混合共同因子分析模型之貝氏推論

Bayesian Inference for Mixtures of Common Factor Analyzers

王婉倫(Wan-Lun Wang) ； 徐偉舜(Wei-Shun Hsu)

《智慧科技與應用統計學報》 11卷1期 (2013/06) Pp. 1-24

摘要

混合因子分析（MFA）方法對於高維度資料之基於模型的密度估計和分群是一個很自然可以聯想到的應用工具，尤其在樣本數相對於變數的維度來得小時更是如此。然而，當群集個數不小時，MFA之因子共變異數矩陣的參數個數會相當的大，造成估計上的困難。為了進一步減少參數的個數，混合共同因子分析（MCFA），其為MFA 的精簡延伸法，最近已被發展出來分析高維度資料。在本文中，我們採用貝氏分析方法來配適MCFA，更明確地說，貝氏方法將基於感興趣的後驗分佈之隨機抽樣來執行參數估計和後驗推論。對於模型的參數，採用共軛和弱訊息先驗分佈以確保得到合適的參數後驗分配。我們利用有效的馬可夫鏈蒙地卡羅（Markov chain Monte Carlo；MCMC）技術，其結合了資料擴增法來填補隱藏變數以及吉布斯抽樣法來生成參數。同時，估計隱藏因子和新個體分類的技術也被探討。模擬研究和實例分析顯示了我們的方法在實際應用上提供令人滿意的結果。

關鍵字

共同因子負荷量 ； Gibbs抽樣 ； 馬可夫鏈蒙地卡羅 ； 混合共同因子分析器 ； 混合因子分析器

並列摘要

The mixtures of factor analyzers (MFA) approach is a natural tool for model-based density estimation and clustering of high-dimensional data, especially when the number of observations is not relatively large than their dimension. However, the number of parameters in the component-covariance matrices of MFA is quite large when the number of clusters is not small. To further reduce the number of parameters, mixtures of common factor analyzers (MCFA) have recently been developed as a parsimonious extension of the MFA to analyze high-dimensional data. In this paper, we adopt a fully Bayesian approach, more specifically a treatment that carries out estimation and inference based on stochastic sampling of the posterior distributions of interest, to fitting the MCFA. Natural conjugate and weakly informative priors on the distributions of model parameters are introduced to ensure proper posterior distributions of parameters. We provide an efficient Markov chain Monte Carlo (MCMC) technique which incorporates data augmentation for imputation of latent variables with Gibbs sampler for generation of parameters. The techniques for estimation of latent factors and classification of new objects are also investigated. Simulation studies and a real-data example demonstrate that our methodology performs satisfactorily.

並列關鍵字

common factor loadings ； Gibbs sampler ； Markov chain Monte Carlo ； MCFA ； MFA