基於Penalized Spline的信賴帶之比較與改良

• Main Authors: 游博安, Yu, Po An
• Language: 中文
• Published: 國立政治大學
• Subjects: B-Spline; Penalized Spline; 信賴帶; 混合效應模型; 無母數方法; B-spline; Penalized spline; Confidence band; Mixed model; Nonparametric
• Online Access: http://thesis.lib.nccu.edu.tw/cgi-bin/cdrfb3/gsweb.cgi?o=dstdcdr&i=sid=%22G1023540161%22.

迴歸分析中，若變數間有非線性(nonlinear)的關係，此時我們可以用B-spline線性迴歸，一種無母數的方法，建立模型。Penalized spline是B-spline方法的一種改良，其想法是增加一懲罰項，避免估計函數時出現過度配適的問題。本文中，考慮三種方法:(a) Marginal Mixed Model approach, (b) Conditional Mixed Model approach, (c) 貝氏方法建立信賴帶，其中，我們對第一二種方法內的估計式作了一點調整，另外，懲罰項中的平滑參數也是我們考慮的問題。我們發現平滑參數確實會影響信賴帶，所以我們使用cross-validation來選取平滑參數。在調整的cross-validation下，Marginal Mixed Model的信賴帶估計不平滑的函數效果較好，Conditional Mixed Model的信賴帶估計平滑函數的效果較好，貝氏的信賴帶估計函數效果較差。 === In regression analysis, we can use B-spline to estimate regression function nonparametrically when the regression function is nonlinear. Penalized splines have been proposed to improve the performance of B-splines by including a penalty term to prevent over-fitting. In this article, we compare confidence bands constructed by three estimation methods: (a) Marginal Mixed Model approach, (b) Conditional Mixed Model approach, and (c) Bayesian approach. We modify the first two methods slightly. In addition, the selection of smoothing parameter of penalization is considered. We found that the smoothing parameter affects confidence bands a lot, so we use cross-validation to choose the smoothing parameter. Finally, based on the restricted cross-validation, Marginal Mixed Model performs better for less smooth regression functions, Conditional Mixed Model performs better for smooth regression functions and Bayesian approach performs badly.