臺灣博碩士論文加值系統

在資料分析的過程中，通常有多種的統計方法或模型可以採用，然而不同的統計方法或模型通常具有不同的配適能力與預測精準度，如何從中選擇適當的方法或模型因而是一個重要的問題。在這篇論文中，我們探討如何從一組曲線配適或曲面配適方法中選擇一個適當的方法，尤其著重在非線性方法間的選擇。其中最棘手的部分在於如何建立一個可以衡量非線性方法及複雜配適方法的統計機制，使其可以公平地比較各種配適方法，進而從中選出一個最好的資料分析方法。

本篇論文主要是以Ye (1998)所提出的廣義自由度為基礎。在曲線配適問題中，雖然已經有很多的準則可以用來選擇配適曲線的方法，然而這些準則大多只適用於比較線性方法間之優劣。在此論文中我們提出一個準則使其可以公平地衡量一組樣條函數平滑參數選取方法的表現，進而從中選出最適當的平滑參數選取方法。此外，我們進一步將廣義自由度的想法推廣到空間預測的曲面配適問題上，並提出一個地理統計模型選取的一般準則。所提出的準則不僅可以用來選擇各種空間預測方法，也適用具空間相關雜訊的迴歸變數選擇問題。此篇論文在曲線配適及曲面配適問題上所提的新方法，除了透過模擬實驗來呈現其優越性，也同時以理論證明這些方法皆具有漸近最佳的性質。

In the process of data analysis, there are usually a number of candidate statistical methods (models) that can be used, and different methods (models) generally have different performances under different situations. In this thesis, we focus on model selection in curve and surface fitting. We develop a general rule to fairly assess among candidate curve or surface fitting methods regardless of whether the fitting procedures are complex and whether the corresponding estimates are linear, nonlinear, or even discontinuous.

Based on the concept of generalized degrees of freedom (GDF) (Ye 1998), we propose an improved Cp method to select among a class of selection criteria in spline smoothing. In addition, a general methodology for geostatistical model selection is proposed by further generalizing GDF to spatial prediction. The proposed method not only can be used to select among various spatial prediction methods, but also can be applied to the variable selection problem in spatial regression. The validities of the proposed model selection methods for curve and surface fitting are justified both numerically and theoretically.

Contents
1 Introduction ------ 1
1.1 Model Selection ------ 1
1.2 Curve Fitting ------ 2
1.2.1 Cubic Splines ------ 2
1.2.2 Choice of the Smoothing Parameter ------ 5
1.3 Surface Fitting ------ 6
1.3.1 Kriging ------ 7
1.3.2 Thin-Plate Splines ------ 9
1.4 Dissertation Organization ------ 11
2 Model Selection and Generalized Degrees of Freedom ------ 12
2.1 Model Selection in Regression ------ 12
2.2 Generalized Degrees of Freedom ------ 14
2.3 Estimating the Generalized Degrees of Freedom ------ 15
2.3.1 Stein''s Unbiased Risk Estimate ------ 16
2.3.2 Parametric Bootstrap ------ 16
2.3.3 Data Perturbation ------ 17
3 An Improved Cp Criterion for Spline Smoothing ------ 19
3.1 Introduction ------ 19
3.2 The Proposed Method ------ 20
3.2.1 Adaptive Cp ------ 22
3.2.2 Asymptotic Optimality of Adaptive Cp ------ 23
3.3 Simulation Study ------ 24
3.4 Discussion ------ 27
4 Geostatistical Model Selection ------ 30
4.1 Introduction ------ 30
4.2 Geostatistical Models and Spatial Prediction ------ 32
4.3 Generalized Degrees of Freedom for Spatial Model Selection ------ 35
4.3.1 Generalized Degrees of Freedom ------ 36
4.3.2 Data Perturbation ------ 38
4.3.3 Optimal Model Selection for Spatial Prediction ------ 40
4.3.4 Estimation of Noise Variance ------ 43
4.4 Simulation Study ------ 45
4.5 Application ------ 58
4.6 Discussion ------ 62
5 Summary ------ 64
Bibliography ------ 66
Appendix ------ 73

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