奇异线性模型参数估计的相对效率

提出了奇异线性模型中参数β的最佳线性无偏估计(BLUE)相对于最小二乘估计(LSE)的一种新的相对效率,并给出了该相对效率的下界,最后讨论了该相对效率与广义相关系数的关系.     

单输入单输出大规模动力系统模型的简化方法

提出的简化单输入单输出大规模动力系统的一种新方法是系统在等式约束最小二乘法的一种推广.这种方法是一种投影方法,其投影依赖于奇异分解和Krylov子空间.通过平移算子,使得降阶模型与原模型的前r+i模准确地匹配,剩余的高阶模利用拉格朗日乘子法进行等式约束最小二乘的形式逼近原模.通过拉格朗日乘子法来求解具有约束条件的最小二乘问题,让推导出来的用于模型简化的投影变换矩阵更为简便.     

一种有偏估计与最小二乘估计的两种新的相对效率

考察了线性回归模型的回归系数的一类有偏估计,在均方误差矩阵准则下将其与最小二乘估计(LSE)进行比较,导出了这类有偏估计相对于LSE的两种新的相对效率的上、下界.     

基于对偶的不精确交替方向乘子法求解核范数正则化最小二乘问题

数据时代的所有事物都可以用数据描述记录.在数据分析中,对部分缺失数据补充,即矩阵补全问题.此类问题已有一定的研究,如通过求解核范数正则化最小二乘问题来达到所需效果.该文从对偶问题出发,使用交替方向乘子法(ADMM)来求解.在一定假设条件下,讨论了不精确对偶交替方向乘子法(dADMM)的全局收敛性.数值试验中,通过与原问题交替方向乘子法(pADMM)进行比较,验证了该算法的优越性.     

误差为球对称分布的非线性回归

本文讨论了误差为球对称分布的非线性回归模型,给出了最小二乘估计量(LSE)的随机展开式、偏差、方差和偏度,残差和拟合误差的偏差、方差,并讨论其近似置信域。     

广义G-M模型参数估计的一种相对效率

对于广义G—M模型，如果最小二乘估计（LSE）与最佳线性无偏估计（BLUE）相等，就可以用LSE代替BLUE反之，用LSE代替BLUE就要蒙受一些损失．有时，这种损失可能是很大的，因而研究这种损失的大小就显得颇为重要．本文提出了一种新的相对效率，并给出了该相对效率的上下界，最后讨论了该相对效率与广义相关系数的关系．     

随机效应线性模型中回归系数和参数的 MMLUE 的稳健性

直观地说,这里的稳健性是指统计推断关于线性模型即假设条件具有相对稳定性,这就是说,当模型假设发生某种微小变化时,相应的统计推断也只有微小改变.例如Zyskind 针对固定效应线性模型 Y=Xβ+ε,讨论了线性可估函数 c'β的最小二乘估计(LSE)关于协方差阵的稳健性.我们知道,在假设ε～N(0,(?)~2I)的情形下,c'β的最小均方线性无偏估计(MMLUE)与其 LSE 相同.但在实际中,我们不可能要求一个     

随机删失数据非线性回归模型的最小一乘估计

研究了随机删失数据非线性回归模型的最小一乘(LAD)估计问题, 证明了LAD估计量的渐近性质, 包括相合性、依概率有界性和渐近正态性等. 模拟结果显示对删失数据回归问题, LAD估计仍比最小二乘估计(LSE)稳健.     

一类线性约束矩阵不等式及其最小二乘问题

本文主要考虑一类线性矩阵不等式及其最小二乘问题,它等价于相应的矩阵不等式最小非负偏差问题.之前相关文献提出了求解该类最小非负偏差问题的迭代方法,但该方法在每步迭代过程中需要精确求解一个约束最小二乘子问题,因此对规模较大的问题,整个迭代过程需要耗费巨大的计算量.为了提高计算效率,本文在现有算法的基础上,提出了一类修正迭代方法.该方法在每步迭代过程中利用有限步的矩阵型LSQR方法求解一个低维矩阵Krylov子空间上的约束最小二乘子问题,降低了整个迭代所需的计算量.进一步运用投影定理以及相关的矩阵分析方法证明了该修正算法的收敛性,最后通过数值例子验证了本文的理论结果以及算法的有效性.     

协方差改进估计的优良性：计算机模拟结果

对于相依线性回归方程组成的系统.本文对它的回归系数的协方差改进估计(CIE)及其两步估计(TCIE)与最小二乘估计(LSE)进行了计算机模拟比较.模拟结果揭示了这种改进估计的估计的统计优良性.     

THE NESTED TOTAL LEAST SQUARES PROBLEM:FORMULATION,SOLUTIONS AND PERTURBATION ANALYSIS

In this paper, we propose the nested totoal least squatres problem (NTLS), which is an extension of the equality constrained least squares problem (LSE). The formulation of the NTLS problem is given and the solution set of the NTLS problem is obtained. The least squares residuals and the minimal norm correction matrices of the NTLS solution are provided and a perturbation analysis of the NTLS solutions is given.     

Zur Kondition des linearen Ausgleichsproblems mit linearen Gleichungen als Nebenbedingungen

Summary F.L. Bauer has treated in several papers [1, 3, 4] the condition related to the solution of linear equations and to the algebraic eigenvalue problem. We study the condition for the linear least squares problem with linear equality constraints (problem LSE). A perturbation theory of problem LSE is presented and three condition numbers are defined. Problem LSE includes the linear least squares problem (problem LS). There are examples with identical solution of problem LSE and of problem LS. Sometimes the condition of problem LSE is better and sometimes the condition of problem LS is better. Several numerical tests illustrate the theory.

Herrn Prof. Dr. Dr. F.L. Bauer zum 60. Geburtstag gewidmet     

Algebraic methods for equality constrained least squares problems in commutative quaternionic theory

This paper, by means of two matrix representations of a commutative quaternion matrix, studies the relationship between the solutions of commutative quaternion equality constrained least squares (LSE) problems and that of complex and real LSE problems and derives two algebraic methods for finding the solutions of equality constrained least squares problems in commutative quaternionic theory.     

On perturbations of some constrained subspaces

Perturbation bounds of subspaces, such as eigen-spaces, singular subspaces, and canonical subspaces, have been extensively studied in the literature. In this paper, we study perturbations of some constrained subspaces of 1×2, 2×1, and 2×2 block matrices, in which only one of the sub-matrices can be changed. Such problems rise from the least squares–total least squares problem, the constrained least squares problem, and the constrained total least squares problem.     

An algorithm for linear least squares problems with equality and nonnegativity constraints

We present a new algorithm for solving a linear least squares problem with linear constraints. These are equality constraint equations and nonnegativity constraints on selected variables. This problem, while appearing to be quite special, is the core problem arising in the solution of the general linearly constrained linear least squares problem. The reduction process of the general problem to the core problem can be done in many ways. We discuss three such techniques.The method employed for solving the core problem is based on combining the equality constraints with differentially weighted least squares equations to form an augmented least squares system. This weighted least squares system, which is equivalent to a penalty function method, is solved with nonnegativity constraints on selected variables.Three types of examples are presented that illustrate applications of the algorithm. The first is rank deficient, constrained least squares curve fitting. The second is concerned with solving linear systems of algebraic equations with Hilbert matrices and bounds on the variables. The third illustrates a constrained curve fitting problem with inconsistent inequality constraints.     

Accuracy and Stability of the Null Space Method for Solving the Equality Constrained Least Squares Problem

The null space method is a standard method for solving the linear least squares problem subject to equality constraints (the LSE problem). We show that three variants of the method, including one used in LAPACK that is based on the generalized QR factorization, are numerically stable. We derive two perturbation bounds for the LSE problem: one of standard form that is not attainable, and a bound that yields the condition number of the LSE problem to within a small constant factor. By combining the backward error analysis and perturbation bounds we derive an approximate forward error bound suitable for practical computation. Numerical experiments are given to illustrate the sharpness of this bound.     

A primal-dual interior-point algorithm for nonlinear least squares constrained problems

This paper extends prior work by the authors on solving nonlinear least squares unconstrained problems using a factorized quasi-Newton technique. With this aim we use a primal-dual interior-point algorithm for nonconvex nonlinear programming. The factorized quasi-Newton technique is now applied to the Hessian of the Lagrangian function for the transformed problem which is based on a logarithmic barrier formulation. We emphasize the importance of establishing and maintaining symmetric quasi-definiteness of the reduced KKT system. The algorithm then tries to choose a step size that reduces a merit function, and to select a penalty parameter that ensures descent directions along the iterative process. Computational results are included for a variety of least squares constrained problems and preliminary numerical testing indicates that the algorithm is robust and efficient in practice.     

Exploiting additional structure in equality constrained optimization by structured SQP secant algorithms

In 1988, Tapia (Ref. 1) developed and analyzed SQP secant methods in equality constrained optimization taking explicitly the additive structure of the problem setting into account. In this paper, we extend Tapia's augmented scale Lagrangian secant method to the case where additional structure coming from the objective function is available. Using the example of nonlinear least squares with equality constraints, we demonstrate these ideas and develop a convergence theory proving local and q-superlinear convergence for this kind of structured SQP-algorithms.This research was supported by the Studienstiftung des Deutschen Volkes.     

等式约束加权线性最小二乘问题的解法

1 引言 在实际应用中常会提出解等式约束加权线性最小二乘问题 min||b-Ax||_M,(1.1) x∈C~n s.t.Bx=d, 其中B∈C~(p×n),A∈C~(q×n),d∈C~p,b∈C~q,M∈C~(q×q)为Hermite正定阵. 对于问题(1.1),目前已有多种解法,见文[1—3).本文将利用广义逆矩阵的知识,给出(1.1)的通解及迭代解法.本文中关于矩阵广义逆与投影算子(矩阵)的记号基本上与文[4]的相同.例如,A~+表示A的MP逆,P_L表示到子空间L上的正交投影算子,λ_(max)(MAY)表示矩阵M~(1/2)AY的最大特征值.我们还要用到广义BD逆的概念: 设A∈C~(n×n),L为C~n的子空间,则称A_(L)~(+)=P_L(AP_L+P_L⊥)~+为A关于L的广义BD逆.     

A simple and efficient algorithm for fused lasso signal approximator with convex loss function

We consider the augmented Lagrangian method (ALM) as a solver for the fused lasso signal approximator (FLSA) problem. The ALM is a dual method in which squares of the constraint functions are added as penalties to the Lagrangian. In order to apply this method to FLSA, two types of auxiliary variables are introduced to transform the original unconstrained minimization problem into a linearly constrained minimization problem. Each updating in this iterative algorithm consists of just a simple one-dimensional convex programming problem, with closed form solution in many cases. While the existing literature mostly focused on the quadratic loss function, our algorithm can be easily implemented for general convex loss. We also provide some convergence analysis of the algorithm. Finally, the method is illustrated with some simulation datasets.