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Clusters, outliers, and regression: fixed point clusters

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  • Hennig, Christian

Abstract

Fixed point clustering is a new stochastic approach to cluster analysis. The definition of a single fixed point cluster (FPC) is based on a simple parametric model, but there is no parametric assumption for the whole dataset as opposed to mixture modeling and other approaches. An FPC is defined as a data subset that is exactly the set of non-outliers with respect to its own parameter estimators. This paper concentrates upon the theoretical foundation of FPC analysis as a method for clusterwise linear regression, i.e., the single clusters are modeled as linear regressions with normal errors. In this setup, fixed point clustering is based on an iteratively reweighted estimation with zero weight for all outliers. FPCs are non-hierarchical, but they may overlap and include each other. A specification of the number of clusters is not needed. Consistency results are given for certain mixture models of interest in cluster analysis. Convergence of a fixed point algorithm is shown. Application to a real dataset shows that fixed point clustering can highlight some other interesting features of datasets compared to maximum likelihood methods in the presence of deviations from the usual assumptions of model based cluster analysis.

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  • Hennig, Christian, 2003. "Clusters, outliers, and regression: fixed point clusters," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 183-212, July.
    • Handle: RePEc:eee:jmvana:v:86:y:2003:i:1:p:183-212
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    References listed on IDEAS

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    1. Hennig, Christian & Christlieb, Norbert, 2002. "Validating visual clusters in large datasets: fixed point clusters of spectral features," Computational Statistics & Data Analysis, Elsevier, vol. 40(4), pages 723-739, October.
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    5. Fair, Ray C & Jaffee, Dwight M, 1972. "Methods of Estimation for Markets in Disequilibrium," Econometrica, Econometric Society, vol. 40(3), pages 497-514, May.
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    1. D'Urso, Pierpaolo & Santoro, Adriana, 2006. "Fuzzy clusterwise linear regression analysis with symmetrical fuzzy output variable," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 287-313, November.
    2. Yao, Weixin & Wei, Yan & Yu, Chun, 2014. "Robust mixture regression using the t-distribution," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 116-127.
    3. Müller, Christine H. & Garlipp, Tim, 2005. "Simple consistent cluster methods based on redescending M-estimators with an application to edge identification in images," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 359-385, February.
    4. Bai, Xiuqin & Yao, Weixin & Boyer, John E., 2012. "Robust fitting of mixture regression models," Computational Statistics & Data Analysis, Elsevier, vol. 56(7), pages 2347-2359.
    5. Rainer Schlittgen, 2011. "A weighted least-squares approach to clusterwise regression," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 95(2), pages 205-217, June.
    6. L. A. García‐Escudero & A. Gordaliza & R. San Martín & S. Van Aelst & R. Zamar, 2009. "Robust linear clustering," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 301-318, January.
    7. Francesca Torti & Domenico Perrotta & Marco Riani & Andrea Cerioli, 2019. "Assessing trimming methodologies for clustering linear regression data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 13(1), pages 227-257, March.
    8. García-Escudero, L.A. & Gordaliza, A. & Mayo-Iscar, A. & San Martín, R., 2010. "Robust clusterwise linear regression through trimming," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3057-3069, December.
    9. Neykov, N. & Filzmoser, P. & Dimova, R. & Neytchev, P., 2007. "Robust fitting of mixtures using the trimmed likelihood estimator," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 299-308, September.

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