Algorithms and statistical analysis for linear structured weighted total least squares problem

Weighted total least squares(WTLS)have been regarded as the standard tool for the errors-in-variables(EIV)model in which all the elements in the observation vector and the coefficient matrix are contaminated with random errors.However,in many geodetic applications,some elements are error-free and some random observations appear repeatedly in different positions in the augmented coefficient matrix.It is called the linear structured EIV(LSEIV)model.Two kinds of methods are proposed for the LSEIV model from functional and stochastic modifications.On the one hand,the functional part of the LSEIV model is modified into the errors-in-observations(EIO)model.On the other hand,the stochastic model is modified by applying the Moore-Penrose inverse of the cofactor matrix.The algorithms are derived through the Lagrange multipliers method and linear approximation.The estimation principles and iterative formula of the parameters are proven to be consistent.The first-order approximate variance-covariance matrix(VCM)of the parameters is also derived.A numerical example is given to compare the performances of our proposed three algorithms with the STLS approach.Afterwards,the least squares(LS),total least squares(TLS)and linear structured weighted total least squares(LSWTLS)solutions are compared and the accuracy evaluation formula is proven to be feasible and effective.Finally,the LSWTLS is applied to the field of deformation analysis,which yields a better result than the traditional LS and TLS estimations.

An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

Properties of the total least squares estimation

Through theoretical derivation, some properties of the total least squares estimation are found. The total least squares estimation is the linear transformation of the least squares estimation, and the total least squares estimation is unbiased. The condition number of the total least squares estimation is greater than the least squares estimation, so the total least squares estimation is easier to be affected by the data error than the least squares estimation. Then through the further derivation, the relationships of solutions, residuals and unit weight variance estimations between the total least squares and the least squares are given.

ON THE ACCURACY OF THE LEAST SQUARES AND THE TOTAL LEAST SQUARES METHODS

Consider solving an overdetermined system of linear algebraic equations by both the least squares method (LS) and the total least squares method (TLS). Extensive published computational evidence shows that when the original system is consistent. one often obtains more accurate solutions by using the TLS method rather than the LS method. These numerical observations contrast with existing analytic perturbation theories for the LS and TLS methods which show that the upper bounds for the LS solution are always smaller than the corresponding upper bounds for the TLS solutions. In this paper we derive a new upper bound for the TLS solution and indicate when the TLS method can be more accurate than the LS method.Many applied problems in signal processing lead to overdetermined systems of linear equations where the matrix and right hand side are determined by the experimental observations (usually in the form of a lime series). It often happens that as the number of columns of the matrix becomes larger, the

Penalized total least squares method for dealing with systematic errors in partial EIV model and its precision estimation

When the total least squares(TLS)solution is used to solve the parameters in the errors-in-variables(EIV)model,the obtained parameter estimations will be unreliable in the observations containing systematic errors.To solve this problem,we propose to add the nonparametric part(systematic errors)to the partial EIV model,and build the partial EIV model to weaken the influence of systematic errors.Then,having rewritten the model as a nonlinear model,we derive the formula of parameter estimations based on the penalized total least squares criterion.Furthermore,based on the second-order approximation method of precision estimation,we derive the second-order bias and covariance of parameter estimations and calculate the mean square error(MSE).Aiming at the selection of the smoothing factor,we propose to use the U curve method.The experiments show that the proposed method can mitigate the influence of systematic errors to a certain extent compared with the traditional method and get more reliable parameter estimations and its precision information,which validates the feasibility and effectiveness of the proposed method.

A Quadratic Constraint Total Least-squares Algorithm for Hyperbolic Location

A novel algorithm for source location by utilizing the time difference of arrival (TDOA) measurements of a signal received at spatially separated sensors is proposed. The algorithm is based on quadratic constraint total least-squares (QC-TLS) method and gives an explicit solution. The total least-squares method is a generalized data fitting method that is appropriate for cases when the system model contains error or is not known exactly, and quadratic constraint, which could be realized via Lagrange multipliers technique, could constrain the solution to the location equations to improve location accuracy. Comparisons of performance with ordinary least-squares are made, and Monte Carlo simulations are performed. Simulation results indicate that the proposed algorithm has high location accuracy and achieves accuracy close to the Cramer-Rao lower bound (CRLB) near the small TDOA measurement error region.

Total least-squares EIO model,algorithms and applications

A functional model named EIO(Errors-In-Observations) is proposed for general TLS(total least-squares)adjustment. The EIO model only considers the correction of the observation vector, but doesn't consider to correct all elements in the design matrix as the EIV(Errors-In-Variables) model does, furthermore, the dimension of cofactor matrix is much smaller. Iterative algorithms for the parameter estimation and their precise covariance matrix are derived rigorously, and the computation steps are also presented. The proposed approach considers the correction of the observations in the coefficient matrix, and ensures their agreements in every matrix elements. Parameters and corrections can be solved at the same time.An approximate solution and a precise solution of the covariance matrix can be achieved by corresponding algorithms. Applications of EIO model and the proposed algorithms are demonstrated with several examples. The results and comparative studies show that the proposed EIO model and algorithms are feasible and reliable for general adjustment problems.

Linear-regression models and algorithms based on the Total-Least-Squares principle

In classical regression analysis, the error of independent variable is usually not taken into account in regression analysis. This paper presents two solution methods for the case that both the independent and the dependent variables have errors. These methods are derived from the condition-adjustment and indirect-adjustment models based on the Total-Least-Squares principle. The equivalence of these two methods is also proven in theory.

Perturbation Analysis for the Matrix-Scaled Total Least Squares Problem

In this paper, we extend matrix scaled total least squares (MSTLS) problem with a single right-hand side to the case of multiple right-hand sides. Firstly, under some mild conditions, this paper gives an explicit expression of the minimum norm solution of MSTLS problem with multiple right-hand sides. Then, we present the Kronecker-product-based formulae for the normwise, mixed and componentwise condition numbers of the MSTLS problem. For easy estimation, we also exhibit Kronecker-product-free upper bounds for these condition numbers. All these results can reduce to those of the total least squares (TLS) problem which were given by Zheng et al. Finally, two numerical experiments are performed to illustrate our results.

An Improved Algorithm of Grounding Grids Corrosion Diagnosis Based on Total Least Square Method

A new model considering corrosion property for grounding grids diagnosis is proposed,which provides reference solutions of ambiguous branches.The constraint total least square method based on singular value decomposition is adopted to improve the effectiveness of grounding grids' diagnosis algorithm.The improvement can weaken the influence of the model's error,which results from the differences between design paper and actual grid.Its influence on touch and step voltages caused by the interior resistance of conductors is taken into account.Simulation results show the validity of this approach.

Application of Total Least Square Algorithm in Static Decoupling of NGMIMU

The linear coupling of Non-gyro Micro Inertial Measurement Unit (NGMIMU) is akind of system error that affects the accuracy of measurement seriously. In this article, theauthor puts forward a new linear decoupling algorithm which simultaneously considers the error ofstandard input signal and output of accelerators when the coupling parameters are calculated. TheTotal Least Square (TLS) solutions of coupling parameters own the minimum characteristic to theinput and output values. Then these parameters are used to reconstruct the outputs of acceleratorsso as to realize the decoupling. The emulation result show that the ratio of decoupling error isless than 8 percent and verify the feasibility of this algorithm.

Novel passive localization algorithm based on weighted restricted total least square

A novel multi-observer passive localization algorithm based on the weighted restricted total least square （WRTLS） is proposed to solve the bearings-only localization problem in the presence of observer position errors. Firstly, the unknown matrix perturbation information is utilized to form the WRTLS problem. Then, the corresponding constrained optimization problem is transformed into an unconstrained one, which is a generalized Rayleigh quotient minimization problem. Thus, the solution can be got through the generalized eigenvalue decomposition and requires no initial state guess process. Simulation results indicate that the proposed algorithm can approach the Cramer-Rao lower bound （CRLB）, and the localization solution is asymptotically unbiased.

基于总体最小二乘-旋转不变算法的地表核磁共振信号参数估计

在地表核磁共振(SNMR)找水系统中,根据SNMR信号的参数能够预估地下含水层的储水量、导电性以及孔隙结构等信息。然而在实际应用中探测现场采集的SNMR信号十分微弱,易受到环境噪声干扰,导致无法直接获取SNMR信号的参数。针对这一问题,该文提出基于总体最小二乘-旋转不变法(TLS-ESPRIT)的地表核磁共振信号参数估计方法。基于谐波噪声与SNMR信号的相似信号特征构成一个由多个正弦衰减信号叠加的混合信号模型,使用TLS-ESPRIT将混合信号参数提取问题转换为旋转不变矩阵的广义特征值求解,从而获得SNMR信号的拉莫尔频率和弛豫时间,并结合最小二乘法求得其初始振幅和相位。仿真信号和实测信号实验结果表明此方法能够估计出混有随机噪声和工频谐波噪声的SNMR信号的参数,相比传统的谐波建模方法,在参数提取精度上效果更好。

基于位姿参数估计的多视角点云配准方法

传统的点云配准算法通过两点云数据之间的特征实现对应点配对,这种方法要求点云具有明确的特征,且存在计算量大、匹配时间长、配准精度低等问题,而ICP算法虽然应用广泛,但对初始值敏感。对此,提出了一种基于位姿参数估计的多视角点云配准方法(PPE-ICP)。首先通过分析误差的分布特性可证明误差极小值存在,使用A∗搜索算法寻找误差极小值,降低误差传播的影响,为后续的参数估计提供较好的初值;其次将总体最小二乘估计引入点云配准,在不依赖点云数据的同时,使用少量参考点就能获得点云从目标坐标系到东北天坐标系的转换矩阵,完成点云位姿矫正,结合迭代最近点算法(ICP),实现点云精确配准。通过与FGR-ICP、FPFH-ICP、NDT-ICP、RANSAC-TrICP和KSS-ICP这5种方法在公开数据集和自制实验装置收集到的点云上进行对比实验,点云数据量为20000点时实现配准只需6.55 s,极大地降低了大数据量下点云配准的时间成本,在实地点云配准中平移误差最大不超过0.03 m,旋转误差控制在0.07°。实验结果表明,PPE-ICP对相似变换、残缺点云和低重复率具有较强的鲁棒性,在多视角点云配准中具有较高的配准效率和配准精度。

基于转向架航向角的既有铁路平面线形识别方法研究

轨道平面线形识别是解算轨道不平顺的基础。针对外挂式动态轨道检查仪的实际需要,提出一种基于转向架航向角的铁路平面线形识别方法。首先,通过分段试拟合判定准则自动识别各测点所属线元类型和特征点数量;然后,建立基于转向架航向角的平面线形特征点优化模型;最后,结合云模型改进传统遗传算法的搜索策略提出云遗传算法,实现铁路平面线形精确识别。研究表明,平面线路曲线特征点识别偏差在4 m以内,线形识别准确度均达到97%以上,满足外挂式动态轨道检查仪对轨道不平顺解算的精度要求。

基于TLS的改进子空间投影算法

针对经典MUSIC算法在信源相干、低信噪比、小快拍数等非理想环境下性能失效的问题,提出了一种改进的基于TLS的加权子空间投影算法。首先对阵列接收的数据协方差矩阵进行重构处理,以达到解相干目的;其次充分利用子空间信息,基于总体最小二乘拟合方法对特征值进行拟合修正,基于修正MUSIC算法思想,利用校正后的噪声特征值和信号特征值分别对噪声子空间和信号子空间进行加权处理,得到改进后的噪声子空间和信号子空间,并将两者结合得到新的空间谱函数;最后进行谱峰搜索,完成信号源的波达方向估计。仿真结果表明,改进后的算法既适用于相干信号环境,在低信噪比、小快拍数及信号入射角度间隔较小的情况下,又能有效估计出信源的波达方向。

预应力混凝土梁桥短线匹配法的抗差估计方法

针对预应力混凝土梁桥短线匹配法线形控制问题,充分考虑节段梁控制点测量值中可能出现的随机误差与粗差,提出一种附有条件限制的总体最小二乘抗差估计方法,适用于短线匹配法线形控制;推导了解算方法,并给出了具体的解算步骤;以黄茅海跨海通道工程6×60 m连续刚构中的某一跨为基准进行了数据模拟。结果表明:该方法可以实时地修正施工中各类误差导致的预制线形变化,有效地阻止了施工误差与测量误差在后续节段梁中的不断积累;与既有算法相比,所提算法的计算结果准确度最高,不仅在终端节段末端与基准线形的差值均值最小,而且在代表整体线形连续性的偏差平方和均值上与最小二乘法的结果相近,优于传统的选点控制方法。

一种多频多系统周跳探测与修复方法

为满足多频多系统精密单点定位(PPP)数据预处理的实际需求,提出了一种周跳探测与修复方法。针对多频多系统PPP数据处理中各卫星的信号数量不同,利用最小二乘模糊度降相关平差(LAMBDA)原理构建模糊度组合。通过仅使用相位观测值估计的倾斜电子总量(STEC)探测不敏感周跳和漏判周跳。对于周跳修复,将伪距和载波相位的观测值在历元间作差求得周跳的浮点解和协方差阵,然后应用LAMBDA算法搜索周跳整数解并进行修复。通过静态、动态以及磁暴环境下周跳探测和修复实验,验证了所提方法的有效性,多频条件下周跳修复的正确率达到100%。

基于车载式激光雷达的隧道变形动态巡检技术研究

[目的]针对我国隧道变形监测手段实时性差、成本高、效率低的现状,须有效提升隧道变形监测效率,对此提出基于车载式激光雷达的隧道变形监控量测方法。[方法]利用激光雷达传感器扫描收集隧道全断面三维数据,通过整体最小二乘法对隧道点云数据进行平差及曲面拟合,根据拟合的隧道断面椭圆参数进行形变分析,实现隧道变形情况的快速动态巡检。[结果及结论]试验结果表明,该方法能够快速准确地获取隧道全断面数据,结合整体最小二乘法对数据进行处理分析可有效获取隧道变形情况,基本实现在不影响隧道正常交通的情况下有效完成隧道变形动态实时监测。

求解自回归模型参数的最小二乘法

在自回归模型参数的解算中,针对增广矩阵中不同位置上同一观测值有不同的改正数这一问题,本文提出了AR模型参数估计的最小二乘法。首先对AR模型进行等价变换,重新组成新的函数模型,两次运用最小二乘方法分别求解误差矩阵与未知参数,最后给出了参数协因数阵的一阶近似估计公式。算例结果表明,本文方法在求解AR模型参数时具有可行性。采用本文方法,将总体最小二乘问题的求解转换为最小二乘问题,计算简单且易于编程实现。同时,该方法有效地减小了计算量,从而降低了求解的复杂程度。