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    题名: 基於 multi-resolution B-spline basis 之二維曲面估計
    Estimate of the two-dimensions surface based on multi-resolution B-spline basis
    作者: 林哲宇
    Lin, Zhe-Yu
    贡献者: 黃子銘
    林哲宇
    Lin, Zhe-Yu
    关键词: B-spline迴歸
    節點選取
    曲面估計
    B-spline regression
    Knot selection
    Surface estimation
    日期: 2020
    上传时间: 2020-05-05 11:56:46 (UTC+8)
    摘要: 本研究是根據Yuan[12]提出的Multi-resolution B-spline basis節點放置方法對於節點的篩選做進一步的改良，以擾動項ε的變異數σ^2為標準，採用向後刪除的概念提出方法一及方法二，也將其改良後的方法透過張量積擴展到二維曲面的估計，以copula密度函數作為迴歸函數，與核迴歸中的局部線性迴歸的結果進行比較。 依模擬結果，方法一會因為σ的估計膨脹而篩選掉過多節點，方法二受σ的估計影響較小，估計效果較佳，也較為穩定。 在以copula密度函數作為迴歸函數的模擬實驗中，較不平滑的迴歸函數使用方法二來估計，估計效果較佳；較平滑的迴歸函數使用局部線性迴歸來估計為最佳，但如果在σ的估計上能更好，方法二的估計效果可能優於局部線性迴歸。
    This thesis is based on the multi-resolution B-spline basis knots placement method proposed by Yuan[12] to further improve the selection of knots. Based on the variation of the disturbance term, Method 1 and Method 2,are proposed using the concept of backward deletion. The improved method is also extended to the estimation of the two-dimensional surface. through the tensor product. Simulation studies have been carried out to compare the performance of Methods 1 and 2, and local linear regression. According to the results of simulation studies, Method 1 tends to filter out too many knots because of the large estimation error of σ,and Method 2 is less affected by the estimation of σ. The estimation based on Method 2 is more accurate and stable. In the studies where Methods 1 and 2, and local linear regression are compared, Method 2 outperforms local linear regression when the regression function is less smooth. When the regression function is smooth, local linear regression performs better than Method 2. However,if the estimation of σ can be better, the estimation accuracy of Method 2 may be better than local linear regression.
    參考文獻: [1] William S. Cleveland and Susan J. Devlin. Locally weighted regression: An approach to regression analysis by local fitting. Journal of the American Statistical Association, 83(403):596–610, 1988.

    [2] WS Cleveland. Lowess: A program for smoothing scatterplots by robust locally
    weighted regression. The American Statistician, 35:54, 1981.

    [3] Jerome H. Friedman. Multivariate adaptive regression splines. Ann. Statist., 19(1):1–67, 1991.

    [4] David L. B. Jupp. Approximation to data by splines with free knots. SIAM Journal on Numerical Analysis, 15(2):328–343, 1978.

    [5] 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.

    [6] Mary J. Lindstrom. Penalized estimation of free-knot splines. Journal of Computational and Graphical Statistics, 8(2):333–352, 1999.

    [7] Satoshi Miyata and Xiaotong Shen. Adaptive free-knot splines. Journal of Computational and Graphical Statistics, 12(1):197–213, 2003.

    [8] M. R. Osborne, B. Presnell, and B. A. Turlach. Knot selection for regression splines via the lasso. In Dimension Reduction, Computational Complexity, and Information, pages 44–49. America, Inc, 1998.

    [9] Abe Sklar. Fonctions de r´epartition "a n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Universit´e de Paris, 8:229–231, 1959.

    [10] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society (Series B), 58:267–288, 1996.

    [11] Wannes Van Loock, Goele Pipeleers, J. Schutter, and Jan Swevers. A convex optimization approach to curve fitting with b-splines. IFAC Proceedings Volumes (IFAC-PapersOnline), 18:2290–2295, 2011.

    [12] Yuan Yuan, Nan Chen, and Shiyu Zhou. Adaptive b-spline knot selection using multi-resolution basis set. IIE Transactions, 45:1263–1277, 2013.

    [13] 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
    106354016
    資料來源: http://thesis.lib.nccu.edu.tw/record/#G0106354016
    数据类型: thesis
    DOI: 10.6814/NCCU202000415
    显示于类别:[統計學系] 學位論文

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