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Mixtures of modified \(t\)-factor analyzers for model-based clustering, classification, and discriminant analysis. (English) Zbl 1204.62098

Summary: A novel family of mixture models is introduced based on modified \(t\)-factor analyzers. Modified factor analyzers were recently introduced within the Gaussian context and our work presents a more flexible and robust alternative. We introduce a family of mixtures of modified \(t\)-factor analyzers that uses this generalized version of the factor analysis covariance structure. We apply this family within three paradigms: model-based clustering; model-based classification; and model-based discriminant analysis. In addition, we apply a recently published Gaussian analogue to this family [P.D. McNicholas and T.B. Murphy, Stat. Comput. 18, 285–296 (2008)] under model-based classification and discriminant analysis paradigms for the first time. Parameter estimation is carried out within the alternating expectation-conditional maximization framework and the Bayesian information criterion is used for model selection. Two real data sets are used to compare our approach to other popular model-based approaches; in these comparisons, the chosen mixtures of modified \(t\)-factor analyzers models perform favourably. We conclude with a summary and suggestions for future work.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H25 Factor analysis and principal components; correspondence analysis

Software:

R; PGMM
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[1] Aitken, A.C., On Bernoulli’s numerical solution of algebraic equations, Proceedings of the royal society of Edinburgh, 46, 289-305, (1926) · JFM 52.0098.05
[2] Anderson, E., The irises of the gaspe peninsula, Bulletin of the American iris society, 59, 2-5, (1935)
[3] Andrews, J.L., McNicholas, P.D., to appear. Extending mixtures of multivariate t-factor analyzers. Statistics and Computing, doi:10.1007/s11222-010-9175-2. · Zbl 1255.62171
[4] Andrews, J.L.; McNicholas, P.D.; Subedi, S., Model-based classification via mixtures of multivariate t-distributions, Computational statistics and data analysis, 55, 1, 520-529, (2011) · Zbl 1247.62151
[5] Banfield, J.D.; Raftery, A.E., Model-based gaussian and non-Gaussian clustering, Biometrics, 49, 3, 803-821, (1993) · Zbl 0794.62034
[6] Besag, J.; Green, P.; Higdon, D.; Mengersen, K., Bayesian computation and stochastic systems, Statistical science, 10, 1, 3-41, (1995) · Zbl 0955.62552
[7] Böhning, D.; Dietz, E.; Schaub, R.; Schlattmann, P.; Lindsay, B., The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family, Annals of the institute of statistical mathematics, 46, 373-388, (1994) · Zbl 0802.62017
[8] Bouveyron, C.; Girard, S.; Schmid, C., High-dimensional data clustering, Computational statistics and data analysis, 52, 1, 502-519, (2007) · Zbl 1452.62433
[9] Celeux, G.; Govaert, G., Gaussian parsimonious clustering models, Pattern recognition, 28, 781-793, (1995)
[10] Dasgupta, A.; Raftery, A.E., Detecting features in spatial point processes with clutter via model-based clustering, Journal of the American statistical association, 93, 94-302, (1998) · Zbl 0906.62105
[11] Dean, N.; Murphy, T.B.; Downey, G., Using unlabelled data to update classification rules with applications in food authenticity studies, Journal of the royal statistical society series C, 55, 1, 1-14, (2006) · Zbl 1490.62155
[12] Dempster, A.P.; Laird, N.M.; Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm, Journal of the royal statistical society series B, 39, 1, 1-38, (1977) · Zbl 0364.62022
[13] Fisher, R.A., The use of multiple measurements in taxonomic problems, Annals of eugenics, 7, 2, 179-188, (1936)
[14] Forina, M.; Armanino, C.; Castino, M.; Ubigli, M., Multivariate data analysis as a discriminating method of the origin of wines, Vitis, 25, 189-201, (1986)
[15] Fraley, C.; Raftery, A.E., Model-based clustering, discriminant analysis, and density estimation, Journal of the American statistical association, 97, 458, 611-631, (2002) · Zbl 1073.62545
[16] Fraley, C., Raftery, A.E., 2006. MCLUSTversion 3 for R: normal mixture modeling and model-based clustering, Technical Report 504, Department of Statistics, University of Washington. Minor revisions January 2007 and November 2007.
[17] Ghahramani, Z., Hinton, G.E., 1997. The EM algorithm for factor analyzers, Technical Report CRG-TR-96-1, University Of Toronto, Toronto.
[18] Hubert, L.; Arabie, P., Comparing partitions, Journal of classification, 2, 193-218, (1985)
[19] Karlis, D.; Meligkotsidou, L., Finite mixtures of multivariate Poisson distributions with application, Journal of statistical planning and inference, 137, 6, 1942-1960, (2007) · Zbl 1116.60006
[20] Karlis, D.; Santourian, A., Model-based clustering with non-elliptically contoured distributions, Statistics and computing, 19, 73-83, (2009)
[21] Keribin, C., Estimation consistante de l’ordre de modèles de mélange, Comptes rendus de l’académie des sciences Série I mathématique, 326, 2, 243-248, (1998) · Zbl 0954.62023
[22] Keribin, C., Consistent estimation of the order of mixture models, Sankhyā, the Indian journal of statistics series A, 62, 1, 49-66, (2000) · Zbl 1081.62516
[23] Lagrange, J.L., Méchanique analitique, (1788), Chez le Veuve Desaint Paris
[24] Leroux, B.G., Consistent estimation of a mixing distribution, The annals of statistics, 20, 1350-1360, (1992) · Zbl 0763.62015
[25] Lindsay, B.G., 1995. Mixture models: theory, geometry and applications, in: ‘NSF-CBMS Regional Conference Series in Probability and Statistics’, vol. 5. Institute of Mathematical Statistics, Hayward, California.
[26] McLachlan, G.J., Discriminant analysis and statistical pattern recognition, (1992), John Wiley & Sons New Jersey
[27] McLachlan, G.J.; Bean, R.W.; Jones, L.B.-T., Extension of the mixture of factor analyzers model to incorporate the multivariate t-distribution, Computational statistics and data analysis, 51, 11, 5327-5338, (2007) · Zbl 1445.62053
[28] McLachlan, G.J.; Krishnan, T., The EM algorithm and extensions, (1997), Wiley New York · Zbl 0882.62012
[29] McLachlan, G.J.; Peel, D., Robust cluster analysis via mixtures of multivariate t-distributions, (), 658-666
[30] McLachlan, G.J.; Peel, D., Finite mixture models, (2000), John Wiley & Sons New York · Zbl 0963.62061
[31] McLachlan, G.J., Peel, D., 2000b. Mixtures of factor analyzers. In: Proceedings of the Seventh International Conference on Machine Learning. Morgan Kaufmann, San Francisco, pp. 599-606.
[32] McNicholas, P.D., Model-based classification using latent Gaussian mixture models, Journal of statistical planning and inference, 140, 5, 1175-1181, (2010) · Zbl 1181.62095
[33] McNicholas, P.D.; Murphy, T.B., Parsimonious Gaussian mixture models, Statistics and computing, 18, 285-296, (2008)
[34] McNicholas, P.D.; Murphy, T.B., Model-based clustering of longitudinal data, The Canadian journal of statistics, 38, 1, 153-168, (2010) · Zbl 1190.62120
[35] McNicholas, P.D., Murphy, T.B., 2010b. Model-based clustering of microarray expression data via latent Gaussian mixture models. Bioinformatics 26. (21), 2705-2712.
[36] McNicholas, P.D.; Murphy, T.B.; McDaid, A.F.; Frost, D., Serial and parallel implementations of model-based clustering via parsimonious Gaussian mixture models, Computational statistics and data analysis, 54, 3, 711-723, (2010) · Zbl 1464.62131
[37] Meng, X.-L.; Rubin, D.B., Maximum likelihood estimation via the ECM algorithm: a general framework, Biometrika, 80, 267-278, (1993) · Zbl 0778.62022
[38] Meng, X.-L.; van Dyk, D., The EM algorithm—an old folk song sung to a fast new tune (with discussion), Journal of the royal statistical society series B, 59, 511-567, (1997) · Zbl 1090.62518
[39] R Development Core Team 2010. R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria URL: \(\langle\)http://www.R-project.org〉.
[40] Rand, W.M., Objective criteria for the evaluation of clustering methods, Journal of the American statistical association, 66, 846-850, (1971)
[41] Roeder, K.; Wasserman, L., Practical Bayesian density estimation using mixtures of normals, Journal of the American statistical association, 92, 894-902, (1997) · Zbl 0889.62021
[42] Schwarz, G., Estimating the dimension of a model, The annals of statistics, 6, 31-38, (1978)
[43] Spearman, C., The proof and measurement of association between two things, American journal of psychology, 15, 72-101, (1904)
[44] Tipping, T.E.; Bishop, C.M., Mixtures of probabilistic principal component analysers, Neural computation, 11, 2, 443-482, (1999)
[45] Tipping, T.E.; Bishop, C.M., Probabilistic principal component analysers, Journal of the royal statistical society series B, 61, 611-622, (1999) · Zbl 0924.62068
[46] Titterington, D.M.; Smith, A.F.M.; Makov, U.E., Statistical analysis of finite mixture distributions, (1985), John Wiley & Sons Chichester · Zbl 0646.62013
[47] Woodbury, M.A., Inverting modified matrices, statistical research group, memographic report no. 42, (1950), Princeton University Princeton, New Jersey
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