本文關(guān)鍵詞： 貝葉斯懲罰回歸 正則化參數(shù) 嶺回歸 lasso回歸 變量選擇　出處：《西南交通大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：懲罰回歸在統(tǒng)計(jì)學(xué)中是一種至關(guān)重要的回歸系數(shù)估計(jì)和變量選擇方法,而正則化參數(shù)在懲罰回歸中起著平衡損失函數(shù)和正則項(xiàng)的作用。因此,在懲罰回歸的擬合過(guò)程中,選擇一個(gè)合適的正則化參數(shù)顯得極其重要。在深入學(xué)習(xí)貝葉斯理論知識(shí)、研究懲罰回歸的基礎(chǔ)上,從貝葉斯分析角度出發(fā),通過(guò)尋找懲罰回歸中各個(gè)組成部分與貝葉斯模型中先驗(yàn)函數(shù)、似然函數(shù)之間的對(duì)應(yīng)關(guān)系,相應(yīng)地,給出懲罰回歸中所選擇的正則化參數(shù)的貝葉斯估計(jì)表達(dá)式。具體地,文章在懲罰回歸中的響應(yīng)變量服從正態(tài)分布,回歸參數(shù)服從指數(shù)族分布族中的某一個(gè)成員假設(shè)條件下,將懲罰回歸中的損失函數(shù)表示成貝葉斯模型中的似然函數(shù),正則項(xiàng)可表示成貝葉斯模型中的先驗(yàn)函數(shù),通過(guò)貝葉斯公式,將損失函數(shù)和正則項(xiàng)以貝葉斯模型的形式結(jié)合起來(lái),于是形成了一個(gè)關(guān)于回歸系數(shù)的后驗(yàn)分布的貝葉斯模型。這個(gè)過(guò)程在找出懲罰回歸和貝葉斯模型之間各個(gè)部分之間的對(duì)應(yīng)關(guān)系的同時(shí),相應(yīng)地,得到懲罰回歸中的正則化參數(shù)的貝葉斯估計(jì)表達(dá)式。將其推廣到一般的懲罰回歸(嶺回歸和lasso回歸),得到它們具體的正則化參數(shù)估計(jì)表達(dá)式。從貝葉斯角度得到的正則化參數(shù)的估計(jì)表達(dá)式包含有響應(yīng)變量和回歸系數(shù)分布中參數(shù),因此文章又重點(diǎn)探討了關(guān)于分布中未知參數(shù)(位置參數(shù)和刻度參數(shù))的貝葉斯估計(jì)值。最后,通過(guò)實(shí)例分析,在數(shù)據(jù)集滿足一定的條件下,和現(xiàn)有的嶺參數(shù)選擇方法(廣義交叉驗(yàn)證法、嶺跡法)分析比較,文中所探討的方法,和廣義交叉驗(yàn)證法相比降低了計(jì)算的復(fù)雜度,和嶺跡法相比較,在一定意義上給出了貝葉斯意義下的統(tǒng)計(jì)解釋。
[Abstract]:Penalty regression is a very important method of regression coefficient estimation and variable selection in statistics, and regularization parameters play a role of balance loss function and regular term in penalty regression. In the fitting process of penalty regression, it is very important to choose a suitable regularization parameter. On the basis of studying the Bayesian theory and punishment regression, we proceed from the perspective of Bayesian analysis. By looking for the corresponding relation between each component of penalty regression and the prior function and likelihood function in Bayesian model, the corresponding relation is obtained. The Bayesian estimation expression of the regularized parameters selected in the penalty regression is given. Specifically, the response variables in the penalty regression are obeyed from the normal distribution. Under the assumption of a member of exponential family distribution, the loss function in penalty regression is expressed as the likelihood function in Bayesian model, and the regular term is expressed as a prior function in Bayesian model. By using Bayesian formula, the loss function and the regular term are combined in the form of Bayesian model. A Bayesian model of a posterior distribution of regression coefficients is formed, which finds out the corresponding relationship between penalty regression and each part of Bayesian model, and at the same time, the corresponding relationship between each part of the penalty regression and Bayesian model is found. The Bayesian estimation expression of regularization parameters in penalty regression is obtained and generalized to general penalty regression (Ridge regression and lasso regression). Their specific regularization parameter estimation expressions are obtained. The estimated expressions of regularization parameters obtained from Bayesian perspective contain parameters in the distribution of response variables and regression coefficients. Therefore, the Bayesian estimation of unknown parameters (position parameter and scale parameter) in the distribution is discussed. Finally, through an example analysis, the data set satisfies certain conditions. Compared with the existing ridge parameter selection methods (generalized cross validation method, ridge trace method), the method discussed in this paper reduces the computational complexity compared with the generalized cross validation method and is compared with the ridge trace method. The statistical explanation in Bayesian sense is given in a certain sense.
【學(xué)位授予單位】：西南交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O212.8

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