A class of generalized ridge estimator for high-dimensional linear regression

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楊思芃(Szu-peng Yang) 查詢紙本館藏

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統計研究所

論文名稱

(A class of generalized ridge estimator for high-dimensional linear regression)

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摘要(中)

此篇論文建立在多元線性迴歸(Multiple linear regression)模型之上。在這個模型之下，一般常用的最小平方估計量（Least square estimator）並不適合用在變數個數大的情況，會產生共線性（Collinearity）的問題，特別是在變數個數大於樣本數的時候。Hoerl和Kennard在1970年提出了Generalized ridge迴歸方法。在理論上，Generalized ridge估計量可以解決最小平方估計量的共線性問題。其後，也有許多人討論過特殊型式的Generalized ridge估計量。但是，當變數個數增大的時候，需要估計的參數也隨之增加，導致其實行上的困難，因此大多只考慮樣本數大於變數個數的情形。我們在此篇論文提出了一個在高變數個數之下也能運作的Generalized ridge估計量的特殊型。除此之外，此估計量在貝氏理論中也具有適當的解釋，更可以與先驗資訊做連結，藉此取得較佳的估計。在此篇論文中，我們做了顯著性檢定、模擬資料以及實際資料分析。資料分析中，一般的ridge估計量被拿來與我們提出的估計量做比較，而我們提出的估計量以均方差(Mean square error)來說表現得比ridge估計量來得好。

摘要(英)

In multiple linear regression, the least square estimator is inappropriate for high-dimensional regressors, especially for p≥n. Consider the linear regression model. The generalized ridge estimator has been considered by many authors under the usual p

關鍵字(中)

★ 脊迴歸
★ 高維度資料

關鍵字(英)

★ Ridge regression
★ High dimensional
★ Generalized ridge regression

論文目次

摘要 I

Abstract II

致謝辭 III

Contents IV

List of Tables V

List of Figures VI

1. Introduction 1

2. Background 3

3. Method 7

3.1 Proposed idea 7

3.2 Proposed estimator 8

3.3 Estimation of by cross validation 8

3.4 Significance testing 10

4. Theory 12

4.1 Baysian interpretation 12

4.2 MSE calculation 13

5. Simulation 20

5.1 Model design 20

5.2 MSE for fixed λ and Δ 21

5.3 MSE for estimated and 23

5.4 Significance testing 26

6. Data Analysis 30

6.1 Non-small cell lung cancer data 30

6.2 Numerical results 31

7. Conclusion and Discussion 36

Reference 37

Appendix 39

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指導教授

江村剛志(Takeshi Emura)

審核日期

2014-7-24

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