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研究生:張仙芳
研究生(外文):Sen-Fang Chang
論文名稱:加權多項式迴歸模型之D最適設計-泛函逼近法
論文名稱(外文):D-optimal designs for weighted polynomial regression - a functional-algebraic approach
指導教授:張福春張福春引用關係
指導教授(外文):Fu-Chuen Chang
學位類別:碩士
校院名稱:國立中山大學
系所名稱:應用數學系研究所
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:2004
畢業學年度:92
語文別:英文
論文頁數:14
中文關鍵詞:遞迴算法泰勒級數隱函數定理加權多項式迴歸.矩陣近似D之最適設計有理函數
外文關鍵詞:weighted polynomial regression.recursive algorithmTaylor seriesrational functionapproximate D-optimal designmatriximplicit function theorem
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  • 下載下載:5
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在此論文中,我們主要探討的是有關於加權多項式迴歸模型D之最適設計的問題,其中加權函數限定為大於零的函數,在[m_0+a,m_0-a]內去建構此最適設計。我們發現在此區間內,如果加權函數一次微分除以加權函數自己本身為有理函數且a趨近於零時,則建構此D之最適設計的問題可以轉為解微分方程的問題,在解微分方程過程中,利用矩陣在代數上的相關知識,以泰勒展開式去逼近矩陣中的未知參數,而在泰勒展開式中的係數部分,我們提供了一個遞迴演算法來估計它們,因此,從這個線性系統中,我們可以估算出那些以D之最適設計的實驗點為零根的多項式之係數。
This paper is concerned with the problem of computing theapproximate D-optimal design for polynomial regression with weight function w(x)>0 on the design interval I=[m_0-a,m_0+a]. It is shown that if w''(x)/w(x) is a rational function on I and a is close to zero, then the problem of constructing D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix including a finite number of auxiliary unknown constants, which can be approximated by a Taylor expansion. We provide a recursive algorithm to compute Taylor expansion of these constants. Moreover, the D-optimal
interior support points are the zeros of a polynomial which has coefficients that can be computed from a linear system.
Contens:
Abstract..................................................ii
1. Introduction...........................................1
2. The differential equation..............................2
3. Taylor expansion.......................................5
4. Examples...............................................7
References................................................12
Appendix..................................................14
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