伯氏多項式在貝氏部份單調迴歸之研究

Bayesian partial monotone regression by using Bernstein polynomial

廖偉廷(Wei-Ting Liao)

指導教授 ： 吳裕振

中原大學/理學院/應用數學研究所/碩士(2010年)

https://doi.org/10.6840/cycu201000170

摘要

迴歸曲線的問題是非常重要的，尤其是在統計上的迴歸研究，根據其自變數和應變數的關係，不但可以幫助我們評估過去的利弊，也能推測未來的走向，再計算其迴歸函數來摒除誤差並分析出所有的可能性，來讓我們做出最佳的選擇。

在本篇論文中，主要是探討在條件限制下的迴歸曲線，其條件限制為部份單調的迴歸曲線，並通用於具有單調性質的遞增或遞減函數。在此，利用伯氏多項式可以逼近所有連續函數的特性，只需找出其係數條件，即可讓函數符合我們所要的規則，同時也易於做出事前分佈的給定，因此我們採用伯氏多項式來描述我們的圖形。但也因為其函數比一般線性迴歸還要來的複雜許多，若是使用M.L.E.法來做估計也會較為複雜。因此，我們提供了一個用來估計部份單調迴歸曲線的方法，就是先利用貝氏估計法去估計函數後，再用馬可夫鏈蒙地卡羅演算法來計算事後分佈。

在統計模擬方面，我們可以利用各種統計軟體來做估計，其中我們使用了統計程式R 去做不同樣本數的資料模擬並寫入第四章的模擬計算裡，再經由我們利用獨立演算法去計算後，寫出詳細的步驟供大家參考。我們不但完整的介紹出模型的建立及理論的應用，也比較了程式估計出來的函數值和實際函數的差異程度，最後再做其結論。

關鍵字

馬可夫鏈蒙地卡羅演算法 ； 伯氏多項式 ； 部份單調迴歸曲線 ； 貝氏估計法

並列摘要

The topic of linear regression is very important so far away. Especially when we are doing the linear research on the statistic by the relationship between the dependent variable and the independent variable that can help us to evaluate the past and to predict the forecast. Besides, we can also calculate the function of linear regression to get rid of error and to analysis all of the possibility, then make the best choice by estimator.

The main in the thesis, is to study the curve line of regression under the restrict condition which is called partial linear regression. The curve line can be used in the functions which are having the property with monotone increasing
or monotone decreasing. We can adopt Bernstein polynomial to apply our regression functions because of the reason that the property of Bernstein polynomial can be limited all of continuous functions. All we have to do is
finding the coefficient of Bernstein polynomial that can conform to our conditions of partial linear regression and infer to posterior easier. But it is harder to estimate it by using M.L.E.; these functions are more complicating than others general linear regression functions.

Therefore, we provide a method to estimate the partial linear regression. The method is using Markov Chain Monte Carlo (M.C.M.C.) to calculate the posterior of regression data after using Bayesian inference to estimate the function.

We have already written down all of detail by IMA algorithm steps in the “Section 4” in the thesis. Above all, we not only introduce the model complete but also explain the application of theorem. Besides, we simulate the program
by software R of statistic with different numbers of data, then calculate the estimated result to compare with real data for our conclusion.

並列關鍵字

Bayesian inference ； Markov Chain Monte Carlo ； Partial monotone linear regression ； Bernstein polynomial

參考文獻

王佐剛(2008), “Bayesian regression with isotonic random bernstein polynomials”, Department of Applied Mathematics, Chung-Yuan Christian University, master thesis.

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