# 臺灣博碩士論文加值系統

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 在本文中，為分析具遺失值的多變量資料，我們提出一個以多變量t分佈為基礎的因子分析模式，稱為tFA模型，其概念是傳統因子模式的一種穩健延伸。在估計過程中，為了使解析表示式的推導更為容易，我們結合兩種輔助排列矩陣用以表示各觀測值觀察得到與觀察不到的成份。假設遺失值為隨機出現的機制，我們發展出兩種EM型態的演算法，來求取參數的估計值，並且對遺失值進行資料插補。本文也有研究從不完整資料預測因子分數的技術，並透過分析一組不完整雞骨頭的資料來示例所提出的方法。
 In this thesis, we present a robust extension of factor analysis model based on the multivariate t distribution, called the tFA model, for analyzing multivariate data with missing values . To facilitate derivation of analytical expression in the estimating procedure, two auxiliary indicator matrices are incorporated into the model for the determination of observed and missing components of each observation. Under the missing at random mechanism, we develop two EM-type algorithms, which are performed to estimate the parameters and conduct a single imputation for each missing value. The technique for prediction of factor scores from incomplete data is also investigated. The proposed methodologies are illustrated through the analysis of chicken-bone data set where missing values are inherently present.
 1.緒論.........................12.文獻回顧......................3 2.1多變量t分佈.................3 2.2演算法簡介..................3 2.3AECM演算法.................4 2.4遺失訊息資料架構.............53.模型與估計.....................7 3.1因子分析模型.................7 3.1.1t因子分析模型............7 3.2EM演算法....................10 3.2.1完整資料架構.............10 3.2.2具遺失訊息架構............11 3.3AECM演算法..................13 3.3.1完整資料架構..............13 3.3.2具遺失訊息架構............15 3.4因子旋轉與因子分數估算..........16 3.4.1因子旋轉.................16 3.4.2因子分數.................174.資料分析........................18 4.1實例分析.....................18 4.2EM演算法與AECM演算法的差異......215.結論............................25參考文獻...........................26附錄..............................29
 1.Andrews, J.L., McNicholas, P.D., 2010. Extending mixtures of multivariate t\$-\$factor analyzers. Statistics and Computing. To appear.2.zzalini, A., Capitaino A., 2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t\$-\$distribution. Journal of the Royal Statistical Society. Series B 65, 367-389.3.Dempster, A.P., Laird, N.M., Rubin, D.B., 1977. Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society. Series B 39, 1-38.4. Escobar, M.D., West, M., 1995. Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association 90, 577-88.5.Fokou’e, E., Titterington, D.M., 2003. Mixtures of factor analysers. Bayesian estimation and inference by stochastic simulation. Machine Learning 50, 73-94.6. Ghahramani, Z., Hinton, G.E., 1997. The EM algorithm for mixtures of factor analyzers (Tech. Report No. CRG\$-\$TR\$-\$96\$-\$1), University of Toronto.7. Greselin, F., Ingrassia S., 2010. Constrained monotone EM algorithms for mixtures of multivariate t distributions. Statistics and Computing 20, 9-22.8.Hurley, C., 2004. Clustering visualizations of multivariate data. Journal of Computational and Graphical Statistics 13, 788-8069.Kotz, S, Nadarajah, S., 2004. Multivariate t Distributions and Their Applications. Cambridge University Press. Statistics and Computing 20, 9-2210. Lin, T.I. 2009. Maximum likelihood estimation for multivariate skew normal mixture models. Journal of Multivariate Analysis 100, 257-265.11. Lin, T.I 2010. Robust mixture modeling using multivariate skew t distributions. Statistics and Computing 20, 343-356.12.Lin, T.I., Ho, H.J., Shen, P.S., 2009. Computationally efficient learning of multivariate t mixture models with missing information. Computational Statistics 24, 375\$-\$392.13.Lin, T.I., Lee, J.C., Ni, H.F., 2004. Bayesian Analysis of mixture modelling using the multivariate t distribution. Statistics and Computing 14, 119-130.14. Liu, C.H., Rubin D.B., 1994. The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika 81, 633-48.15. Liu C.H., Rubin D.B., 1995. ML estimation of the t distribution using EM and its extensions, ECM and ECME. Statistica Sinica 5, 19-39.16.McLachlan, G.J., Bean, R.W., Jones, B.T., 2007. Extension of the mixture of factor analyzers model to incorporate the multivariate t-distribution. Computational Statistics and Data Analysis 51, 5327-5338.17. McLachlan, G.J., Krishnan, T., 2008. The EM Algorithm and Extensions, 2nd edn, John Wiley and Sons, New York.18. Meng, X.L., Rubin, D.B., 1993. Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80, 267-7819. Meng, X.L., van Dyk, D., 1997. The EM algorithm – an old folk-song sung to a fast new tune. Journal of the Royal Statistical Society. Series B 59, 511-567.20. R Development Core Team, 2008. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna.21. Richard A.Johnson,Dean W.Wichen, 2007. Applied Multivariate Statistical Analysis 496,517.22.Schwarz, G., 1978. Estimating the dimension of a model. THe Annals of Statistics 6, 461-464.23. Shoham, S., 2002. Robust clustering by deterministic agglomeration EM of mixtures of multivariate t\$-\$distributions. Pattern Recogn 35, 1127\$-\$1142.24.Wang, H.X., Hu, Z., 2009. On EM Estimation for Mixture of Multivariate t-Distributions. Neural Processing Letters 30, 243-256.25. Zhao, J.H., Yu, P.L.H., Jiang Q., 2008. ML estimation for factor analysis: EM or non-EM? Statistics and Computing 18, 109-123.
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