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Robust linear clustering

Author

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  • L. A. García‐Escudero
  • A. Gordaliza
  • R. San Martín
  • S. Van Aelst
  • R. Zamar

Abstract

Summary. Non‐hierarchical clustering methods are frequently based on the idea of forming groups around ‘objects’. The main exponent of this class of methods is the k‐means method, where these objects are points. However, clusters in a data set may often be due to certain relationships between the measured variables. For instance, we can find linear structures such as straight lines and planes, around which the observations are grouped in a natural way. These structures are not well represented by points. We present a method that searches for linear groups in the presence of outliers. The method is based on the idea of impartial trimming. We search for the ‘best’ subsample containing a proportion 1−α of the data and the best k affine subspaces fitting to those non‐discarded observations by measuring discrepancies through orthogonal distances. The population version of the sample problem is also considered. We prove the existence of solutions for the sample and population problems together with their consistency. A feasible algorithm for solving the sample problem is described as well. Finally, some examples showing how the method proposed works in practice are provided.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jorssb:v:71:y:2009:i:1:p:301-318
    DOI: 10.1111/j.1467-9868.2008.00682.x
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    References listed on IDEAS

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    1. Van Aelst, Stefan & (Steven) Wang, Xiaogang & Zamar, Ruben H. & Zhu, Rong, 2006. "Linear grouping using orthogonal regression," Computational Statistics & Data Analysis, Elsevier, vol. 50(5), pages 1287-1312, March.
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    6. Hennig, Christian, 2003. "Clusters, outliers, and regression: fixed point clusters," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 183-212, July.
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    Citations

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    Cited by:

    1. Antonio Punzo & Paul. D. McNicholas, 2017. "Robust Clustering in Regression Analysis via the Contaminated Gaussian Cluster-Weighted Model," Journal of Classification, Springer;The Classification Society, vol. 34(2), pages 249-293, July.
    2. Hu, Hao & Yao, Weixin & Wu, Yichao, 2017. "The robust EM-type algorithms for log-concave mixtures of regression models," Computational Statistics & Data Analysis, Elsevier, vol. 111(C), pages 14-26.
    3. 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.
    4. 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.
    5. Gao, Jinxin & Hitchcock, David B., 2010. "James-Stein shrinkage to improve k-means cluster analysis," Computational Statistics & Data Analysis, Elsevier, vol. 54(9), pages 2113-2127, September.
    6. Luca Greco, 2022. "Robust fitting of mixtures of GLMs by weighted likelihood," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(1), pages 25-48, March.
    7. Luca Greco & Antonio Lucadamo & Claudio Agostinelli, 2021. "Weighted likelihood latent class linear regression," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(2), pages 711-746, June.
    8. Andrea Cerioli & Domenico Perrotta, 2014. "Robust clustering around regression lines with high density regions," 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. 8(1), pages 5-26, March.
    9. 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.
    10. Stefan Aelst & Ruben H. Zamar, 2019. "Comments on: Data science, big data and statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 360-362, June.
    11. Luis García-Escudero & Alfonso Gordaliza & Carlos Matrán & Agustín Mayo-Iscar, 2010. "A review of robust clustering methods," 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. 4(2), pages 89-109, September.

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