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    题名: 傘型迴歸函數估計
    Estimation of umbrella shaped regression function
    作者: 林似蓉
    贡献者: 黃子銘
    林似蓉
    关键词: 傘型迴歸函數
    樣條函數
    節點
    umbrella shaped regression function
    spline
    knot
    日期: 2011
    上传时间: 2012-10-30 10:58:21 (UTC+8)
    摘要: 傘型迴歸函數是類似傘的形狀的迴歸函數，只要符合先上升後下降的趨勢皆為傘型迴歸函數。無母數迴歸函數中最常見的方法之一是樣條(Splines)迴歸函數。樣條為充分平滑分段多項式函數，而節點(knots)為平滑多項式函數連接的地方。在本論文中，將節點以等距離擺放並以AIC(Akaike information criterion)值得到合理的節點數。用三種方法的樣條迴歸函數去估計傘型函數。第一種為RSPL(restrictted spline regression)，也就是有形狀限制時的樣條迴歸函數。第二種是CSPL(concave spline regression)，是參考Meyer寫的樣條迴歸函數，此樣條迴歸函數為凹函數(concave function)。最後一種則稱SPL(spline regression)，為沒有形狀限制也不是凹函數的樣條函數。以IMSE為評估標準，IMSE越小，則代表此方法估計的越好。由模擬結果，在估計先上升後下降的函數時，用RSPL的方法去估計會得到最小的IMSE；而在估計凹函數時，則是CSPL會得到最小的IMSE。利用RSPL和SPL兩個方法估計由中央氣象局蒐集最近13年(1998-2010)的月均溫資料並探討最近幾年的月均溫資料趨勢是否有改變。未來假如需要估計傘型函數時，則可利用本篇所述的方法去估計。
    In this thesis, we consider the problem of estimating a regression function assuming the regression function is unimodal. The proposed method is to model the regression function as linear combination of B-spline basis functions with equally spaced knots, and the number of knots is determined using AIC (Akaike information criterion). Specific constraints are placed on the coefficients of basis functions to ensure that estimated regression function is unimodal. The coefficients are estimated using least square method.

    The proposed method is refered as RSPL and is compared with two other methods: SPL and CSPL, where SPL is similar to RSPL except that the coefficients of basis functions are estimated without any constraints, and CSPL gives concave regression function estimates. Simulation results show that RSPL outperforms SPL and CSPL when the true regression function is unimodal but not concave, and CSPL outperforms RSPL and SPL when the true regression function is concave. Also, RSPL is applied to temperature data to estimate temperature trend within one year.
    參考文獻: [1] H. Akaike. A new look at the statistical model identification. Institute of Statistical
    Mathematics, Minato-ku, Tokyo, Japan, 19 , Issue: 6:716– 723, 1974.
    [2] Wolfgang Härdle. Applied nonparametric regression. Cambridge University Press,
    1990.
    [3] Luke Keele. Semiparametric Regression for the Social Sciences. Wiley, Chichester,
    UK, 2008. ISBN 978-0470319918.

    [4] E. Mammen and C. Thomas-agnan. Smoothing splines and shape restrictions. Scan-
    dinavian Journal of Statistics, 26:239–252, 1998.
    [5] Mary C. Meyer. Inference using shape-restricted regression splines. The Annals of Applied Statistics, 2(3):1013–1033, 2008
    [6] Satoshi Miyata and Xiaotong Shen. Free-knot splines and adaptive knot selection.
    J. Japan Statist. Soc., Vol. 35 No. 2:303–324, 2005.
    [7] Michael R. Osborne, Brett Presnell, and Berwin A. Turlach. Knot selection for
    regression splines via the LASSO. In Computing Science and Statistics. Dimen-sion Reduction, Computational Complexity and Information. Proceedings of the 30th
    Symposium on the Interface, pages 44–49, 1998.
    [8] J. O. Ramsay. Monotone regression splines in action (C/R: p442-461). Statistical
    Science, 3:425–441, 1988.
    [9] Larry L. Schumaker. Spline Functions: Basic Theory. Cambridge University Press,
    2007.
    [10] Gideon Schwarz. Estimating the dimension of a model. The Annals of Statistics,
    6:461–464, 1978.
    [11] E. V. Shikin and Alexander I. Plis. Handbook on Splines for the User. CRC Press,
    1995.
    [12] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal
    Statistical Society (Series B), 58:267–288, 1996.
    描述: 碩士
    國立政治大學
    統計研究所
    99354015
    100
    資料來源: http://thesis.lib.nccu.edu.tw/record/#G0099354015
    数据类型: thesis
    显示于类别:[統計學系] 學位論文

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