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研究生:蕭愛齡
研究生(外文):Ai-ling Hsiao
論文名稱:利用Bernstein多項式來研究二元迴歸
論文名稱(外文):Binary regression with Bernstein polynomials
指導教授:趙一峰趙一峰引用關係張憶壽張憶壽引用關係
指導教授(外文):I-Feng ChaoI-Shou Chang
學位類別:碩士
校院名稱:國立中央大學
系所名稱:數學研究所
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:2008
畢業學年度:96
語文別:中文
論文頁數:19
中文關鍵詞:馬可夫鏈蒙地卡羅法logistic 迴歸模型Bernstein 多項式
外文關鍵詞:MCMCBernstein polynomiallogistic regression model
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傳統上,針對二元資料之分析多採用 logistic 迴歸模型。但此模型在事件發生之條件機率上有單調函數之限制,因此我們利用Bernstein 多項式來表達事件發生之條件機率,因而於本文中提出一個藉由Bernstein 多項式所建構的貝氏迴歸模型。在貝氏方法中,我們將先驗分佈建立在Bernstein 多項式的次數和係數所組成的參數空間上,並對統計推論所需的後驗分佈用MCMC 的方法做
抽樣。最後,在相同的模型與方法下,比較在不同樣本數及先驗分佈下的模擬結果;其次,對於logistic 迴歸模型的限制,經由模擬顯示本文所提出的貝氏迴歸有較小的均方誤差。
Data analysis of binary response variables are often conducted by logistic regression model. Logistic regression model assumes that the conditional probability function of success is a monotonic function. In order to eliminate this sometimes unnecessary monotone restriction, we propose to use Bernstein polynomials to model the conditional probability of success. As a Bayesian approach, we put a prior on the space of Bernstein polynomials having values in [0,1] through their coe cients. The sample from the posterior distribution for inference purpose is obtained by MCMC methods. We conduct simulation studies to examine the e ects of sample size and priors, to indicate that the numerical performance of this method is generally good and to show that our model performs better than the logistic regression model when
the regression function is not monotone.
1 緒論.............................................................1
2 模型
2.1 介紹Bernstein 多項式及Bernstein-Weierstrass 定理.....3
2.2 由Bernstein 多項式所建構的二元迴歸模型................4
2.3 演算法......................................................6
3 模擬研究
3.1 有效模擬次數........................................10
3.2 平均均方誤差........................................11
3.3 模擬結果與討論....................................13
4 結論...........................................................18
參考文獻......................................................19
[1] Chang, I. S., Chien, L. C., Hsiung, C. A., Wen, C. C., and Wu, Y. J. (2007). Shape restricted regression with random Bernstein polynomials. In Complex Dataset and Inverse Problems. IMS Lecture Notes Monograph Series , 54, 187-202
[2] 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
[3] Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2004). Bayesian data analysis, 2nd ed. Chapman and Hall, Boca Raton.
[4] Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711-732.
[5] Resnick, S. I. (1999). A Probability Path. Birkhauser, Boston.
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