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題名 基於B-spline的密度函數估計之節點選取之準則
Knot selection criteria for density function estimation based on B-spline
作者 江瑞濬
Chiang, Jui-Chun
貢獻者 黃子銘
Huang, Tzee-Ming
江瑞濬
Chiang, Jui-Chun
關鍵詞 密度函數估計
樣條函數近似
節點選取
交叉驗證
Density estimation
Spline approximation
Knot selection
Cross validation
日期 2024
上傳時間 5-Aug-2024 13:59:28 (UTC+8)
摘要 本論文在B-spline的背景下進行密度估計，藉由類似帶寬選擇(bandwidth selection)的概念並提出一種挑選節點位置的準則，以估計出較平滑的機率密度函數。節點選取之準則主要透過「抽樣的留一最小平方交叉驗證」(sample leave-one-out least square cross validation) 挑選兩個調節參數並進行估計。通過本文分析不同模擬資料下的結果顯示，此挑選節點位置的準則在估計機率密度函數部分表現良好，因為平均下來的「積分均方誤差」(Integrated Squared Error)數值較小。
In this thesis, the problem of density estimation based on spline approximation is considered. A procedure for determining knot positions is proposed. The procedure involve two tunning parameters which are determined using sample leave-one-out cross validation. The simulation results indicate that the knot selection procedure performs well since the averages of integrated squared errors are small.
參考文獻 [1] Bowman,A.W.(1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika, 71, 353-360. [2] C.D.Boor.(1978). A partical guide to splines. Springer New York. [3] D.Ruppert.(2002). Selecting the number of knots for penalized splines. Journal of Computational and Graphical Statistics, 11(4), 735-757 [4] E.Halpern.(1973). Bayesian spline regression when the number of knots is unknown. Journal of the Royal Statistical Society, B, 35, 347-360. [5] E.Parzen.(1962). On estimation of a probability density function and mode. Ann. Math. Statist. , 33(3), 1065-1076 [6] Hongmei Kang, Falai Chen, Yusheng Li, Jiansong Deng, and Zhouwang Yang. (2015). Knot calculation for spline fitting via sparse optimization. Computer-Aided Design, 58, 179–188. [7] I.J.Schoenbreg.(1983). Contributions to the problem of approximation of equidistant data by analytic functions. Quart.~Appl.~Math., 112-144 [8] J.S.Horne and E.O.Garton.(2006). Likelihood cross-validation versus least squares cross-validation for choosing the smoothing parameter in kernel home-range analysis. The Journal of Wildlife Management, 70, 641–648. [9] L.Piegl and W.Tiller.(1996). The NURBS Book. Springer, 81-116 [10] M.Stone.(1974). Cross-validatory choice and assessment of statistical predictions. Journal of the Royal Statistical Society , 36(2), 111-147. [11] M.P.Wand and M.C.Jones.(1995). Kernel Smoothing. Chapman and Hall. [12] Nicolas Molinari, Jean-François Durand, and Robert Sabatier.(2004). Bounded optimal knots for regression splines. Computational statistics and data analysis, 45(2), 159–178. [13] Paul, H.E. and Brian, D.M.(1996). Flexible smoothing with b-splines and penalties. Statistical science, 89–102. [14] Peter Hall and Huang,Li-Shan.(2001). Nonparametric kernel regression subject to monotonicity constraints. Ann. Statist, 29(3), 624-647. [15] Randall,L.E.(1988). Spline smoothing and nonparametric regression. Marcel Dekker. [16] Seymour, Geisser.(1975). The predictive sample reuse method with applications. Journal of the American Statistical Association, 70(350), 320-328. [17] Silverman,B.W.(1986). Density estimation for statistics and data analysis. Chapman and Hall, London, United Kingdom.
描述 碩士
國立政治大學
統計學系
111354011
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0111354011
資料類型 thesis
dc.contributor.advisor 黃子銘zh_TW
dc.contributor.advisor Huang, Tzee-Mingen_US
dc.contributor.author (Authors) 江瑞濬zh_TW
dc.contributor.author (Authors) Chiang, Jui-Chunen_US
dc.creator (作者) 江瑞濬zh_TW
dc.creator (作者) Chiang, Jui-Chunen_US
dc.date (日期) 2024en_US
dc.date.accessioned 5-Aug-2024 13:59:28 (UTC+8)-
dc.date.available 5-Aug-2024 13:59:28 (UTC+8)-
dc.date.issued (上傳時間) 5-Aug-2024 13:59:28 (UTC+8)-
dc.identifier (Other Identifiers) G0111354011en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/152776-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 111354011zh_TW
dc.description.abstract (摘要) 本論文在B-spline的背景下進行密度估計，藉由類似帶寬選擇(bandwidth selection)的概念並提出一種挑選節點位置的準則，以估計出較平滑的機率密度函數。節點選取之準則主要透過「抽樣的留一最小平方交叉驗證」(sample leave-one-out least square cross validation) 挑選兩個調節參數並進行估計。通過本文分析不同模擬資料下的結果顯示，此挑選節點位置的準則在估計機率密度函數部分表現良好，因為平均下來的「積分均方誤差」(Integrated Squared Error)數值較小。zh_TW
dc.description.abstract (摘要) In this thesis, the problem of density estimation based on spline approximation is considered. A procedure for determining knot positions is proposed. The procedure involve two tunning parameters which are determined using sample leave-one-out cross validation. The simulation results indicate that the knot selection procedure performs well since the averages of integrated squared errors are small.en_US
dc.description.tableofcontents 1. 研究背景及目的 1 2. 文獻回顧 2 3. 研究方法 5 4. 模擬資料分析 11 5. 研究結論及建議 24 6. 參考文獻 27zh_TW
dc.format.extent 1028734 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0111354011en_US
dc.subject (關鍵詞) 密度函數估計zh_TW
dc.subject (關鍵詞) 樣條函數近似zh_TW
dc.subject (關鍵詞) 節點選取zh_TW
dc.subject (關鍵詞) 交叉驗證zh_TW
dc.subject (關鍵詞) Density estimationen_US
dc.subject (關鍵詞) Spline approximationen_US
dc.subject (關鍵詞) Knot selectionen_US
dc.subject (關鍵詞) Cross validationen_US
dc.title (題名) 基於B-spline的密度函數估計之節點選取之準則zh_TW
dc.title (題名) Knot selection criteria for density function estimation based on B-splineen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Bowman,A.W.(1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika, 71, 353-360. [2] C.D.Boor.(1978). A partical guide to splines. Springer New York. [3] D.Ruppert.(2002). Selecting the number of knots for penalized splines. Journal of Computational and Graphical Statistics, 11(4), 735-757 [4] E.Halpern.(1973). Bayesian spline regression when the number of knots is unknown. Journal of the Royal Statistical Society, B, 35, 347-360. [5] E.Parzen.(1962). On estimation of a probability density function and mode. Ann. Math. Statist. , 33(3), 1065-1076 [6] Hongmei Kang, Falai Chen, Yusheng Li, Jiansong Deng, and Zhouwang Yang. (2015). Knot calculation for spline fitting via sparse optimization. Computer-Aided Design, 58, 179–188. [7] I.J.Schoenbreg.(1983). Contributions to the problem of approximation of equidistant data by analytic functions. Quart.~Appl.~Math., 112-144 [8] J.S.Horne and E.O.Garton.(2006). Likelihood cross-validation versus least squares cross-validation for choosing the smoothing parameter in kernel home-range analysis. The Journal of Wildlife Management, 70, 641–648. [9] L.Piegl and W.Tiller.(1996). The NURBS Book. Springer, 81-116 [10] M.Stone.(1974). Cross-validatory choice and assessment of statistical predictions. Journal of the Royal Statistical Society , 36(2), 111-147. [11] M.P.Wand and M.C.Jones.(1995). Kernel Smoothing. Chapman and Hall. [12] Nicolas Molinari, Jean-François Durand, and Robert Sabatier.(2004). Bounded optimal knots for regression splines. Computational statistics and data analysis, 45(2), 159–178. [13] Paul, H.E. and Brian, D.M.(1996). Flexible smoothing with b-splines and penalties. Statistical science, 89–102. [14] Peter Hall and Huang,Li-Shan.(2001). Nonparametric kernel regression subject to monotonicity constraints. Ann. Statist, 29(3), 624-647. [15] Randall,L.E.(1988). Spline smoothing and nonparametric regression. Marcel Dekker. [16] Seymour, Geisser.(1975). The predictive sample reuse method with applications. Journal of the American Statistical Association, 70(350), 320-328. [17] Silverman,B.W.(1986). Density estimation for statistics and data analysis. Chapman and Hall, London, United Kingdom.zh_TW