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.

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.

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.

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.

基于WTLS的井筒变形监测算法及方案设计

针对基于最小二乘的常规井筒变形监测算法中,系数矩阵往往包含观测值误差影响,难以解算出井筒变形最优解,提出一种抗差加权整体最小二乘算法。该算法通过同时对待求参数和系数矩阵进行估计,克服常规最小二乘系数矩阵受观测值随机误差影响;同时结合IGGⅢ三段权函数抗差因子,调整各观测值权函数,消除观测值粗差对最终变形估计值的影响,得到最优的井筒变形值。最后分别采用模拟和实测井筒数据对算法进行了验证,结果表明:当观测值含有随机误差时,整体最小二乘算法优于最小二乘,但都无法抵御观测粗差影响;抗差加权整体最小二乘算法可以有效抵御观测粗差,修正观测随机误差影响,且精度优于抗差最小二乘;采用抗差加权整体最小二乘算法对实测井筒变形数据进行解算结果与实际相符,可以增强变形监测结果的可靠性,提高井筒变形量精度。

基于Rayleigh商的不等精度二次曲线拟合的WTLS迭代解法

针对独立不等精度离散点的二次曲线拟合问题,以系数矩阵元素的一阶误差传播得到的方差为权倒数,采用加权总体最小二乘估计方法求解拟合参数,将加权总体最小二乘问题转化为Rayleigh商问题,从而只需求一正定矩阵的特征值和特征向量,便可通过迭代计算得到待估参数的解。该方法性能稳定且计算量较小,是针对WTLS问题的一种相对简捷高效的计算方法。

一种基于PEIV模型的GNSS高程拟合解法

采用二次曲面函数拟合全球卫星导航系统(GNSS)高程时,系数矩阵中含有观测坐标,所以系数矩阵同样存在误差,故采用总体最小二乘拟合GNSS高程更加合理。但系数矩阵不同位置有相同的元素,为保证这些相同元素有相同的改正数,本文提出了基于部分变量含误差(PEIV)模型的GNSS高程拟合解法,该方法可以保证相同元素的改正数一致。最后通过模拟算例和实例算例分析发现,本文解法与加权总体最小二乘(WTLS)解算结果一致,验证了本文解法的可行性。

加权总体最小二乘法在三维框架转换中的应用

国际地球参考框架转换是小角度直角坐标转换,一般采用布尔莎七参数模型通过最小二乘方法求解。针对坐标转换中存在系数矩阵有常数元素和重复元素的情况,提出适用于ITRF转换的加权结构总体最小二乘方法。该方法考虑坐标先验精度,构造了顾及系数矩阵结构性的矩阵,确保重复元素具有相同的改正数,并使用拉格朗日乘数法迭代求解。实际数据解算转换参数结果显示,该算法的精度和准确度优于传统方法。

An iterative algorithm of NWTLS-EC for three dimensional-datum transformation with large rotation angle

The Gauss-Markov （GM） model and the Errors-in-Variables （EIV） model are frequently used to perform 3D coordinate transformations in geodesy and engineering surveys. In these applications, because the observation errors in original coordinates system are also taken into account, the latter is more accurate and reasonable than the former. Although the Weighted Total Least Squares （WTLS） technique has been intro- duced into coordinate transformations as the measured points are heteroscedastic and correlated, the Variance- Covariance Matrix （VCM） of observations is restricted by a particular structure, namely, only the correlations of each points are taken into account. Because the 3D datum transformation with large rotation angle is a non- linear problem, the WTLS is no longer suitable in this ease. In this contribution, we suggested the nonlinear WTLS adjustments with equality constraints （NWTLS-EC） for 3D datum transformation with large rotation an- gle, which removed the particular structure restriction on the VCM. The Least Squares adjustment with Equality （LSE） constraints is employed to solve NWTLS-EC as the nonlinear model has been linearized, and an iterative algorithm is proposed with the LSE solution. A simulation study of 3D datum transformation with large rotation angle is given to insight into the feasibility of our algorithm at last.

一种基于LS-WTLS的稳健平面拟合方法

为了解决平面数据点位精度差异性及平面模型常数项解算精度较低的问题,提出了一种基于最小二乘-加权总体最小二乘(LS-WTLS)的稳健平面拟合方法。该方法采用加权最小二乘模型与稳健估计IGGⅢ方案相结合的方式对平面模型误差项参量进行解算,然后通过设置阈值剔除粗差数据,利用最小二乘法对平面模型常数项进行解算,以此进一步提高了平面模型各参量的解算精度。结果表明,新方法相对于最小二乘(LS)法、总体最小二乘(TLS)法、LS-TLS法、IGGⅢ-LS-TLS法,其单位权中误差分别提高了53.6%,195.0%,47.5%和5.1%,平面拟合精度分别提高了49.4%,179.3%,48.7%和46.99%,表现出了良好的抗粗差干扰能力。该研究验证了新方法在平面拟合领域的优越性和可靠性。

基于WTLS的部分最小二乘粗差探测法

针对平差时EIV模型中所含粗差的问题,提出了EIV模型的粗差探测法,利用变形后的加权总体最小二乘法依次对假设不含粗差的一组观测向量进行平差计算,然后通过平差后所得到的单位权方差估计值的大小对观测向量进行分组,从而达到粗差定位的目的。通过算例分析比较,得出在观测值独立等精度时,本文的方法可以有效的发现观测数据所存在的粗差。

基于y≈A(x)ξ函数模型的WTLS方法在GPS高程拟合中的应用

在GPS高程拟合中,用一般的最小二乘原理来处理拟合数据,效果往往不太理想。将y≈Aξ的函数模型通过加权整体最小二乘平方差方法处理拟合数据,能够得到更好的估计参数,高程异常值拟合精度也相应提高。

部分待估参数具有先验随机性的WTLS平差

部分待估参数具有先验随机信息,且误差方程系数矩阵含有观测误差,是一类新的平差问题。本文构造了部分待估参数含有先验随机信息的加权整体最小二乘平差函数模型,推导该模型参数估计与精度评定公式,给出计算步骤,适用于一般情形。实例对比分析证明,该算法正确可靠,迭代收敛速度较优。

基于改进无迹卡尔曼滤波的锂电池荷电状态估计

锂电池是一种广泛应用的能源器件,高效的荷电状态估计是锂电池安全管理的基础。为了提高荷电状态估计的精度,提出了一种联合估计方法。首先,采用递归限制总体最小二乘法辨识模型参数,解决了传统递推最小二乘法存在辨识偏差导致准确性降低的问题;接着,提出了基于权重优化的无迹卡尔曼滤波算法,提高了荷电状态估计的精度;最后,引入北京公交动态压力测试工况的仿真实例对锂电池放电状态进行建模,并通过与3种先进方法进行比较,验证了所提方法在精度和收敛速度方面的优越性。

非均匀采样潜艇耐压壳体测量数据处理方法

在役潜艇的建造和修理需要检测其耐压壳体的截面圆度,因现场条件限制或多设备融合测量,测量采样点经常呈现非均匀性,导致最小二乘圆拟合的结果与实际情况不符,研究旨在减小非均匀采样带来的影响,提高拟合圆的精度。基于非均匀采样定权提出一种非均匀采样加权总体最小二乘圆拟合方法,同时考虑观测向量和系数矩阵都有扰动,用于潜艇耐压壳体测量数据的处理。通过对4种不同预设形状的采样分析数值实验表明,该方法能提高非均匀采样情况下圆拟合的精度,但当随机误差加大至1.5 mm,该方法不能完全优于最小二乘法。非均匀采样加权总体最小二乘圆拟合法计算得到的圆心偏差及半径偏差都比最小二乘法的结果更小,具有较好的工程实用价值,可用于在役潜艇耐压壳体圆度测量非均匀采样时的圆拟合计算。

Robust structured total least squares algorithm for passive location

A new approach called the robust structured total least squares(RSTLS) algorithm is described for solving location inaccuracy caused by outliers in the single-observer passive location. It is built within the weighted structured total least squares(WSTLS)framework and improved based on the robust estimation theory.Moreover, the improved Danish weight function is proposed according to the robust extremal function of the WSTLS, so that the new algorithm can detect outliers based on residuals and reduce the weights of outliers automatically. Finally, the inverse iteration method is discussed to deal with the RSTLS problem. Simulations show that when outliers appear, the result of the proposed algorithm is still accurate and robust, whereas that of the conventional algorithms is distorted seriously.

水质氨氮、水质总磷校准曲线加权问题探讨

探讨了水质氨氮、水质总磷校准曲线应用普通最小二乘法(ordinary least square,OLS)和加权最小二乘法(weighted least square,WLS)拟合后对测定结果的不同影响。基于分光光度法测定水质氨氮、水质总磷的检测方法,对一年内两项目分别累计的6条校准曲线进行异方差检验,在检验有异方差现象后,通过加权最小二乘法对曲线进行拟合,对比WLS和OLS两种曲线方程在同一吸光度下推算得值与标准样品值的相对误差绝对值|δ|高低。两项目OLS曲线|δ|值在大多数浓度点都低于WLS曲线|δ|值。在0<|δ|<1范围内两项目OLS曲线|δ|值个数明显较多,|δ|>1范围内两项目WLS曲线|δ|值个数明显较多。针对实验室水质氨氮、水质总磷两项目,即使校准曲线存在异方差,通过OLS拟合的曲线也能较准确地推算结果。

半变异函数模型参数估计的综合定权法

针对半变异函数参数估计时点对数法定权未能顾及滞后距的影响,可能无法反应滞后距较小几个点重要性的问题,对多项式回归法的权重进行优化。通过平行四边形法则综合点对数和滞后距的影响,给出一种综合定权法。以球面模型为例,通过不同定权方法估计半变异函数模型参数,实验结果表明综合定权法各点权重更合理,克里金估值也更接近实测值。

一种基于双影响因子的软基沉降时序模型

为进一步提高软基沉降预测精度,对利用钻探取样、单轴固结实验、渗透—三轴固结联合实验、经验公式建立的固结沉降时序模型以及通过稳健加权总体最小二乘算法拟合得到的区块沉降时序模型进行组合,得到一种以时间为自变量、基于固结沉降和区块沉降双影响因子的软基沉降时序模型。将模型应用于广州市南沙区某软基地面监测点沉降预测中,结果表明该模型较好地反演了软基沉降规律,预测误差指标均小于其他模型,预测准确性和可靠性最高。

加权总体最小二乘在三维激光标靶拟合中的应用

针对平面标靶和球标靶拟合中最小二乘方法和总体最小二乘方法存在的问题,提出了一种加权总体最小二乘的拟合方法,根据点云激光反射强度以及对系数阵A列向量部分修正引入权阵P、PX、P0,建立较LS方法和TLS方法更加合理的模型,并给出相应的迭代算法。经实例计算证明加权总体最小二乘方法更加合理,可以获得更高精度的参数解。