貝氏分析使用伯氏多項式對遞增迴歸分析之研究

Bayesian Regression with Isotonic Random Bernstein Polynomials

王佐剛(Tso-Kang Wang)

指導教授 ： 吳裕振

中原大學/理學院/應用數學研究所/碩士(2009年)

https://doi.org/10.6840/cycu200900828

摘要

迴歸的問題，自古以來是相當重要的，如線性迴歸，本篇論文主要估計遞增迴歸的曲線，利用貝氏方法，因為伯氏多項式可逼近所有的連續函數，而且我們找到係數條件，即可得到此伯氏多項式去遞增，而且可逼近所有的遞增連續函數，而且對於此迴歸曲線的事前分佈給定，相當容易，對於計算事後分佈有莫大的幫助，所以模型用伯氏多項式來描述遞增的曲線，所以比線性迴歸較為複雜許多，其 M.L.E. 估計也比較困難，所以根據以上的理由，因此我們採用貝氏方法，而且計算其事後分佈，用馬可夫鍊蒙地卡羅 (M.C.M.C.) 之方法來計算，在這篇論文有完整介紹其模型演算法和理論之推論。演算法程式用軟體 Matlab，而且其所得到的估計，相當不錯。

主要理論在第4節。在 M.L.E. 方法，計算相當複雜，未來我們可以做為研究的主題，在和我們估計比較，當然本篇論文也可推廣到二維曲面迴歸之估計，這都是非常好的研究，也是作為我們未來研究的發展方向。

關鍵字

伯氏多項式 ； 馬可夫鍊蒙地卡羅

並列摘要

The problem of regression which alike linear regression is very important since remotest time.
This paper is about using Bayesian method to estimate increasing regression curve.Because of continuous functions be approached by Bernstein polynomails which can be found the coefficient to show that it is increasing and approach all of the insreasing continuous functions.On the other hand, it's easy to have the prior and is helpful in calculate the posterior by using Bernstein polynomials.So the model using Bernstein polynomials to describe the graph of increasing curve is more complicate than using linear regression,without saying, it's also more difficult to estimate the M.L.E.

Above all,we decide to use Bayesian method and calculate the posterior by using Markov Chain Monte Carlo(M.C.M.C.) method. We have introduction on model algorithm and the theorem of reduction completely in this paper.And using the package software Matlab writing the program to get the estimator can be very nice.

The theorem is in 4. It's really complicate by using M.L.E. mothod.We may use this to be our research title in the future.Compare with our estimation,the paper also can be expended to estimation of 2-dimention surface regression,all of them can be direction of research in our future.

並列關鍵字

Markov Chain Monte Carlo ； Bernstein polynomails

參考文獻

regression and concave regression with random Bernstein

polynomials. (Preprint).

estimator of a monotone regression function. Bernoulli 12,

Green, P. G. (1995). Reversible jump Markov chain Monte Carlo

computation and Bayesian model determination. Biometrika 82,