本文選題：高維數(shù)據(jù) + Huber函數(shù)��； 參考：《北京交通大學》2017年碩士論文

【摘要】：高維數(shù)據(jù)在很多科學領域都會遇到,比如說在信息科學、生物學、經(jīng)濟學等領域.高維數(shù)據(jù)給現(xiàn)代統(tǒng)計方法和優(yōu)化計算帶來很大的挑戰(zhàn).傳統(tǒng)的回歸方法不能有效的進行分析.考慮到高維數(shù)據(jù)容易導致共線性問題及高維數(shù)據(jù)的誤差可能是重尾的,基于最小二乘的線性回歸方法不能有效分析這樣的數(shù)據(jù),于是我們引入了加彈性網(wǎng)懲罰的分位數(shù)回歸模型,這個模型結合了二次正則和LASSO收縮的優(yōu)點,既能解決共線性問題又能實現(xiàn)變量篩選.另外,由于彈性網(wǎng)懲罰特有的結構特點,使得模型有分組的效果,即高度相關的變量將會同時被選進模型或被剔除出模型.由于分位數(shù)損失函數(shù)具有凸性但不具有可微性,不利于模型求解,于是通過Huber光滑函數(shù),將分位數(shù)損失函數(shù)光滑化,得到加彈性網(wǎng)懲罰的光滑化分位數(shù)回歸(SQEN).我們還驗證了SQEN估計值具有統(tǒng)計性質(zhì).為了有效估計SQEN的值,我們引入一個有效的迭代算法:SQEN—MM算法,并建立算法的全局收斂性.在文章的最后,我們通過數(shù)值實驗進一步驗證我們提出的方法的有效性.本文分為六章,第一章介紹了研究背景和研究現(xiàn)狀;第二章引入加彈性網(wǎng)懲罰的光滑分位數(shù)回歸模型(SQEN);第三章介紹了SQEN模型估計值的統(tǒng)計性質(zhì);第四章引入求解SQEN模型的最優(yōu)化算法并證明算法的全局收斂性;第五章通過數(shù)值實驗驗證算法的有效性;第六章對全文進行了總結,并對未來要進一步研究的工作進行了展望.
[Abstract]:High-dimensional data are encountered in many fields of science, such as information science, biology, economics and so on.High dimensional data bring great challenges to modern statistical methods and optimization calculation.The traditional regression method can not be effectively analyzed.Considering that high-dimensional data can easily lead to collinearity problems and that the error of high-dimensional data may be heavy-tailed, the linear regression method based on least squares can not effectively analyze such data.So we introduce a quantile regression model with penalty of elastic net. This model combines the advantages of quadratic regularization and LASSO contraction and can solve the collinearity problem as well as variable selection.In addition, due to the unique structural characteristics of the elastic network, the model has the effect of grouping, that is, the highly relevant variables will be selected into the model or removed from the model at the same time.Because the quantile loss function is convexity but not differentiable, it is difficult to solve the model. Therefore, the quantile loss function is smoothed by Huber smooth function, and the smoothing quantile with elastic net penalty is obtained.We also verify the statistical properties of SQEN estimators.In order to estimate the value of SQEN effectively, we introduce an effective iterative algorithm: SQEN-MM algorithm, and establish the global convergence of the algorithm.At the end of the paper, we further verify the effectiveness of the proposed method by numerical experiments.This paper is divided into six chapters, the first chapter introduces the research background and research status, the second chapter introduces the smooth quantile regression model with penalty of elastic net, the third chapter introduces the statistical properties of the estimated value of the SQEN model.In chapter 4, the optimization algorithm for solving SQEN model is introduced and the global convergence of the algorithm is proved; in the fifth chapter, the validity of the algorithm is verified by numerical experiments; in chapter 6, the full text is summarized, and the future research work is prospected.
【學位授予單位】：北京交通大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O212

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