臺灣博碩士論文加值系統

在這篇論文中, 貝氏迴歸為兩個或更多的變數, 提出事前樣本空間為高維度的伯氏多項式, 因為多元伯氏多項式可逼近任意連續函數[ 可參考 Altomare 和Campiti (1994) ]。從我們以前的文獻的檢驗, 這一做法並沒有被考慮過圖形的概念; 我們給一個方法, 如何生成函數是單調的和計算事後分配, 使用馬爾可夫鏈蒙特卡羅方法。這些事前空間我們可考慮幾何的資訊, 如: 凸性, 在二個變數的實值函數, y 固定對x 是遞增且x 固定對y 是遞增, y 固定對x 是遞增且x 固定對y 是凹口向上, 和x 固定對y 是凹口向上且y 固定對x 是凹口向上等圖形, 而且可選擇到平滑的曲線, 並證明事前空間可以足夠大, 幾乎涵蓋所有的連續函數。

In this paper, Bayes regression for two or more variables is proposed using priors on multivariate Bernstein polynomials, since multivariate Bernstein polynomials can be used to approximate to an arbitrary continuous function of several variables [ see Altomare and Campiti (1994) ]. From our Literature survey, this approach has not been considered before; so far what has been done was any for functions which are monotone and generated sampling from the posterior distribution using Markov chain Monte Carlo methods. These priors easily take
into consideration geometric information like convexity, increasing in x for fixed y and increasing in y for fixed x , increasing in x for fixed y and convex in y for fixed x , convex in x for fixed y and convex in y for fixed x , as well select only smooth function, can have large enough support, and can be easily specified and generated. Simulation studies
to evaluate the performance of these Bayes methods.

中文摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II
誌謝. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
目錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV
1. 介紹1
2. 高維度伯氏多項式的性質2
2.1 高維度伯氏多項式的幾何圖形與係數之關係. . . . . 2
2.2 高維度伯氏多項式的逼近定理. . . . . . . . . . . . 3
2.2.1 機率方法. . . . . . . . . . . . . . . . . . . 3
2.2.2 分析方法. . . . . . . . . . . . . . . . . . . 4
2.2.3 不同形式的等價高維度伯氏多項式逼近定理5
3. 貝氏迴歸及其模型13
4. 高維度伯氏事前樣本空間所能提供的大小(support) 15
5. 貝氏推論16
5.1 事前分佈. . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 演算法. . . . . . . . . . . . . . . . . . . . . . . . . 17
6. 討論18
參考文獻. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

[1] Altomare, F., and Campiti, M. (1994). Korovkin-type Approximation Theory
and its Application. Math. Mon. 105 447-451.
[2] Chang, I. S., Chien, L. C., Hsiung, C. A., Wen, C. C., and Wu Y. J. (2006).
Shape restricted regression with random Bernstein polynomials. Accepted for
the Vardi Volume, IMS Lecture Notes-Monograph Series.
[3] Chang, I. S., Hsiung, C. A.,Wu, Y. J., and Yang, C. C. (2005). Bayesian survival
analysis using Bernstein polynomials. Scandinavian journal of statistics, 32,
447-466. (Authors are arranged in alphabetic order)(SCI).
[4] Cheney,E.W. (1998). Introduction to approximation theory. Providence, RI.
[5] Dette, H. and Scheder, R. (2006). Strictly monotone and smooth nonparametric
regression for two or more variables. The canadian journal of statistics, Vol. 34,
No. 4. P.535-561.
[6] Gebhardt, F. (1970). An algorithm for monotone regression with one or more
independent variables. Biometrika, 57:263-271.
[7] Green, P. G. (1995). Reversible jump Markov chain Monte Carlo computation
and Bayesian model determination. Biometrika 82 711-732.
[8] Lorentz, C. G. (1986). Bernstein Polynomials. Chelsea, New York.
[9] Wu, Y. J. andWang, S. F. (2008). Bernstein polynomial in statistic application.
Master thesis, Chung Yuan Christian University.