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研究生:吳雅貞
研究生(外文):Wu Ya Chan
論文名稱:比較兩種保持邊界特性的M平滑方法
論文名稱(外文):Compare two kinds of Edges-Preserving M-Smoother
指導教授:朱至剛朱至剛引用關係
指導教授(外文):Chu,Chih-Kang
學位類別:碩士
校院名稱:國立東華大學
系所名稱:應用數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:中文
論文頁數:30
中文關鍵詞:無母數迴歸M-平滑方法區域極大絕對極大
外文關鍵詞:Nonparametric regressionRobust M-estimationlocal maximumglobl maximum
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在隨機取樣的無母數迴歸分析模型中,迴歸函數核估計方法算是最受歡迎的估計方法。然而,它有一個缺點。亦即,在斷點近傍內的估計效果不盡理想。1998年Chu, Glad,Godtliebsen, and Marron提出保持邊界特性的平滑方法(Edges-preserving smoother,稱為保持邊界特性的M平滑方法),理論上它是基於穩固的最大概似估計法(Robust M-estimation)的概念,並且在計算上它應用局部值的特性。大體來說,保持邊界特性的M平滑方法都能有令人滿意的結果。
本文內容係提出兩種保持邊界特性的M平滑方法,即區域極大方法與絕對極大方法。並且比較此兩種方法在估計斷點的效果。由模擬結果發現,當迴歸函數存在很多斷點時,區域極大方法會比絕對極大方法有更好的估計效果。但是,當迴歸函數斷點很少時,絕對極大方法也能很清楚的反應斷點所在位置。所以在斷點很少時,絕對極大方法也不失為一個好的估計方法。
此外,為了降低保持邊界特性的M平滑估計值的均方誤差,在參數的選取上我們提出了建議和比較。最後我們選取一組最佳參數h和g,運用到兩種保持邊界特性的M平滑方法上,使得此兩種方法能夠發揮最大功效。

In the case of the random design nonparametric regression,
the kernel menthod is the most popular regression function estimator . Howerver,there is a drawback to the kernel method.
That is, it is lower efficiency when the estimator within the
neighborhood of the jump point. A new edges-preserving smoother,
that is called "edges-preserving M-smoother", was proposed by Chu,Glod, Godtliebsen, and Marron (1998). It is based on robust
M-estimatior and using local minima property. In most cases the
edges-preserving M-smoother has a pleasing result.
The contents of this thesis is is to propose two kinds of
edges-preserving M-smoother: the method of local maximum and the
method of global maximum. Then, compare the estimative efficiency at the jump points. Simulation studie demonstrate that the method of local maximum has better estimative
efficiency than the method of globl maximum when the regression
function has too much jumps. But the method of global maximum can also show the position of the jump points when the
regression function has less jumps. So when the regression
function has less jumps, the method of the global maximum is also a good estimative method.
Besides, to reduce the mean squared error of the edges-preserving M-smoother, we give the recommendtion and comparisons for the choice of parameters h and g. At the end of the thesis, we choose a pair h and g, which is the best parameters. For applying to two kinds of edges-preserving M-smoother,they can make two methods arrive to the maximal elliciency.

目錄
1. 簡介
1.1 迴歸分析簡介.....................1
1.2 無母數迴歸分析簡介...............2
2. 迴歸模型及估計量
2.1無母數迴歸分析.....................5
( Nonparametric Regression)}
2.2 兩種保持邊界特性的$M$平滑方法 ....6
(Two kinds of Edges-Preserving M-Smoother)}
2.3 M函數
討論Sp函數的組成可反應斷點..........7
解釋為什麼可以用a來估計m(x).........7
2.4 參數的選擇........................8
3 模擬研究
3.1 比較兩種保持邊界特性的方法.........10
3.2 模擬研究...........................10

(1).C.K.,GodtChuliebsen,~F. and Marron,~J,S.(1998),"Edge-Preserving Smoothers for Image Processing,"{\sl Journal of the American Statistical Assciation},93,526-556.(2).C.K. and Marron,~J.S.(1991),"Choosing a Kernel regression estimator,"{\sl Statistical Science},6,404-436.
(3).Hardle,~W.(1990),"it Applied Nonparametric Regression,"{\sl Cambridge University Press,New York}.
(4).Cleveland,~W.S.(1979),"Robuest locally weighted regression and smoothing scatter plots,"{\sl Jounral of the American Statistical Association},74,829-836.
(5).Eubak,~R.L(1988),"Spline Smoothing and Nonparmetric Regression",New York:{\sl Marcel Dekker}.
(60.Fan,~J(1933)," Local linear regression smoothers and their minimax efficiencies,"{\sl Annals of Statistics},21,196-216.
(7).Fan,~J.and Gijbels,~I.(1992)," Variable bandwith and local linear regression smoothers,"{\sl Annals of Statistics},20,2008-2036.
(8).Gasser,~T.and Mueller,~H.G.(1979),"Kernel estimation of regression functions,"in smoothing Techniques for Curve
Estimation(T.Gasser and M.Rosenblatt,eds),268,Heidelberg:S
pringer.
(9)Wu,J.S.and Chu,C.K.(1992),"Double smoothing for kernel estimators in nonparametric regression,"{\sl Journal of Nonparametrics,Statistics},1,375-386.

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