臺灣博碩士論文加值系統

因子分析（FA）是一種典型的資料縮減方法，目的是在尋找較低維度的變數，用以解釋相互之間有潛在關係存在的變數。本文將常態因子模型延伸，假設其潛在因子與未知的誤差皆為偏斜t分佈模型，並將此模型稱為偏斜t因子（STFA）模型。此模型對違反常態假設的潛在因子展現了穩健性也提供了捕捉偏斜且厚尾的觀測數據較彈性的選擇。我們發展了一個計算上便利的EM-type演算法去迭代計算出參數的最大概似估計值。最後，我們透過一組實例分析闡述所提出的方法的實用性且結果會較優於現有的一些其他的方法。

Factor analysis(FA) is a classical data reduction technique that seeks a potentially lower number of unobserved variables accounting for most correlation among the observed variables. This thesis presents an extension of the FA model by assuming jointly a restricted version of multivariate skew t distribution for the latent factors and unobservable errors, called the skew-t FA model. The proposed model shows robustness to violations of normality assumptions of the underlying latent factors and provides flexibility in capturing extra skewness as well as heavier tails of the observed data. A computationally feasible EM-type algorithm is developed for computing maximum likelihood estimates of the parameters. The usefulness of the proposed methodology is illustrated by a real-life example and result also demonstrates its better performance over various existing methods.

1. Introduction 1
2. Preliminaries 4
2.1. The restricted multivariate skew normal distribution 4
2.2. The restricted multivariate skew t distribution 5
3. Skew-t factor analysis model 8
3.1. Model formulation 8
3.2. Maximum likelihood estimation via the ECM algorithm 9
4. Provision of standard errors 15
5. Numerical illustration 17
6. Conclusion 25
A. Proof of Proposition 1 26
B. Proof of Proposition 3 27
C. Proof of Proposition 4 28
D. Derivations of the score vector 31

Akaike, H. (1973) Information theory and an extension of the maximum likelihood principle. In 2nd Int. Symp. on Information Theory (Edited by B. N. Petrov and F. Csaki), 267--281. Akademiai Kiado, Budapest.

Anderson, T.W. (2003) An Introduction to Multivariate Statistical Analysis, third ed. Wiely, New York.

Azzalini, A. (1985) A class of distributions which includes the normal ones. Scandinacian Journal of Statistics. 12, 171--178.

Azzalini, A. (2005). The skew-normal distribution and related multivariate families. Scand. J. Statist. 32, 159--188.

Azzalini, A. and Capitanio, A. (1999) Statistical applications of the multivariate skew normal distribution. J. Roy. Statist. Soc. Ser. B 61, 579--602.

Azzalini, A. and Capitaino, A. (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J. R. Stat. Soc. Ser. B 65, 367--389.

Azzalini, A. and Dalla Valle, A. (1996) The multivariate skew-normal distribution. Biometrika 83, 715--726.

Azzalini, A. and Genton, M.G. (2008) Robust likelihood methods based on the skew-t and related distributions. Int. Statist. Rev., 76, 106--129.

Baek, J. and McLachlan, G.J. (2011) Mixtures of common t-factor analyzers for clustering high-dimensional microarray data. Bioinformatics 27, 1269--1276.

Baek, J., McLachlan, G.J., and Flack, L. (2010) Mixtures of factor analyzers with common factor loadings: applications to the
clustering and visualisation of high-dimensional data. IEEE Transactions on Pattern Analysis and Machine Intelligence 32,
1298--1309.

Basilevsky, A. (2008) Statistical Factor Analysis and Related Methods: Theory and Applications. John Wiley & Sons, Inc.

Branco, M.D. and Dey, D.K. (2001) A general class of multivariate skew-elliptical distributions. Journal of Multivariate Analysis., 79, 99--113.

Cook, R.D. and Weisberg, S. (1994) An Introduction to Regression Graphics. Wiley, New York.

Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977) Maximum likelihood from incomplete data via the EM algorithm (with
discussion). J. R. Stat. Soc. Ser. B 39, 1--38.

Efron, B. and Hinkley, D.V. (1978) Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher Information (with discussion). Biometrika 65 457--487.

Efron, B. and Tibshirani, R. (1986) Bootstrap method for standard errors, confidence intervals, and other measures of statistical accuracy. Stat. Sci. 1, 54--77.

Fokoue, E. and Titterington, D.M., (2003) Mixtures of factor analyzers. Bayesian estimation and inference by stochastic simulation. Mach. Learning 50, 73--94.


Ghahramani, Z. and Hinton, G. (1997) The EM algorithm for mixtures of factor analyzers. Technical Report CRG-TR-96-1, University of Toronto.

Gupta, A.K. and Huang, W.J. (2002) Quadratic forms in skew normal variates. J. Math. Anal. Appl. 273, 558--564.

Kotz, S. and Nadarajah, S. (2004) Multivariate t distributions and their applications. Cambridge University Press.

Healy, M.J.R. (1968) Multivariate normal plotting. App. Statist. 17, 157--161.

Ho, H.J., Lin, T.I., Chang, H.H., Haase, H.B., Huang, S. and Pyne, S. (2012) Parametric modeling of cellular state transitions as measured with flow cytometry different tissues. BMC Bioinformatics 13 (Suppl 5):S5

Jamshidian, M. (1997) An EM algorithm for ML factor analysis with missing data. In Berkane, M. (Ed.) Latent Variable Modeling and Applications to Causality, (pp. 247-258). Springer Verlag, New York.

Jones, M.C. and Faddy, M.J. (2003) A skew extension of the t-distribution, with applications. J. Roy. Statist. Soc. Ser. B, 65, 159--174.

Lachos, V.H., Ghosh, P., and Arellano-Valle, R.B. (2010) Likelihood based inference for skew normal independent linear
mixed models. Statistica Sinica. 20, 303--322.

Lange, K.L., Little, R.J.A. and Taylor, J.M.G. (1989) Robust statistical modeling using the t distribution. J. Amer. Statist. Assoc. 84, 881--896.

Lee, S., McLachlan, G. (2013a) Finite mixtures of multivariate skew t-distributions: some recent and new results Statist. Comput. DOI 10.1007/s11222-012-9362-4.

Lee, S.X. and McLachlan, G.J. (2013b). On mixtures of skew normal and skew t-distributions. Adv. Data Anal. Classif. Doi 10.1007/s11634-013-0132-8.

Lin, T.I., Ho, H.J. and Chen, C.L. (2009) Analysis of multivariate skew normal models with incomplete data. J. Multivariate Anal. 100, 2337--2351.

Lin, T.I., Lee, J.C. and Ho, H.J. (2006) On fast supervised learning for normal mixture models with missing information.
Pattern Recognit. 39, 1177--1187.

Lin, T.I. (2010) Robust mixture modeling using multivariate skew t distributions. Stat. Comput. 20, 343--356.

Lin, T.I. and Lin, T.C. (2011) Robust statistical modelling using the multivariate skew t distribution with complete and
incomplete data. Stat. Model. 11, 253--277.

Lin, T.I., Lee, J.C. and Ho, H.j. (2006) On fast supervised learning for normal mixture models with missing information,
Pattern Recog. 39, 1177--1187.

Lin, T.I., Lee, J.C. and Hsieh, W.J. (2007) Robust mixture modeling using the skew t distribution. Stat. Comput. 17, 81--92.

Lopes, H. F. and West M. (2004) Bayesian model assessment in factor analysis. Stat. Sin. 14, 41--67.

Louis, T.A. (1982) Finding the observed information when using the EM algorithm. J. R. Stat. Soc. Ser. B 44, 226--232.

McLachlan, G.J., Bean, R.W. and Jones, L.B.T. (2007) Extension of the mixture of factor analyzers model to incorporate the multivariate t-distribution. Comput. Stat. Data Anal. 51, 5327--5338.

McLachlan, G.J. and Peel, D. (2000) Mixture of factor analyzers. In Proceedings of the Seventeenth International
Conference on Machine Learning, P. Langley (Ed.). San Francisco:Morgan Kaufmann, 599--606.

McLachlan, G.J.,Peel, D. and bean, R.W. (2003) Modelling high-dimensional data by mixtures of factor analyzers. Computational Statistics and Data Analysis. 41, 379--388.

Meng, X.L. and Rubin, D.B. (1993) Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80, 267--278.

Pyne S., Hu, X., Wang, K., Rossin, E., Lin, T.I., Maier, L.M., Baecher-Allan, C., McLachlan, G.J., Tamayo, P., Hafler,
D.A., De Jager, P.L. and Mesirov, J.P. (2009) Automated high-dimensional flow cytometric data analysis. Proc. Natl. Acad.
Sci. USA, 106, 8519--8524.

Rossin, E., Lin, T.I., Ho, H.J., Mentzer, S.J. and Pyne, S. (2011) A framework for analytical characterization of monoclonal antibodies based on reactivity profiles in different tissues. Bioinformatics 27, 2746--2753.

Sahu, S.K., Dey, D.K. and Branco, M.D. (2003) A new class of multivariate skew distributions with application to Bayesian regression models. Can. J. Statist. 31, 129--150.

Schwarz, G. (1978) Estimating the dimension of a model. Ann. Statist. 6, 461--464.

Spearman, C. (1904) General intelligence, objectively determined and measured. Am. J. Psy. 15, 201--293.

Titterington, D.M., Smith, A.F.M. and Markov, U.E. (1985) Statistical analysis of finite mixture distributions. Wiely, New
York.

Wang, K., McLachlan, G.J., Ng, S.K. and Peel, D. (2009) EMMIX-skew (R package version 1.0-12): EM Algorithm for Mixture of
Multivariate Skew Normal/t Distributions.

Wang, W.L. and Lin, T.I. (2013) An efficient ECM algorithm for maximum likelihood estimation in mixtures of t-factor analyzers. Comput. Statist. 28, 751--769.

Zacks, S. (1971) The Theory of Statistical Inference. John Wiley, New York.

Zhang, J., Li, J. and Liu, C. (2013) Robust factor analysis using the multivariate t-distribution. unpublished manuscript.