本文選題：穩(wěn)健 + 混合模型��； 參考：《昆明理工大學(xué)》2017年碩士論文

【摘要】：在統(tǒng)計學(xué)中影響統(tǒng)計結(jié)果的重要因素有兩個:一是觀測數(shù)據(jù),二是對總體某些特性(分布、獨立性等)的假設(shè).當(dāng)觀測數(shù)據(jù)中存在一些不能很好的代表總體的異常點或者研究總體不滿足一些傳統(tǒng)的統(tǒng)計方法對總體某些特性的假設(shè)時,就會出現(xiàn)問題甚至導(dǎo)致錯誤的結(jié)論.這個時候,一些更為穩(wěn)健的統(tǒng)計方法、更為穩(wěn)健的分布類型更能體現(xiàn)出在處理這類問題上的優(yōu)勢,t分布、Laplace分布、Pearson type Ⅶ分布等一些包含異常點的"厚尾分布"對異常值和偏離均值較多的厚尾數(shù)據(jù)都不是特別敏感,是一種很不錯的穩(wěn)健分布類型,同時也體現(xiàn)出了穩(wěn)健統(tǒng)計方法的特點:即使存在少量異常點,對與理想分布的偏離所引起的結(jié)果影響也不是很大;存在較多的異常點也不至于導(dǎo)致錯誤的結(jié)論.隨著社會的發(fā)展,我們生活中各個領(lǐng)域的數(shù)據(jù)也越來越復(fù)雜、多樣,這時勢必要對這些異質(zhì)的總體進行聚類分析,混合模型應(yīng)運而生,用不同的參數(shù)和比例的分布來擬合不同的幾類數(shù)據(jù).大量異方差數(shù)據(jù)的存在違背了傳統(tǒng)回歸模型中方差齊次性的假設(shè),為了有效的控制方差,在處理異方差數(shù)據(jù)的問題上,我們多采用聯(lián)合均值與方差模型,現(xiàn)在我們也可以將模型方法進行推廣,使適用范圍更加廣泛,把同質(zhì)總體中的聯(lián)合均值與方差模型推廣到異質(zhì)總體的混合模型中.進一步地,當(dāng)考慮混合數(shù)據(jù)的分類情況未知時,我們還可以引入混合專家系統(tǒng),對混合比例進行建模,應(yīng)用Logistic回歸對影響混合比例的未知參數(shù)進行估計.本文主要基于t分布、Laplace分布、Pearson type Ⅶ分布三種穩(wěn)健的分布應(yīng)用EM算法對異質(zhì)總體的混合聯(lián)合位置與尺度模型的未知參數(shù)進行極大似然估計,主要內(nèi)容有:第一,基于t分布下,建立混合聯(lián)合位置與尺度參數(shù)的模型,應(yīng)用EM算法、極大似然估計、Gauss-Newton迭代算法對模型中的未知參數(shù)進行估計,并通過Monte Carlo模擬方法驗證所提出估計方法的有效性.然后試著把所提出的估計方法與實際生活聯(lián)系起來,解決一些實際問題.第二,基于Laplace分布下,建立混合聯(lián)合位置與尺度參數(shù)的模型,應(yīng)用EM算法、極大似然估計、Gauss-Newton迭代算法對模型中的未知參數(shù)進行估計,并通過Monte Carlo模擬方法驗證所提出估計方法的有效性.然后試著把所提出的估計方法與實際生活聯(lián)系起來,解決一些實際問題.第三,基于Pearson type Ⅶ分布下,建立混合聯(lián)合位置與尺度參數(shù)的模型,.應(yīng)用EM算法、極大似然估計、Gauss-Newton迭代算法對模型中的未知參數(shù)進行估計,并通過Monte Carlo模擬方法驗證所提出估計方法的有效性.然后試著把所提出的估計方法與實際生活聯(lián)系起來,解決一些實際問題.第四,基于Laplace分布下,在混合專家系統(tǒng)中,建立混合聯(lián)合位置與尺度參數(shù)的模型,應(yīng)用MM算法、EM算法、極大似然估計、Gauss-Newton迭代算法對模型中的未知參數(shù)進行估計,并通過Monte Carlo模擬方法驗證所提出估計方法的有效性.然后試著把所提出的估計方法與實際生活聯(lián)系起來,解決一些實際問題.
[Abstract]:There are two important factors affecting statistical results in Statistics: one is the observation data, and the two is the assumption of the general characteristics (distribution, independence, etc.). When there are some exceptions in the observation data that can not be well represented as a whole, or the study generally does not meet the assumptions of some traditional statistical methods on some of the overall characteristics of the general statistical method, At this time, some more robust statistical methods, the more robust distribution types can reflect the advantages of dealing with these problems, t distribution, Laplace distribution, Pearson type VII distribution and other "thick tail distribution" containing abnormal points, and the heavy tailed data with more deviations from the mean value. It is not particularly sensitive, it is a very good robust distribution type, and it also embodies the characteristics of robust statistical methods: even if there is a small number of anomaly points, the effects on the deviations from the ideal distribution are not very large; there are many anomalies that do not lead to the wrong conclusions. With the development of society, we live in our lives. The data in various fields are becoming more and more complex and diverse. At this time, we must cluster analysis of these heterogeneous groups. The mixed model comes into being and fits different kinds of data with different parameters and proportions. The existence of a large number of heteroscedasticity data is contrary to the hypothesis of the traditional Chinese homogeneity in the traditional return model, in order to be effective. In the control of variance, we use the combined mean and variance model in dealing with the problem of heteroscedasticity. Now we can also extend the model method to make the scope of application more extensive, and extend the joint mean and variance model in homogeneity to the mixture model of heterogeneous population. When the class situation is unknown, we can also introduce a hybrid expert system to model the mixture ratio and estimate the unknown parameters that affect the mixture ratio by Logistic regression. This paper is based on three robust distributions of t distribution, Laplace distribution, Pearson type VII distribution and the mixed joint location and scale of EM algorithm for heterogeneous population. The main content of the unknown parameters of the model is maximum likelihood estimation. The main contents are as follows: first, based on the t distribution, the model of mixed joint position and scale parameter is established. EM algorithm, maximum likelihood estimation, Gauss-Newton iterative algorithm are used to estimate the unknown parameters in the model, and the Monte Carlo simulation method is used to verify the proposed estimation method. Then, we try to connect the proposed method with real life and solve some practical problems. Second, based on the Laplace distribution, the model of mixed joint position and scale parameter is established. The EM algorithm, maximum likelihood estimation, Gauss-Newton iterative algorithm are used to estimate the unknown parameters in the model, and the Monte Carlo module is used. This method verifies the effectiveness of the proposed method. Then we try to connect the proposed method with real life and solve some practical problems. Third, based on the Pearson type VII distribution, a model of mixed joint position and scale parameters is established. The EM algorithm, maximum likelihood estimation, and Gauss-Newton iterative algorithm are applied to the model. The unknown parameters are estimated, and the effectiveness of the proposed method is verified by the Monte Carlo simulation method. Then, we try to connect the proposed method with real life and solve some practical problems. Fourth, based on the Laplace distribution, the mixed joint location and the scale parameter model is established in the mixed expert system. The MM algorithm, EM algorithm, maximum likelihood estimation, Gauss-Newton iterative algorithm are used to estimate the unknown parameters in the model, and the effectiveness of the proposed estimation method is verified by the Monte Carlo simulation method. Then, the proposed estimation method is linked to the actual life, and some practical problems are solved.

【學(xué)位授予單位】：昆明理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O212.1

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本文編號：1804930