preferences by rankings in order to segment them by hierarchical cluster...
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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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