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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 Ming en_US dc.contributor.author (Authors) 游博安 zh_TW dc.contributor.author (Authors) Yu, Po An en_US dc.creator (作者) 游博安 zh_TW dc.creator (作者) Yu, Po An en_US dc.date (日期) 2015 en_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) G1023540161 en_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) 102354016 zh_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 11.1 Penalized Spline 22 三種建立信賴帶的方法 32.1 使用Volume of Tube Formula建立信賴帶 32.1.1 基於Marginal Mixed Model的信賴帶 42.1.2 基於Conditional Mixed Model的信賴帶 52.2 貝氏信賴帶 63 模擬與比較 73.1 固定平滑參數的信賴帶 73.2 cross-validation選取平滑參數的信賴帶比較 143.3 調整的cross-validation選取平滑參數的信賴帶比較 164 結論與建議 184.1 結論 184.2 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/#G1023540161 en_US dc.subject (關鍵詞) B-Spline zh_TW dc.subject (關鍵詞) Penalized Spline zh_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-spline en_US dc.subject (關鍵詞) Penalized spline en_US dc.subject (關鍵詞) Confidence band en_US dc.subject (關鍵詞) Mixed model en_US dc.subject (關鍵詞) Nonparametric en_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 Spline en_US dc.type (資料類型) thesis en 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
