不定最小二乘问题和近似因子模型的相关研究

发布时间：2018-05-02 06:22

本文选题：最小二乘问题 + 不定最小二乘问题　；参考：《重庆大学》2016年博士论文

【摘要】：不定最小二乘问题不仅可以作为一般最小二乘问题的一种推广形式,同时H?估计和控制,风险敏感估计和控制(risk-sensitive estiamtion and control),有限记忆自适应滤波(finite memory adaptive filtering)等控制论问题都可以统一的转化成不定最小二乘问题来研究.因此关于不定最小二乘问题展开研究是有价值和意义的.另外,我们还对因子模型作了一些讨论.由于严格因子模型不能很好的刻画变量之间的联系,而且在高维的情况下,现有的样本量不足以给出一个好的估计,这些因素促使关于近似因子模型的研究变得日益活跃起来.第二章是预备知识部分,我们首先给出一些基本定义,再在此基础上给出条件数的统一定义形式.另外,一些必要的引理和概念也在这里给出,同时我们还证明了一些文章中要使用的基本定理.在第三章和第四章中,我们讨论了不定最小二乘问题及其线性约束情况的条件数.基于数据空间上定义的乘积范数,我们首先给出其条件数的统一定义形式,然后在此新的框架下建立不定最小二乘问题条件数的精确表达式.作为其特殊情况,我们还给出了范数型,混合型和分量型条件数的具体表达式.在2-范数的情况下,我们给出了几种具有存储和计算优势的各类最小二乘问题条件数的等价形式.在第三章中,根据不定最小二乘问题与整体最小二乘问题之间的联系,我们从不定最小二乘问题的角度重新推导了整体最小二乘问题的条件数表达式.在第四章,我们给出了推导各类线性最小二乘问题及其约束情况的条件数表达式所使用的统一技巧.我们的方法不再借助于奇异值分解,不需要各种复杂的构造技巧,具有一定的普适性.同时,我们还研究了系数矩阵具有特殊结构的各类最小二乘问题,建立了相应的结构型条件数的表达式.针对大型矩阵精确计算条件数过于耗时的问题,我们提出了基于概率统计方法的几种估计条件数的算法.最后,我们通过随机数值实验来检验这几种算法估计条件数的效率,另外还对不同条件数之间的差异做了比较.第五章,我们研究了近似因子模型的估计问题.由于目前关于近似因子模型的惩罚似然估计方法不能保证误差协方差矩阵的正定性,而且现有文献中基于极大似然方法估计协方差矩阵通常是使用ADMM方法来保证协方差阵的正定性.但是ADMM方法需要选取惩罚参数,而惩罚参数又对实际计算中算法的收敛速度有着不可忽视的影响.为了克服这些困难,我们提出了一种新的算法,该算法能在保证误差协方差矩阵正定性的同时,还有着较快的收敛速度.通过数值模拟,我们发现保证误差协方差矩阵的正定性可以提高近似因子模型的估计效率和预测精度,而且根据协方差矩阵的分解公式,我们的算法还可以给出很好的协方差矩阵的估计.
[Abstract]:The indeterminate least squares problem can not only be used as a generalized form of the general least squares problem, but also can be used as a generalized form of the general least squares problem. Estimation and control, risk sensitive estimation and control, risk-sensitive estiamtion and control, finite memory adaptive filtering, finite memory adaptive filtering and so on, can be uniformly transformed into indefinite least squares problems to be studied. Therefore, it is valuable and meaningful to study the indefinite least squares problem. In addition, we also discuss the factor model. Because the strict factor model can not depict the relation between variables well, and in the case of high dimension, the existing sample size is not enough to give a good estimate, these factors make the research on approximate factor model become more and more active. In the second chapter, we give some basic definitions, and then give the unified definition form of conditional number. In addition, some necessary lemmas and concepts are also given here, and we also prove some basic theorems to be used in this paper. In chapter 3 and chapter 4, we discuss the indefinite least squares problem and the number of conditions for its linear constraints. Based on the product norm defined on the data space, we first give the unified definition form of the condition number, and then establish the exact expression of the condition number of the indefinite least squares problem under this new frame. As a special case, we also give the concrete expressions of the condition numbers of norm type, mixed type and component type. In the case of 2-norm, we give some equivalent forms of condition numbers for various least squares problems with the advantages of storage and computation. In the third chapter, according to the relation between the indeterminate least squares problem and the global least squares problem, we rederive the conditional number expression of the global least squares problem from the point of view of the indefinite least squares problem. In chapter 4, we give a unified technique to derive the conditional number expressions of all kinds of linear least squares problems and their constraints. Our method no longer depends on singular value decomposition, and does not require any complicated construction techniques, so it is universal. At the same time, we study all kinds of least squares problems with special structure of coefficient matrix, and establish the expression of the corresponding structural condition number. In order to solve the problem that the accurate calculation of condition number of large matrix is too time-consuming, we propose several algorithms for estimating condition number based on probability and statistics method. Finally, we use random numerical experiments to test the efficiency of these algorithms to estimate the number of conditions. In addition, we compare the differences between different conditional numbers. In chapter 5, we study the estimation of approximate factor model. Because the present penalty likelihood estimation method for approximate factor model can not guarantee the positive definiteness of error covariance matrix, Moreover, the ADMM method is usually used in the estimation of covariance matrix based on maximum likelihood method in the existing literature to ensure the positive definiteness of covariance matrix. However, the ADMM method needs to select the penalty parameters, and the penalty parameters have an important effect on the convergence rate of the algorithm in the actual calculation. In order to overcome these difficulties, we propose a new algorithm, which can guarantee the positive definiteness of the error covariance matrix and converge fast. Through numerical simulation, we find that ensuring the positive definiteness of error covariance matrix can improve the estimation efficiency and prediction accuracy of approximate factor model, and according to the decomposition formula of covariance matrix, Our algorithm can also give a good estimate of the covariance matrix.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.5

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