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題名 基於Penalized Spline的信賴帶之比較與改良
Comparison and Improvement for Confidence Bands Based on Penalized Spline
作者 游博安
Yu, Po An
貢獻者 黃子銘
Huang, Tzee Ming
游博安
Yu, Po An
關鍵詞 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
日期 2015
上傳時間 24-Aug-2015 10:33:55 (UTC+8)
摘要 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.
參考文獻 Sun, J. (1993), ”Tail Probabilities of the Maxima of Gaussian Random Fields,” The Annals of Probability, 21 (1), 34-71.

Sun, J., and Loader, C. R. (1994), ”Simultaneous Confidence Bands for Linear Regression and Smoothing,” The Annals of Statistics, 22 (3), 1328-1345.

Eilers, P.H. C., and Marx, B. D. (1996), “Flexible Smoothing With B-splines and Penalties” Statistical Science, 11 (2), 89-121.

Hall, P., and Opsomer, J. (2005), “Theory for Penalized Spline Regression,” Biometrika, 92, 105-118.

Crainiceanu, C. Ruppert, D., Carroll, R., Adarsh, J., and Goodner, B. (2007),”Spatially Adaptive Penalized Splines With Heteroscedastic Errors,” Journal of Computational and Graphical Statistics, 16, 265-288.

Li, Y., and Ruppert, D. (2008), “On the Asymptotics of Penalized Splines,” Biometrika, 95 (2), 415-436.

Claeskens, G. Krivobokova, T., and Opsomer, J. (2009), “Asmptotic Properties of Penalized Spline Estimators,” Biometrika, 96 (3), 529-544.

Kauermann, G., Krivibokova, T., and Fahrmeir, L. (2009), “Some Asymptotic Results on Generalized Penalized Spline Smoothing,” Journal of the Royal Statistical Society, Ser. B, 71 (2), 487-503.

Krivobokova, Kneib, and Claeskens. (2010), “Simultaneous Confidence Bands for Penalized Spline Estimators,” Journal of the American Statistical Association, 105-490.
描述 碩士
國立政治大學
統計研究所
102354016
資料來源 http://thesis.lib.nccu.edu.tw/record/#G1023540161
資料類型 thesis
dc.contributor.advisor 黃子銘zh_TW
dc.contributor.advisor Huang, Tzee Mingen_US
dc.contributor.author (Authors) 游博安zh_TW
dc.contributor.author (Authors) Yu, Po Anen_US
dc.creator (作者) 游博安zh_TW
dc.creator (作者) Yu, Po Anen_US
dc.date (日期) 2015en_US
dc.date.accessioned 24-Aug-2015 10:33:55 (UTC+8)-
dc.date.available 24-Aug-2015 10:33:55 (UTC+8)-
dc.date.issued (上傳時間) 24-Aug-2015 10:33:55 (UTC+8)-
dc.identifier (Other Identifiers) G1023540161en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/77918-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計研究所zh_TW
dc.description (描述) 102354016zh_TW
dc.description.abstract (摘要) 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。zh_TW
dc.description.abstract (摘要) 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.en_US
dc.description.tableofcontents 1 緒論 1
1.1 Penalized Spline 2
2 三種建立信賴帶的方法 3
2.1 使用Volume of Tube Formula建立信賴帶 3
2.1.1 基於Marginal Mixed Model的信賴帶 4
2.1.2 基於Conditional Mixed Model的信賴帶 5
2.2 貝氏信賴帶 6
3 模擬與比較 7
3.1 固定平滑參數的信賴帶 7
3.2 cross-validation選取平滑參數的信賴帶比較 14
3.3 調整的cross-validation選取平滑參數的信賴帶比較 16
4 結論與建議 18
4.1 結論 18
4.2 建議 18		zh_TW
dc.format.extent 916427 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G1023540161en_US
dc.subject (關鍵詞) B-Splinezh_TW
dc.subject (關鍵詞) Penalized Splinezh_TW
dc.subject (關鍵詞) 信賴帶zh_TW
dc.subject (關鍵詞) 混合效應模型zh_TW
dc.subject (關鍵詞) 無母數方法zh_TW
dc.subject (關鍵詞) B-splineen_US
dc.subject (關鍵詞) Penalized splineen_US
dc.subject (關鍵詞) Confidence banden_US
dc.subject (關鍵詞) Mixed modelen_US
dc.subject (關鍵詞) Nonparametricen_US
dc.title (題名) 基於Penalized Spline的信賴帶之比較與改良zh_TW
dc.title (題名) Comparison and Improvement for Confidence Bands Based on Penalized Splineen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) Sun, J. (1993), ”Tail Probabilities of the Maxima of Gaussian Random Fields,” The Annals of Probability, 21 (1), 34-71.

Sun, J., and Loader, C. R. (1994), ”Simultaneous Confidence Bands for Linear Regression and Smoothing,” The Annals of Statistics, 22 (3), 1328-1345.

Eilers, P.H. C., and Marx, B. D. (1996), “Flexible Smoothing With B-splines and Penalties” Statistical Science, 11 (2), 89-121.

Hall, P., and Opsomer, J. (2005), “Theory for Penalized Spline Regression,” Biometrika, 92, 105-118.

Crainiceanu, C. Ruppert, D., Carroll, R., Adarsh, J., and Goodner, B. (2007),”Spatially Adaptive Penalized Splines With Heteroscedastic Errors,” Journal of Computational and Graphical Statistics, 16, 265-288.

Li, Y., and Ruppert, D. (2008), “On the Asymptotics of Penalized Splines,” Biometrika, 95 (2), 415-436.

Claeskens, G. Krivobokova, T., and Opsomer, J. (2009), “Asmptotic Properties of Penalized Spline Estimators,” Biometrika, 96 (3), 529-544.

Kauermann, G., Krivibokova, T., and Fahrmeir, L. (2009), “Some Asymptotic Results on Generalized Penalized Spline Smoothing,” Journal of the Royal Statistical Society, Ser. B, 71 (2), 487-503.

Krivobokova, Kneib, and Claeskens. (2010), “Simultaneous Confidence Bands for Penalized Spline Estimators,” Journal of the American Statistical Association, 105-490.		zh_TW