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題名 B-Splines不同節點選擇方法之比較
The comparison between different methods of knots selection for B-Splines
作者 胡子卿
Hu, Zi-Qing
貢獻者 黃子銘
Huang, Tzee-Ming
胡子卿
Hu, Zi-Qing
關鍵詞 弧線函數
節點
估計
Spline
Knots
Estimation
日期 2017
上傳時間 11-Jul-2017 11:26:14 (UTC+8)
摘要 本文以 B-Spline 的框架研究比較兩種不同的節點估計方法。第一種方法是通過最優化特定 的目標函數並結合相對應的選擇標準選擇出最優化的節點組合。第二種方法則基於幾何控制多 邊形的特性將內部節點的選擇過程與幾何圖形聯繫起來,省去了最優化的過程。另外,本文採 用『節點估計時間』與『誤差平方和』(Mean Squared Error)來評價兩種方法的估計結果。通 過分析各種不同模擬數據下兩種方法的表現情況,本文的主要發現是:第一,無論哪種資料, 第二種方法在計算速度上都是大幅領先第一種方法。第二,在數據資料較小的情況下,第一種 方法中由 Lindstrom 提出的算法並不能很好的配飾模型,最後的估計誤差較大。而在數據資料 較多的情形下,誤差與其他方法較為接近。第三,第一種方法中沒有懲罰項的算法在所有驗證 過的數據中,其表現是所有方法中最穩定且估計誤差最小的。這些發現為如何選擇恰當的節點 估計方法提供了很具價值的參考信息
This study compares two different methods of knot selection for B-Spline. The first one chooses the best knots through optimizing specific objective functions and corresponding crite- rion. Based on some properties of geometric control polygon, the second one connects the knot selection process with geometric figures, which avoids the tedious optimization. On the other hand, we use the time for estimation and the mean squared error to evaluate the performance of these two methods. There are three main findings of this study. The first finding is that the calculation speed of second method is much higher than that of the first one. Secondly, the algorithm proposed by Lindstrom in the first method is not stable and its estimation error is larger when the sample size is small. On the contrary, the performance of the algorithm proposed by Lindstrom becomes better as the sample size increases. Thirdly, the performance of the algorithm without penalty term in the first method is always better than the second method.
參考文獻 參考文獻
[1] Gleb Beliakov. Cutting angle method–a tool for constrained global optimization. Opti- mization Methods and Software, 19(2):137–151, 2004.
[2] Clemens Biller. Adaptive bayesian regression splines in semiparametric generalized linear models. Journal of Computational and Graphical Statistics, 9(1):122–140, 2000.
[3] Hermann G Burchard. Splines (with optimal knots) are better. Applicable Analysis, 3(4):309–319, 1974.
[4] Carl De Boor. A practical guide to splines; rev. ed. Applied mathematical sciences. Springer, Berlin, 2001.
[5] DGT Denison, BK Mallick, and AFM Smith. Automatic bayesian curve fitting. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60(2):333–350, 1998.
[6] Paul HC Eilers and Brian D Marx. Flexible smoothing with b-splines and penalties. Statistical science, pages 89–102, 1996.
[7] Randall L Eubank. Spline smoothing and nonparametric regression. Marcel Dekker, 1988.
[8] Jerome H Friedman. Multivariate adaptive regression splines. The annals of statistics, pages 1–67, 1991.
[9] Jerome H Friedman and Bernard W Silverman. Flexible parsimonious smoothing and additive modeling. Technometrics, 31(1):3–21, 1989.
[10] David LB Jupp. Approximation to data by splines with free knots. SIAM Journal on Numerical Analysis, 15(2):328–343, 1978.
[11] Vladimir K Kaishev, Dimitrina S Dimitrova, Steven Haberman, and Richard J Ver- rall. Geometrically designed, variable knot regression splines. Computational Statistics, 31(3):1079–1105, 2016.
[12] Hongmei Kang, Falai Chen, Yusheng Li, Jiansong Deng, and Zhouwang Yang. Knot calculation for spline fitting via sparse optimization. Computer-Aided Design, 58:179–188, 2015.
[13] Charles Kooperberg, Charles J Stone, and Young K Truong. Hazard regression. Journal of the American Statistical Association, 90(429):78–94, 1995.
[14] Thomas CM Lee. Regression spline smoothing using the minimum description length principle. Statistics & probability letters, 48(1):71–82, 2000.
[15] Mary J Lindstrom. Penalized estimation of free-knot splines. Journal of Computational and Graphical Statistics, 8(2):333–352, 1999.
[16] Satoshi Miyata and Xiaotong Shen. Free-knot splines and adaptive knot selection. Journal of the Japan Statistical Society, 35(2):303–324, 2005.
[17] Nicolas Molinari, Jean-François Durand, and Robert Sabatier. Bounded optimal knots for regression splines. Computational statistics & data analysis, 45(2):159–178, 2004.
[18] Finbarr O’sullivan, Brian S Yandell, and William J Raynor Jr. Automatic smoothing of regression functions in generalized linear models. Journal of the American Statistical Association, 81(393):96–103, 1986.
[19] Daryl Pregibon. Logistic regression diagnostics. The Annals of Statistics, pages 705–724, 1981.
[20] Carl Runge. Über empirische funktionen und die interpolation zwischen äquidistanten ordinaten. Zeitschrift für Mathematik und Physik, 46(224-243):20, 1901.
[21] Larry Schumaker. Spline functions: basic theory. Cambridge University Press, 2007.
[22] Michael Smith and Robert Kohn. Nonparametric regression using bayesian variable selec- tion. Journal of Econometrics, 75(2):317–343, 1996.
[23] Michael Smith, Chi-Ming Wong, and Robert Kohn. Additive nonparametric regression with autocorrelated errors. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60(2):311–331, 1998.
[24] Charles J Stone, Mark H Hansen, Charles Kooperberg, Young K Truong, et al. Polynomial splines and their tensor products in extended linear modeling: 1994 wald memorial lecture. The Annals of Statistics, 25(4):1371–1470, 1997.
[25] Wannes Van Loock, Goele Pipeleers, Joris De Schutter, and Jan Swevers. A convex opti- mization approach to curve fitting with b-splines. IFAC Proceedings Volumes, 44(1):2290– 2295, 2011.
[26] Grace Wahba. A Survey of Some Smoothing Problems and the Method of the Generalized Cross-validation for Solving Them. University of Wisconsin, Department of Statistics, 1976.
[27] Grace Wahba. Spline models for observational data. SIAM, 1990.
[28] Yuan Yuan, Nan Chen, and Shiyu Zhou. Adaptive b-spline knot selection using multi- resolution basis set. IIE Transactions, 45(12):1263–1277, 2013.
[29] Shanggang Zhou and Xiaotong Shen. Spatially adaptive regression splines and accurate knot selection schemes. Journal of the American Statistical Association, 96(453):247–259, 2001.
描述 碩士
國立政治大學
統計學系
104354033
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0104354033
資料類型 thesis
dc.contributor.advisor 黃子銘zh_TW
dc.contributor.advisor Huang, Tzee-Mingen_US
dc.contributor.author (Authors) 胡子卿zh_TW
dc.contributor.author (Authors) Hu, Zi-Qingen_US
dc.creator (作者) 胡子卿zh_TW
dc.creator (作者) Hu, Zi-Qingen_US
dc.date (日期) 2017en_US
dc.date.accessioned 11-Jul-2017 11:26:14 (UTC+8)-
dc.date.available 11-Jul-2017 11:26:14 (UTC+8)-
dc.date.issued (上傳時間) 11-Jul-2017 11:26:14 (UTC+8)-
dc.identifier (Other Identifiers) G0104354033en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/110784-
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) 104354033zh_TW
dc.description.abstract (摘要) 本文以 B-Spline 的框架研究比較兩種不同的節點估計方法。第一種方法是通過最優化特定 的目標函數並結合相對應的選擇標準選擇出最優化的節點組合。第二種方法則基於幾何控制多 邊形的特性將內部節點的選擇過程與幾何圖形聯繫起來,省去了最優化的過程。另外,本文採 用『節點估計時間』與『誤差平方和』(Mean Squared Error)來評價兩種方法的估計結果。通 過分析各種不同模擬數據下兩種方法的表現情況,本文的主要發現是:第一,無論哪種資料, 第二種方法在計算速度上都是大幅領先第一種方法。第二,在數據資料較小的情況下,第一種 方法中由 Lindstrom 提出的算法並不能很好的配飾模型,最後的估計誤差較大。而在數據資料 較多的情形下,誤差與其他方法較為接近。第三,第一種方法中沒有懲罰項的算法在所有驗證 過的數據中,其表現是所有方法中最穩定且估計誤差最小的。這些發現為如何選擇恰當的節點 估計方法提供了很具價值的參考信息zh_TW
dc.description.abstract (摘要) This study compares two different methods of knot selection for B-Spline. The first one chooses the best knots through optimizing specific objective functions and corresponding crite- rion. Based on some properties of geometric control polygon, the second one connects the knot selection process with geometric figures, which avoids the tedious optimization. On the other hand, we use the time for estimation and the mean squared error to evaluate the performance of these two methods. There are three main findings of this study. The first finding is that the calculation speed of second method is much higher than that of the first one. Secondly, the algorithm proposed by Lindstrom in the first method is not stable and its estimation error is larger when the sample size is small. On the contrary, the performance of the algorithm proposed by Lindstrom becomes better as the sample size increases. Thirdly, the performance of the algorithm without penalty term in the first method is always better than the second method.en_US
dc.description.tableofcontents 1 前言 1
1.1 弧線函數簡介與研究動機 1
1.2 研究目的 2
1.3 研究框架 3
2 文獻探討 4
3 研究方法 6
3.1 最佳化方法 6
3.1.1 基於GCV和修正後GCV的估計 6
3.1.2 帶懲罰項的估計 7
3.2 GeD方法 9
3.3 固定等距節點 10
4 模擬資料分析 11
4.1 資料產生與模擬流程 11
4.2 實證結果 12
4.2.1 節點選擇比較 12
4.2.2 估計精度比較 17
5 結論與建議 25
5.1 研究結論 25
5.2 研究建議 26
參考文獻 27		zh_TW
dc.format.extent 683719 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0104354033en_US
dc.subject (關鍵詞) 弧線函數zh_TW
dc.subject (關鍵詞) 節點zh_TW
dc.subject (關鍵詞) 估計zh_TW
dc.subject (關鍵詞) Splineen_US
dc.subject (關鍵詞) Knotsen_US
dc.subject (關鍵詞) Estimationen_US
dc.title (題名) B-Splines不同節點選擇方法之比較zh_TW
dc.title (題名) The comparison between different methods of knots selection for B-Splinesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) 參考文獻
[1] Gleb Beliakov. Cutting angle method–a tool for constrained global optimization. Opti- mization Methods and Software, 19(2):137–151, 2004.
[2] Clemens Biller. Adaptive bayesian regression splines in semiparametric generalized linear models. Journal of Computational and Graphical Statistics, 9(1):122–140, 2000.
[3] Hermann G Burchard. Splines (with optimal knots) are better. Applicable Analysis, 3(4):309–319, 1974.
[4] Carl De Boor. A practical guide to splines; rev. ed. Applied mathematical sciences. Springer, Berlin, 2001.
[5] DGT Denison, BK Mallick, and AFM Smith. Automatic bayesian curve fitting. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60(2):333–350, 1998.
[6] Paul HC Eilers and Brian D Marx. Flexible smoothing with b-splines and penalties. Statistical science, pages 89–102, 1996.
[7] Randall L Eubank. Spline smoothing and nonparametric regression. Marcel Dekker, 1988.
[8] Jerome H Friedman. Multivariate adaptive regression splines. The annals of statistics, pages 1–67, 1991.
[9] Jerome H Friedman and Bernard W Silverman. Flexible parsimonious smoothing and additive modeling. Technometrics, 31(1):3–21, 1989.
[10] David LB Jupp. Approximation to data by splines with free knots. SIAM Journal on Numerical Analysis, 15(2):328–343, 1978.
[11] Vladimir K Kaishev, Dimitrina S Dimitrova, Steven Haberman, and Richard J Ver- rall. Geometrically designed, variable knot regression splines. Computational Statistics, 31(3):1079–1105, 2016.
[12] Hongmei Kang, Falai Chen, Yusheng Li, Jiansong Deng, and Zhouwang Yang. Knot calculation for spline fitting via sparse optimization. Computer-Aided Design, 58:179–188, 2015.
[13] Charles Kooperberg, Charles J Stone, and Young K Truong. Hazard regression. Journal of the American Statistical Association, 90(429):78–94, 1995.
[14] Thomas CM Lee. Regression spline smoothing using the minimum description length principle. Statistics & probability letters, 48(1):71–82, 2000.
[15] Mary J Lindstrom. Penalized estimation of free-knot splines. Journal of Computational and Graphical Statistics, 8(2):333–352, 1999.
[16] Satoshi Miyata and Xiaotong Shen. Free-knot splines and adaptive knot selection. Journal of the Japan Statistical Society, 35(2):303–324, 2005.
[17] Nicolas Molinari, Jean-François Durand, and Robert Sabatier. Bounded optimal knots for regression splines. Computational statistics & data analysis, 45(2):159–178, 2004.
[18] Finbarr O’sullivan, Brian S Yandell, and William J Raynor Jr. Automatic smoothing of regression functions in generalized linear models. Journal of the American Statistical Association, 81(393):96–103, 1986.
[19] Daryl Pregibon. Logistic regression diagnostics. The Annals of Statistics, pages 705–724, 1981.
[20] Carl Runge. Über empirische funktionen und die interpolation zwischen äquidistanten ordinaten. Zeitschrift für Mathematik und Physik, 46(224-243):20, 1901.
[21] Larry Schumaker. Spline functions: basic theory. Cambridge University Press, 2007.
[22] Michael Smith and Robert Kohn. Nonparametric regression using bayesian variable selec- tion. Journal of Econometrics, 75(2):317–343, 1996.
[23] Michael Smith, Chi-Ming Wong, and Robert Kohn. Additive nonparametric regression with autocorrelated errors. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 60(2):311–331, 1998.
[24] Charles J Stone, Mark H Hansen, Charles Kooperberg, Young K Truong, et al. Polynomial splines and their tensor products in extended linear modeling: 1994 wald memorial lecture. The Annals of Statistics, 25(4):1371–1470, 1997.
[25] Wannes Van Loock, Goele Pipeleers, Joris De Schutter, and Jan Swevers. A convex opti- mization approach to curve fitting with b-splines. IFAC Proceedings Volumes, 44(1):2290– 2295, 2011.
[26] Grace Wahba. A Survey of Some Smoothing Problems and the Method of the Generalized Cross-validation for Solving Them. University of Wisconsin, Department of Statistics, 1976.
[27] Grace Wahba. Spline models for observational data. SIAM, 1990.
[28] Yuan Yuan, Nan Chen, and Shiyu Zhou. Adaptive b-spline knot selection using multi- resolution basis set. IIE Transactions, 45(12):1263–1277, 2013.
[29] Shanggang Zhou and Xiaotong Shen. Spatially adaptive regression splines and accurate knot selection schemes. Journal of the American Statistical Association, 96(453):247–259, 2001.		zh_TW