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Clustering and classification via cluster-weighted factor analyzers

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  • Sanjeena Subedi
  • Antonio Punzo
  • Salvatore Ingrassia
  • Paul McNicholas

Abstract

In model-based clustering and classification, the cluster-weighted model is a convenient approach when the random vector of interest is constituted by a response variable $$Y$$ and by a vector $${\varvec{X}}$$ of $$p$$ covariates. However, its applicability may be limited when $$p$$ is high. To overcome this problem, this paper assumes a latent factor structure for $${\varvec{X}}$$ in each mixture component, under Gaussian assumptions. This leads to the cluster-weighted factor analyzers (CWFA) model. By imposing constraints on the variance of $$Y$$ and the covariance matrix of $${\varvec{X}}$$ , a novel family of sixteen CWFA models is introduced for model-based clustering and classification. The alternating expectation-conditional maximization algorithm, for maximum likelihood estimation of the parameters of all models in the family, is described; to initialize the algorithm, a 5-step hierarchical procedure is proposed, which uses the nested structures of the models within the family and thus guarantees the natural ranking among the sixteen likelihoods. Artificial and real data show that these models have very good clustering and classification performance and that the algorithm is able to recover the parameters very well. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Sanjeena Subedi & Antonio Punzo & Salvatore Ingrassia & Paul McNicholas, 2013. "Clustering and classification via cluster-weighted factor analyzers," 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. 7(1), pages 5-40, March.
    • Handle: RePEc:spr:advdac:v:7:y:2013:i:1:p:5-40
    DOI: 10.1007/s11634-013-0124-8
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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. Michael P. B. Gallaugher & Salvatore D. Tomarchio & Paul D. McNicholas & Antonio Punzo, 2022. "Multivariate cluster weighted models using skewed distributions," 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. 16(1), pages 93-124, March.
    3. Salvatore Ingrassia & Antonio Punzo, 2020. "Cluster Validation for Mixtures of Regressions via the Total Sum of Squares Decomposition," Journal of Classification, Springer;The Classification Society, vol. 37(2), pages 526-547, July.
    4. Paul D. McNicholas, 2016. "Model-Based Clustering," Journal of Classification, Springer;The Classification Society, vol. 33(3), pages 331-373, October.
    5. Angelo Mazza & Antonio Punzo, 2020. "Mixtures of multivariate contaminated normal regression models," Statistical Papers, Springer, vol. 61(2), pages 787-822, April.
    6. Morris, Katherine & Punzo, Antonio & McNicholas, Paul D. & Browne, Ryan P., 2019. "Asymmetric clusters and outliers: Mixtures of multivariate contaminated shifted asymmetric Laplace distributions," Computational Statistics & Data Analysis, Elsevier, vol. 132(C), pages 145-166.
    7. Gabriele Soffritti, 2021. "Estimating the Covariance Matrix of the Maximum Likelihood Estimator Under Linear Cluster-Weighted Models," Journal of Classification, Springer;The Classification Society, vol. 38(3), pages 594-625, October.
    8. Luis Angel García-Escudero & Alfonso Gordaliza & Francesca Greselin & Salvatore Ingrassia & Agustín Mayo-Iscar, 2018. "Eigenvalues and constraints in mixture modeling: geometric and computational issues," 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. 12(2), pages 203-233, June.
    9. Salvatore D. Tomarchio & Paul D. McNicholas & Antonio Punzo, 2021. "Matrix Normal Cluster-Weighted Models," Journal of Classification, Springer;The Classification Society, vol. 38(3), pages 556-575, October.
    10. Perthame, Emeline & Forbes, Florence & Deleforge, Antoine, 2018. "Inverse regression approach to robust nonlinear high-to-low dimensional mapping," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 1-14.
    11. Salvatore Ingrassia & Antonio Punzo & Giorgio Vittadini & Simona Minotti, 2015. "Erratum to: The Generalized Linear Mixed Cluster-Weighted Model," Journal of Classification, Springer;The Classification Society, vol. 32(2), pages 327-355, July.
    12. Michael P. B. Gallaugher & Paul D. McNicholas, 2019. "On Fractionally-Supervised Classification: Weight Selection and Extension to the Multivariate t-Distribution," Journal of Classification, Springer;The Classification Society, vol. 36(2), pages 232-265, July.
    13. Utkarsh J. Dang & Antonio Punzo & Paul D. McNicholas & Salvatore Ingrassia & Ryan P. Browne, 2017. "Multivariate Response and Parsimony for Gaussian Cluster-Weighted Models," Journal of Classification, Springer;The Classification Society, vol. 34(1), pages 4-34, April.
    14. Maruotti, Antonello & Punzo, Antonio, 2017. "Model-based time-varying clustering of multivariate longitudinal data with covariates and outliers," Computational Statistics & Data Analysis, Elsevier, vol. 113(C), pages 475-496.
    15. Leila Amiri & Mojtaba Khazaei & Mojtaba Ganjali, 2017. "General location model with factor analyzer covariance matrix structure and its applications," 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. 11(3), pages 593-609, September.
    16. Antonio Punzo & Salvatore Ingrassia & Antonello Maruotti, 2021. "Multivariate hidden Markov regression models: random covariates and heavy-tailed distributions," Statistical Papers, Springer, vol. 62(3), pages 1519-1555, June.
    17. Počuča, Nikola & Jevtić, Petar & McNicholas, Paul D. & Miljkovic, Tatjana, 2020. "Modeling frequency and severity of claims with the zero-inflated generalized cluster-weighted models," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 79-93.
    18. Diani, Cecilia & Galimberti, Giuliano & Soffritti, Gabriele, 2022. "Multivariate cluster-weighted models based on seemingly unrelated linear regression," Computational Statistics & Data Analysis, Elsevier, vol. 171(C).
    19. Sanjeena Subedi & Antonio Punzo & Salvatore Ingrassia & Paul McNicholas, 2015. "Cluster-weighted $$t$$ t -factor analyzers for robust model-based clustering and dimension reduction," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(4), pages 623-649, November.
    20. Salvatore Ingrassia & Antonio Punzo & Giorgio Vittadini & Simona Minotti, 2015. "The Generalized Linear Mixed Cluster-Weighted Model," Journal of Classification, Springer;The Classification Society, vol. 32(1), pages 85-113, April.

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