高维二次度量回归模型研究
发布时间：2018-11-19 12:27
【摘要】：在大数据时代,高维数据呈现在基因组和健康科学、经济与金融、天文学与物理学、信号处理与成像等学科领域.其中一个共同特征是预测变量具有稀疏性.选择最相关的预测变量是高维数据回归分析的一个主要研究内容,具有十分重要的应用价值.为此,针对线性回归假设的许多统计方法被提出和广泛研究.然而,在压缩感知、信号处理与亚波长光学成像等实际问题中,响应变量和回归参数是二次关系.所以,本文引入二次度量回归(QMR)模型,研究了其高维情形下的变量选择问题,并建立了相应的优化理论与算法.第二章引入一致正则性概念,给出相应的判定条件,并且用于高维QMR模型的可辨识性研究.第三章针对高维QMR模型的lq(0 q 1)正则最小二乘问题,给出了相应估计的中偏差和弱Oracle性质,得到了解的存在性及其不动点理论.在此基础上,构造了不动点迭代算法,建立了其收敛性结果.最后,通过数值模拟表明该方法的有效性.第四章针对QMR模型的l0约束最小二乘问题,给出了解的存在性及不动点理论,进而构造了稀疏投影梯度算法,并得到该算法的收敛性.最后,通过数值模拟表明l0约束最小二乘方法的有效性.第五章针对高维QMR模型的特殊情形—线性模型,研究了加权l1正则分位数回归问题.使用交替方向乘子法提出了一种快速、有效算法,得到了算法的收敛性.利用该算法和局部线性近似技巧,还构造了一类非凸惩罚的分位数回归估计的计算方法.最后,数值实验表明该算法的有效性.
[Abstract]:In big data's time, high dimensional data were presented in the fields of genome and health science, economics and finance, astronomy and physics, signal processing and imaging. One of the common characteristics is the sparsity of predictive variables. Choosing the most relevant predictive variables is one of the main research contents of high dimensional data regression analysis, which has very important application value. Therefore, many statistical methods for linear regression hypothesis have been proposed and widely studied. However, in practical problems such as compression sensing, signal processing and subwavelength optical imaging, the response variables and regression parameters are quadratic. Therefore, in this paper, the quadratic metric regression (QMR) model is introduced to study the variable selection problem in the case of high dimension, and the corresponding optimization theory and algorithm are established. In chapter 2, the concept of uniform regularity is introduced, and the corresponding criteria are given, which are used to study the identifiability of high-dimensional QMR model. In chapter 3, for the lq (0Q 1) regular least squares problem of high dimensional QMR model, the intermediate deviation and weak Oracle properties of the corresponding estimates are given, and the existence of the solution and its fixed point theory are obtained. On this basis, the fixed point iterative algorithm is constructed and its convergence results are established. Finally, numerical simulation shows the effectiveness of the method. In chapter 4, the existence of solution and the fixed point theory are given for the l0-constrained least square problem of QMR model, and then the sparse projection gradient algorithm is constructed, and the convergence of the algorithm is obtained. Finally, numerical simulation shows the validity of the l 0 constrained least squares method. In chapter 5, the weighted L 1 regular quantile regression problem is studied for the special case of the high dimensional QMR model, the linear model. A fast and effective algorithm is proposed by using alternating direction multiplier method, and the convergence of the algorithm is obtained. By using this algorithm and the local linear approximation technique, a new method of quantile regression estimation for nonconvex penalty is also constructed. Finally, numerical experiments show the effectiveness of the algorithm.
【学位授予单位】：北京交通大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O212.1
本文编号：2342303
[Abstract]:In big data's time, high dimensional data were presented in the fields of genome and health science, economics and finance, astronomy and physics, signal processing and imaging. One of the common characteristics is the sparsity of predictive variables. Choosing the most relevant predictive variables is one of the main research contents of high dimensional data regression analysis, which has very important application value. Therefore, many statistical methods for linear regression hypothesis have been proposed and widely studied. However, in practical problems such as compression sensing, signal processing and subwavelength optical imaging, the response variables and regression parameters are quadratic. Therefore, in this paper, the quadratic metric regression (QMR) model is introduced to study the variable selection problem in the case of high dimension, and the corresponding optimization theory and algorithm are established. In chapter 2, the concept of uniform regularity is introduced, and the corresponding criteria are given, which are used to study the identifiability of high-dimensional QMR model. In chapter 3, for the lq (0Q 1) regular least squares problem of high dimensional QMR model, the intermediate deviation and weak Oracle properties of the corresponding estimates are given, and the existence of the solution and its fixed point theory are obtained. On this basis, the fixed point iterative algorithm is constructed and its convergence results are established. Finally, numerical simulation shows the effectiveness of the method. In chapter 4, the existence of solution and the fixed point theory are given for the l0-constrained least square problem of QMR model, and then the sparse projection gradient algorithm is constructed, and the convergence of the algorithm is obtained. Finally, numerical simulation shows the validity of the l 0 constrained least squares method. In chapter 5, the weighted L 1 regular quantile regression problem is studied for the special case of the high dimensional QMR model, the linear model. A fast and effective algorithm is proposed by using alternating direction multiplier method, and the convergence of the algorithm is obtained. By using this algorithm and the local linear approximation technique, a new method of quantile regression estimation for nonconvex penalty is also constructed. Finally, numerical experiments show the effectiveness of the algorithm.
【学位授予单位】：北京交通大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O212.1
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