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.

Numerical Simulation of Oil-Water Two-Phase Flow in Low Permeability Tight Reservoirs Based on Weighted Least Squares Meshless Method

In response to the complex characteristics of actual low-permeability tight reservoirs,this study develops a meshless-based numerical simulation method for oil-water two-phase flow in these reservoirs,considering complex boundary shapes.Utilizing radial basis function point interpolation,the method approximates shape functions for unknown functions within the nodal influence domain.The shape functions constructed by the aforementioned meshless interpolation method haveδ-function properties,which facilitate the handling of essential aspects like the controlled bottom-hole flow pressure in horizontal wells.Moreover,the meshless method offers greater flexibility and freedom compared to grid cell discretization,making it simpler to discretize complex geometries.A variational principle for the flow control equation group is introduced using a weighted least squares meshless method,and the pressure distribution is solved implicitly.Example results demonstrate that the computational outcomes of the meshless point cloud model,which has a relatively small degree of freedom,are in close agreement with those of the Discrete Fracture Model(DFM)employing refined grid partitioning,with pressure calculation accuracy exceeding 98.2%.Compared to high-resolution grid-based computational methods,the meshless method can achieve a better balance between computational efficiency and accuracy.Additionally,the impact of fracture half-length on the productivity of horizontal wells is discussed.The results indicate that increasing the fracture half-length is an effective strategy for enhancing production from the perspective of cumulative oil production.

LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED FRACTIONAL BROWNIAN MOTION

In this article, we study a least squares estimator （LSE） of θ for the Ornstein- Uhlenbeck process X0=0,dXt=θXtdt＋dBt^ab, t ≥ 0 driven by weighted fractional Brownian motion B^a,b with parameters a, b. We obtain the consistency and the asymptotic distribution of the LSE based on the observation {Xs, s∈[0,t]} as t tends to infinity.

Constrained Weighted Least Squares Location Algorithm Using Received Signal Strength Measurements

Determine the location of a target has gained considerable interest over the past few years. The Received Signal Strength(RSS) measurements and Differential RSS(DRSS) measurements can be converted to distance or distance ratio estimates for constructing a set of linear equations. Based on these linear equations, a constrained weighted least Squares(CWLS) algorithm for target localization is derived. In addition, an iterative technique based on Newton's method is utilized to give a solution. The covariance and bias of the CWLS algorithm is derived using perturbation analysis. Simulation shows that the proposed estimator achieves better performance than existing algorithms with reasonable complexity.

Preconditioned iterative methods for solving weighted linear least squares problems

A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation （GAOR） methods, is proposed to solve linear systems based on a class of weighted linear least squares problems. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned iterative methods is better than that of the original methods. Furthermore, the effectiveness of the proposed methods is shown in the numerical experiment.

ERRATUM TO： LEAST SQUARES ESTIMATION FOR ORNSTEIN-UHLENBECK PROCESSES DRIVEN BY THE WEIGHTED FRACTIONAL BROWNIAN MOTION （ACTA MATHEMATICA SCIENTIA 2016,36B （2） ：394-408）

We give a correction of Theorem 2.2 of Shen, Yin and Yan （2016）.

Recursive weighted least squares estimation algorithm based on minimum model error principle

Kalman filter is commonly used in data filtering and parameters estimation of nonlinear system,such as projectile's trajectory estimation and control.While there is a drawback that the prior error covariance matrix and filter parameters are difficult to be determined,which may result in filtering divergence.As to the problem that the accuracy of state estimation for nonlinear ballistic model strongly depends on its mathematical model,we improve the weighted least squares method(WLSM)with minimum model error principle.Invariant embedding method is adopted to solve the cost function including the model error.With the knowledge of measurement data and measurement error covariance matrix,we use gradient descent algorithm to determine the weighting matrix of model error.The uncertainty and linearization error of model are recursively estimated by the proposed method,thus achieving an online filtering estimation of the observations.Simulation results indicate that the proposed recursive estimation algorithm is insensitive to initial conditions and of good robustness.

A new method of weighted choice in InSAR Least Squares unwrapping

The decorelation phenomena such as Low-SNR radar signal, shadows and layover caused by topography etc, causes phase data discontinuous and makes the result of unwrapping phase inaccurate or completely wrong. Based on the analysis of influencing factors to the weight selection, this paper develops a new method to choose the weights based on the measurement of confidence in frequency domain. Results show that it is more precise and robust than other methods, and can make up for the defect of sub-estimate to the slope of least squares method.

Prediction of earth rotation parameters based on improved weighted least squares and autoregressive model

In this paper, an improved weighted least squares （WLS）, together with autoregressive （AR） model, is proposed to improve prediction accuracy of earth rotation parameters（ERP）. Four weighting schemes are developed and the optimal power e for determination of the weight elements is studied. The results show that the improved WLS-AR model can improve the ERP prediction accuracy effectively, and for different prediction intervals of ERP, different weight scheme should be chosen.

THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

基于WLS-KF的UWB室内定位滤波算法研究

针对室内超宽带(UWB)定位过程中受到非视距误差(NLOS)干扰而导致定位精度下降的问题,提出了基于抗差估计原理的自适应卡尔曼滤波方法,结合加权最小二乘法对测距信息解算得到定位坐标。在通视环境下进行测距,利用测得的数据计算新息向量和协方差,并基于此构建阈值信息,对NLOS环境产生的量测异常值进行判别,在此基础上利用Sage-Husa滤波对系统噪声协方差进行估计。采用加权最小二乘法对测距信息进行处理,得到标签解算坐标的最优估计。通过MATLAB仿真验证算法的可行性和有效性并在室内环境下进行测距、定位试验验证。仿真和实验结果表明,基于抗差估计原理的自适应卡尔曼滤波方法,结合加权最小二乘法能有效识别NLOS误差,且对定位过程中发生的状态突变能有效进行跟踪,解算得到的标签坐标x方向误差1 cm左右,y方向误差2 cm左右,提高了UWB室内定位的精度。

Weighted Least-Squares for a Nearly Perfect Min-Max Fit

In this note, we experimentally demonstrate, on a variety of analytic and nonanalytic functions, the novel observation that if the least squares polynomial approximation is repeated as weight in a second, now weighted, least squares approximation, then this new, second, approximation is nearly perfect in the uniform sense, barely needing any further, say, Remez correction.

基于改进灰狼算法优化WLSSVM的短期风功率预测

为提高风功率短期预测的准确率,提出一种基于改进灰狼算法优化加权最小二乘支持向量机(Weighted Least Squares Support Vector Machine,WLSSVM)的短期风功率预测方法。采用C-C法对风功率时间序列的嵌入维数进行了计算,根据计算结果确定短期风速预测输入量与输出量的关系。利用Tent映射和参数非线性调整策略对灰狼算法进行改进,得到了优化性能更强的改进灰狼优化(Improved Grey Wolf Optimization,IGWO)算法,并利用测试函数验证了IGWO算法能够加快迭代收敛,提高计算精度。采用IGWO算法对WLSSVM的惩罚系数和核参数进行优化,建立基于IGWO-WLSSVM的短期风功率预测模型。采用某风电场春夏两个不同季节的风功率数据进行算例分析,结果表明,所提短期风功率预测结果的平均相对误差、均方根误差和最大相对误差更小,风功率预测精度和预测结果的稳定性均优于其他方法,验证了所提方法的有效性和实用性。

基于WLS-KF算法高精度通信辐射源定位方法

为提高通信辐射源时差定位精度,提出了一种加权最小二乘估计和卡尔曼滤波相结合的算法。首先利用加权最小二乘算法对目标位置进行估计,得到目标位置的粗略估计值,由于多径效应的影响和误差的存在,基站对目标信号的分辨干扰定位解算是有偏的,利用最小均方误差(LMS)算法抑制多径效应的影响;针对俯仰角误差,再通过卡尔曼滤波算法对俯仰角误差进行估计;利用原始测量值减去对定位误差的估计值,重新利用加权最小二乘算法对目标位置进行估计,加权最小二乘算法和卡尔曼滤波算法交互迭代的方法,时差—测向联合定位,提高目标定位精度;最后进行了定位精度分析,推导了定位精度的GDOP,验证了目标的定位精度。通过仿真验证,上述算法有效地降低了通信辐射源目标带来的计算误差,提高了定位精度;同时对目标定位精度的GDOP进行仿真验证。

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.

ADAPTIVE FUSION ALGORITHMS BASED ON WEIGHTED LEAST SQUARE METHOD

Weighted fusion algorithms, which can be applied in the area of multi-sensor data fusion, are advanced based on weighted least square method. A weighted fusion algorithm, in which the relationship between weight coefficients and measurement noise is established, is proposed by giving attention to the correlation of measurement noise. Then a simplified weighted fusion algorithm is deduced on the assumption that measurement noise is uncorrelated. In addition, an algorithm, which can adjust the weight coefficients in the simplified algorithm by making estimations of measurement noise from measurements, is presented. It is proved by emulation and experiment that the precision performance of the multi-sensor system based on these algorithms is better than that of the multi-sensor system based on other algorithms.

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.

A Note on Change Point Detection Using Weighted Least Square

This paper is concerned with the application of weighted least square method in change point analysis. Testing shift in the mean normal observations with time varying variances as well as of a GARCH time series are considered. For both cases, the weighted estimators are given and their asymptotic behaviors are studied. It is also described that how the resampling methods like Monte Carlo and bootstrap may be applied to compute the finite sample behavior of estimators.

Data point selection for weighted least square fitting of cavity decay time constant

For the accurate extraction of cavity decay time, a selection of data points is supplemented to the weighted least square method. We derive the expected precision, accuracy and computation cost of this improved method, and examine these performances by simulation. By comparing this method with the nonlinear least square fitting （NLSF） method and the linear regression of the sum （LRS） method in derivations and simulations, we find that this method can achieve the same or even better precision, comparable accuracy, and lower computation cost. We test this method by experimental decay signals. The results are in agreement with the ones obtained from the nonlinear least square fitting method.

A new method of weight choice in InSAR least squares unwrapping

The de-coherence phenomena such as Low-SNR radar signal, shadows and layover caused by topography, etc. , causing phase data discontinuity, makes the result of unwrapping phase inaccuracy or even completely wrong. Based on the analysis of influencing factors to weight choice, this thesis develops a new method to choose the weights based on the measure of the confidence in the frequency domain. Experiments show that it could overcome the defect of sub-estimate to the slope of least squares method very well, which has a better rationale, stability and performance.