带弹性网惩罚的光滑化分位数回归

发布时间：2018-04-10 11:39

本文选题：高维数据 + Huber函数　；参考：《北京交通大学》2017年硕士论文

【摘要】：高维数据在很多科学领域都会遇到,比如说在信息科学、生物学、经济学等领域.高维数据给现代统计方法和优化计算带来很大的挑战.传统的回归方法不能有效的进行分析.考虑到高维数据容易导致共线性问题及高维数据的误差可能是重尾的,基于最小二乘的线性回归方法不能有效分析这样的数据,于是我们引入了加弹性网惩罚的分位数回归模型,这个模型结合了二次正则和LASSO收缩的优点,既能解决共线性问题又能实现变量筛选.另外,由于弹性网惩罚特有的结构特点,使得模型有分组的效果,即高度相关的变量将会同时被选进模型或被剔除出模型.由于分位数损失函数具有凸性但不具有可微性,不利于模型求解,于是通过Huber光滑函数,将分位数损失函数光滑化,得到加弹性网惩罚的光滑化分位数回归(SQEN).我们还验证了SQEN估计值具有统计性质.为了有效估计SQEN的值,我们引入一个有效的迭代算法:SQEN—MM算法,并建立算法的全局收敛性.在文章的最后,我们通过数值实验进一步验证我们提出的方法的有效性.本文分为六章,第一章介绍了研究背景和研究现状;第二章引入加弹性网惩罚的光滑分位数回归模型(SQEN);第三章介绍了SQEN模型估计值的统计性质;第四章引入求解SQEN模型的最优化算法并证明算法的全局收敛性;第五章通过数值实验验证算法的有效性;第六章对全文进行了总结,并对未来要进一步研究的工作进行了展望.
[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.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212

【相似文献】