在此研究中，我們考慮迴歸模型由許多未知的群函數組成。其中，有些群的解釋變數對反應變數的影響是不顯著的。為了找到合適的模型並提供良好的預測，我們提出一個演算法，透過一組正交基底函數逼近未知的群函數，再結合 Lasso型式懲罰項進行參數估計。因此，我們能夠找出模型中重要的那些群函數，並且偵測出這些群函數與反應變數之間的關係。最後，我們利用模擬實驗與實際資料分析，說明我們所提出的演算法之結果與表現。

The regression model consisting of many unknown group regressor functions is assumed in our work, and some of them can be irrelevant for the response variable. To find an appropriate model for the prediction problem and model interpretation, an algorithm is developed to search the important regressor functions and their related structures through the introduction of basis functions with the Lasso-type penalty. The performance of the proposed algorithm is evaluated under simulation studies and real data analyses.

1 Introduction 1
1.1 Literature Reviews 2
1.1.1 Lasso-penalty 2
1.1.2 Group Lasso and Adaptive Group Lasso 3
1.2 B-spline and Tensor Product B-spline Functions 4
1.3 Tensor Product Orthonormal Basis Spline Function 6
2 Main Problem 9
2.1 Regression Model with Unknown Grouping Regressors 9
2.2 Algorithm 10
3 Simulation, Comparison, and Real Data Analysis 15
3.1 Simulation 15
3.2 Two-step(BIC) vs. other algorithms 20
3.2.1 Two-step(BIC) vs. GAMSEL 20
3.2.2 Two-step(BIC) vs. gam.selection 26
3.3 Real data analysis 29
3.3.1 Real data - Hospital Information 29
3.3.2 Real data - Boston Housing 33
4 Conclusion 37
References 39

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