Bayesian method for sparse principal component analysis

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張元綺(Yuan-Chi Chang) 查詢紙本館藏

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統計研究所

論文名稱

(Bayesian method for sparse principal component analysis)

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摘要(中)

主成分分析(PCA)是一種常見且熱門的降維方法。然而，當我們想通過主成分的載荷係數(loading)進一步對資料與變數之間做推論時，常因載荷係數不為零而難以得到簡易的解釋結果。本文的主要目的是嘗試透過貝氏方法得到 SPCA 標準下的稀疏載荷，其中我們使用一種全部局部收縮先驗(global-local shrinkage prior)模型 MBSP-TPBN。數值模擬和實際數據證明本文提出的方法。

摘要(英)

Principal component analysis (PCA) is a common and popular dimensionality reduc- tion method. However, when we want to make further inferences between data and variables through the loadings of principal components, it is often difficult to obtain simple interpretation results due to all non-zero loadings. The main purpose of this thesis is to try to obtain the sparse loadings under the SPCA criterion based on a Bayesian approach, in which we use a global-local shrinkage prior model MBSP-TPBN. The numerical study and real data demonstrate the proposed method in the article.

關鍵字(中)

★ 全部局部收縮先驗
★ 主成分分析
★ 載荷係數
★ 稀疏性

關鍵字(英)

★ global-local shrinkage prior
★ principal component analysis
★ loadings
★ SPCA
★ sparse

論文目次

1 Introduction 1
2 Literature Review 3
2.1 SparsePrincipalComponentAnalysis .................... 3
2.1.1 Sparse Principal Component Analysis Criterion . . . . . . . . . . 4
2.1.2 AdjustedTotalVariance ........................ 5
2.2 Global-localShrinkagePriors ......................... 6
2.2.1 Multivariate Bayesian Model with Shrinkage Priors . . . . . . . . 7
2.2.2 Three-ParameterBetaNormalFamily ................ 8
2.2.3 TheMBSP-TPBNModel........................ 9
2.3 TheGibbsSampling .............................. 10
2.3.1 The Conditional Distributions of the MBSP-TPBN Model . . . . . 10
3 Proposed Methodology 12
3.1 BayesianMethodforSparsePrincipalComponent . . . . . . . . . . . . . 12
3.1.1 Sparse Principal Component Analysis via the MBSP-TPBN Model 12
4 Numerical Study 15
4.1 Case1.Gaussiandistribution ......................... 16
4.2 Case2.Uniformdistribution ......................... 18
5 Application 20
5.1 Glassdata .................................... 20 iv
5.2 SyntheticControlChartTimeSeriesData .................. 22
6 Conclusion 25
Reference 27

參考文獻

[1] I. T. Jolliffe. Principal Component Analysis. Berlin; New York: Springer-Verlag, 1986.
[2] Hui Zou, Trevor Hastie, and Robert Tibshirani. “Sparse principal component analysis”. In: Journal of computational and graphical statistics 15.2 (2006), pp. 265– 286.
[3] Ian T Jolliffe, Nickolay T Trendafilov, and Mudassir Uddin. “A modified princi- pal component technique based on the LASSO”. In: Journal of computational and Graphical Statistics 12.3 (2003), pp. 531–547.
[4] Sam Roweis. “EM algorithms for PCA and SPCA”. In: Advances in neural infor- mation processing systems 10 (1997).
[5] Michael E Tipping and Christopher M Bishop. “Probabilistic principal compo- nent analysis”. In: Journal of the Royal Statistical Society: Series B (Statistical Method- ology) 61.3 (1999), pp. 611–622.
[6] Cédric Archambeau and Francis Bach. “Sparse probabilistic projections”. In: Ad- vances in neural information processing systems 21 (2008).
[7] Yue Guan and Jennifer Dy. “Sparse probabilistic principal component analysis”. In: Artificial Intelligence and Statistics. PMLR. 2009, pp. 185–192.
[8] Rajiv Khanna et al. “Sparse submodular probabilistic PCA”. In: Artificial Intelli- gence and Statistics. PMLR. 2015, pp. 453–461.
[9] Artin Armagan, Merlise Clyde, and David Dunson. “Generalized beta mixtures of Gaussians”. In: Advances in neural information processing systems 24 (2011).
[10] Prasenjit Ghosh et al. “Asymptotic properties of Bayes risk of a general class of shrinkage priors in multiple hypothesis testing under sparsity”. In: Bayesian Analysis 11.3 (2016), pp. 753–796.
[11] Ray Bai and Malay Ghosh. “High-dimensional multivariate posterior consis- tency under global–local shrinkage priors”. In: Journal of Multivariate Analysis 167 (2018), pp. 157–170.

指導教授

王紹宣(Shao-Hsuan Wang)

審核日期

2022-8-24

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