臺灣博碩士論文加值系統

在一般線性迴歸模型當中，當樣本數大於變數個數時候，我們是以最小平方法來估計迴歸參數。然而當變數個數大於樣本個數時，會造成共線性問題，導致參數無法估計，無法確認個別自變數對依變數有多大影響。為了解決共線性問題，我們透過變數選取來選取重要的變數，選取方法包含主成份分析法 (PCA)及向前選取法 (FS).

我們使用的模型為無母數加成迴歸模型，透過 backfitting algorithm 來估計整個迴歸函數，個別函數則以無母數方法，使用B-Spline 來估計。我們把兩種選取方法應用在無母數加成模型裡，以積分絕對誤差為判斷標準，透過不同變數及不同生成模型類型的模擬，來判斷哪種配適選取方法較合適。模擬結果可以發現，多數情況下，FS的表現比PCA來得好。

In a general linear regression model, when the sample size $n$ is greater than the number of variables $p$, it is common to use the least squares method to estimate the parameters in the regression model. When $n<p$, the parameters in the regression model cannot be estimated due to collinearity, so it is necessary to perform variable selection before estimating the parameters. In this thesis, I compare two variable selection methods in nonparametric additive regression. The first method is based on principal component analysis (PCA), and the second method is based on forward selection (FS). The integrated absolute error is used to evaluate the performance of these two methods in simulation studies. The simulation results show that FS performs better than PCA in most cases.

1 緒論.............5
2 文獻探討.........7
2.1 加成模型及Backfitting Algorithm...7
2.2 B-Spline迴歸模型.........8
3 研究方法........10
3.1 主成份分析法(PCA)........10
3.1.1 定義........10
3.1.2 主成份計算方式.........10
3.1.3 選取標準..............11
3.2 向前選取法(FS)..........12
3.2.1 定義......12
3.2.2 檢定加成模型的變數是否顯著...12
3.2.3 選取標準.......14
4 模擬資料與比較結果........15
4.1 模擬資料......15
4.1.1 實驗一，變數個數p=20,n>p,模擬10次....17
4.1.2 實驗二，變數個數p=200,n=p,模擬10次...21
4.1.3 實驗三，變數個數p=250,n<p,模擬7次....23
4.2 比較結果........26
5 結論與建議........28
2.1 結論...........28
2.2 建議...........28
附錄.......30

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