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Automated learning of t factor analysis models with complete and incomplete data

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  • Wang, Wan-Lun
  • Castro, Luis M.
  • Lin, Tsung-I

Abstract

The t factor analysis (tFA) model is a promising tool for robust reduction of high-dimensional data in the presence of heavy-tailed noises. When determining the number of factors of the tFA model, a two-stage procedure is commonly performed in which parameter estimation is carried out for a number of candidate models, and then the best model is chosen according to certain penalized likelihood indices such as the Bayesian information criterion. However, the computational burden of such a procedure could be extremely high to achieve the optimal performance, particularly for extensively large data sets. In this paper, we develop a novel automated learning method in which parameter estimation and model selection are seamlessly integrated into a one-stage algorithm. This new scheme is called the automated tFA (AtFA) algorithm, and it is also workable when values are missing. In addition, we derive the Fisher information matrix to approximate the asymptotic covariance matrix associated with the ML estimators of tFA models. Experiments on real and simulated data sets reveal that the AtFA algorithm not only provides identical fitting results, as compared to traditional two-stage procedures, but also runs much faster, especially when values are missing.

Suggested Citation

  • Wang, Wan-Lun & Castro, Luis M. & Lin, Tsung-I, 2017. "Automated learning of t factor analysis models with complete and incomplete data," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 157-171.
    • Handle: RePEc:eee:jmvana:v:161:y:2017:i:c:p:157-171
    DOI: 10.1016/j.jmva.2017.07.009
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    References listed on IDEAS

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    1. M. Liu & T.I. Lin, 2015. "Skew-normal factor analysis models with incomplete data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(4), pages 789-805, April.
    2. Walter Ledermann, 1937. "On the rank of the reduced correlational matrix in multiple-factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 2(2), pages 85-93, June.
    3. Wan-Lun Wang & Min Liu & Tsung-I Lin, 2017. "Robust skew-t factor analysis models for handling missing data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(4), pages 649-672, November.
    4. Ibrahim, Joseph G. & Zhu, Hongtu & Tang, Niansheng, 2008. "Model Selection Criteria for Missing-Data Problems Using the EM Algorithm," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1648-1658.
    5. Zhao, Jianhua & Shi, Lei, 2014. "Automated learning of factor analysis with complete and incomplete data," Computational Statistics & Data Analysis, Elsevier, vol. 72(C), pages 205-218.
    6. McLachlan, G.J. & Bean, R.W. & Ben-Tovim Jones, L., 2007. "Extension of the mixture of factor analyzers model to incorporate the multivariate t-distribution," Computational Statistics & Data Analysis, Elsevier, vol. 51(11), pages 5327-5338, July.
    7. Hamparsum Bozdogan, 1987. "Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions," Psychometrika, Springer;The Psychometric Society, vol. 52(3), pages 345-370, September.
    8. Otto Driel, 1978. "On various causes of improper solutions in maximum likelihood factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 43(2), pages 225-243, June.
    9. Wang, Wan-Lun, 2015. "Mixtures of common t-factor analyzers for modeling high-dimensional data with missing values," Computational Statistics & Data Analysis, Elsevier, vol. 83(C), pages 223-235.
    10. Stanley Sclove, 1987. "Application of model-selection criteria to some problems in multivariate analysis," Psychometrika, Springer;The Psychometric Society, vol. 52(3), pages 333-343, September.
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    Cited by:

    1. Christian E. Galarza & Tsung-I Lin & Wan-Lun Wang & Víctor H. Lachos, 2021. "On moments of folded and truncated multivariate Student-t distributions based on recurrence relations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(6), pages 825-850, August.
    2. Wan-Lun Wang & Tsung-I Lin, 2023. "Model-based clustering via mixtures of unrestricted skew normal factor analyzers with complete and incomplete data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 32(3), pages 787-817, September.
    3. Wan-Lun Wang & Tsung-I Lin, 2022. "Robust clustering via mixtures of t factor analyzers with incomplete 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. 16(3), pages 659-690, September.

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