基于方差分量估计的拟合推估及其在GIS误差纠正的应用

拟合推估解算必须首先求得信号向量的方差协方差矩阵,该协方差矩阵一般通过选定的协方差函数,并通过已测点数据进行拟合得到。显然观测噪声的先验方差协方差阵与拟合得到的随机信号的方差协方差矩阵必须相互协调,即观测噪声向量和信号向量的权矩阵所对应的方差因子应该一致,否则将对固定效应和随机效应参数的估计带来系统性的影响。应用方差分量估计来协调拟合推估模型中观测噪声和信号向量的随机模型,并分别从极大似然估计、MINQUE估计、赫尔默特方差分量估计三方面构建了拟合推估模型的方差分量解,最后利用新提出的理论与方法,对一幅实际的扫描地形图进行误差纠正,结果表明基于方差分量估计的拟合推估法能够提高扫描地形图的精度。     

顾及框架点坐标误差的三维基准转换严密模型

框架点坐标是由观测数据通过平差得到的,不可避免地受到观测误差的影响。针对原框架和目标框架坐标均存在误差、非公共点与公共点间存在相关性,以及转换系数矩阵中仅部分元素存在误差的实际情况,提出了同时考虑框架内误差以及转换点间相关性的基准转换严密模型,该模型将公共点和非公共点联合处理,同时计算坐标转换参数和所有点的坐标转换值,推导出了新的严格坐标转换公式,该公式为传统坐标转换公式基础上增加一改正量的形式;进一步,推导了原框架和目标框架坐标的方差不一致情况下的坐标转换模型的自适应解法;最后,利用"陆态网络工程"2000个区域站的实测坐标进行坐标转换验证,结果表明,这种严密模型较传统坐标转换模型具有更高的坐标转换精度。     

Calibration of geoid error models via a combined adjustment of ellipsoidal, orthometric and gravimetric geoid height data

The well-known statistical tool of variance component estimation (VCE) is implemented in the combined least-squares (LS) adjustment of heterogeneous height data (ellipsoidal, orthometric and geoid), for the purpose of calibrating geoid error models. This general treatment of the stochastic model offers the flexibility of estimating more than one variance and/or covariance component to improve the covariance information. Specifically, the iterative minimum norm quadratic unbiased estimation (I-MINQUE) and the iterative almost unbiased estimation (I-AUE) schemes are implemented in case studies with observed height data from Switzerland and parts of Canada. The effect of correlation among measurements of the same height type and the role of the systematic effects and datum inconsistencies in the combined adjustment of ellipsoidal, geoid and orthometric heights on the estimated variance components are investigated in detail. Results give valuable insight into the usefulness of the VCE approach for calibrating geoid error models and the challenges encountered when implementing such a scheme in practice. In all cases, the estimated variance component corresponding to the geoid height data was less than or equal to 1, indicating an overall downscaling of the initial covariance (CV) matrix was necessary. It was also shown that overly optimistic CV matrices are obtained when diagonal-only cofactor matrices are implemented in the stochastic model for the observations. Finally, the divergence of the VCE solution and/or the computation of negative variance components provide insight into the selected parametric model effectiveness.     

非线性模型中方差和协方差分量的估计

采用差分代替微分的方法,并将非线性模型的似然函数分解为函数模型生成的似然函数和正交补似然函数(也是边缘似然函数)的乘积,由正交补似然函数得到非线性模型中严格的和简化的方差和协方差分量估计的迭代公式.很多学者提出的线性模型中方差和协方差分量估计的迭代公式都是本文的特殊情况.     

方差分量估计理论及其在整体大地测量中的应用

本文探讨了整体大地测量平差中的方差分量估计问题,给出了自适应最小二乘配置;不仅较好地解决了各类观测量权比的确定问题,而且能够确定信号的方差因子。此外,文章还对如何更好地确定局部地区扰动位及其泛函的协方差函数作了讨论;最后通过对一个实测网进行了整体平差,并与经典平差结果进行了比较。     

Maximum likelihood estimate of variance components

Summary Using the orthogonal complement likehood function, an iterative procedure for the maximum likelihood estimates of the variance and covariance components is derived. It is shown that these estimates are identical with the reproducing estimates of the locally best invariant quadratic unbiased estimation of variance and covariance components. Successive approximations of the maximum likelihood estimates are given in addition.     

Combination of spatial networks using an estimated covariance matrix

When combining satellite and terrestrial networks, covariance matrices are used which have been estimated from previous data. It can be shown that the least-squares estimator of the unknown parameters using such an estimated covariance matrix is not necessarily the best. There are a number of cases where a more efficient estimator can be obtained in a different way. The problem occurs frequently in geodesy, since in least-squares adjustment of correlated observations estimated covariance matrices are often used. If the general structure of the covariance matrix is known, results can often be improved by a method called covariance adjustment. The statistical model used in least-squares collocation leads to a type of covariance matrix which fits into this framework. It is shown in which way improvements can be made using a modified approach of principal component analysis. As a numerical example the combination of a satellite and a terrestrial network has been computed with varying assumptions on the covariance matrix. It is shown which types of matrices are critical and where the usual least-squares approach can be applied without hesitation. Finally, a simplified representation of covariances for spatial networks by means of a suitable covariance function is suggested. Paper presented at the International Symposium on Computational Methods in Geometrical Geodesy-Oxford, 2–8 September, 1973.     

An alternative method for non-negative estimation of variance components

A typical problem of estimation principles of variance and covariance components is that they do not produce positive variances in general. This caveat is due, in particular, to a variety of reasons: (1) a badly chosen set of initial variance components, namely initial value problem (IVP), (2) low redundancy in functional model, (3) an improper stochastic model, and (4) data’s possibility of containing outliers. Accordingly, a lot of effort has been made in order to design non-negative estimates of variance components. However, the desires on non-negative and unbiased estimation can seldom be met simultaneously. Likewise, in order to search for a practical non-negative estimator, one has to give up the condition on unbiasedness, which implies that the estimator will be biased. On the other hand, unlike the variance components, the covariance components can be negative, so the methods for obtaining non-negative estimates of variance components are not applicable. This study presents an alternative method to non-negative estimation of variance components such that non-negativity of the variance components is automatically supported. The idea is based upon the use of the functions whose range is the set of all positive real numbers, namely positive-valued functions (PVFs), for unknown variance components in stochastic model instead of using variance components themselves. Using the PVF could eliminate the effect of IVP on the estimation process. This concept is reparameterized on the restricted maximum likelihood with no effect on the unbiasedness of the scheme. The numerical results show the successful estimation of non-negativity estimation of variance components (as positive values) as well as covariance components (as negative or positive values).     

最小二乘配置法在GPS高程异常推估中的应用

简述了最小二乘配置法的基本原理,并在水准联测点不多的情况下,直接采用平方根函数作为各随机参数间的协方差函数用于推估GPS高程异常值。由于最小二乘配置法的函数模型中同时考虑了非随机变量和随机变量,使得高程异常值的推估精度更高。实例分析也证明了此模型在精度上优于传统的平面拟合模型和协方差推估模型。残差分析表明此方法更适合于同时存在内插和外推高程异常值的情况。     

基于移动开窗法协方差估计和方差分量估计的自适应滤波

基于移动窗口协方差估计和方差分量估计，提出了一种新的自适应Kalman滤波技术。计算结果证实，该方法能有效地控制观测异常和载体状态扰动异常对动态系统参数估值的影响。     

方差—协方差分量极大似然估计的通用公式

本文由概括平差函数模型出发，按极大似然做估计原则导出了适用于所有平差函数模型的方差分量估计的通用公式，由Ｋ．Ｋｕｂｉｋ和Ｃ．Ｒ．Ｋｏｃｈ所导出的两个公式都是它的特例。     

Least squares adjustment and collocation

Summary For the estimation of parameters in linear models best linear unbiased estimates are derived in case the parameters are random variables. If their expected values are unknown, the well known formulas of least squares adjustment are obtained. If the expected values of the parameters are known, least squares collocation, prediction and filtering are derived. Hence in case of the determination of parameters, a least squares adjustment must precede a collocation because otherwise the collocation gives biased estimates. Since the collocation can be shown to be equivalent to a special case of the least squares adjustment, the variance of unit weight can be estimated for the collocation also. This estimate gives the scale factor for the covariance matrices being used in the collocation. In addition, the methods of testing hypotheses and establishing confidence intervals for the parameters of the least squares adjustment may be applied to the collocation.     

导航解算中的系统误差及其协方差矩阵拟合

利用Kslman滤波进行导航定位计算不得不涉及观测函数模型和动力学模型,而观测函数模型和动力学模型经常含有系统误差或区域性系统误差.本文提出了一种基于移动窗口的函数模型和随机模型系统误差自适应拟合法.基于相同的窗口给出了相应的观测向量和状态预测向量的协方差矩阵估计方法,其协方差矩阵的估计与现有的Sage滤波不同.利用经系统误差修正后的观测向量和状态预测向量及相应的协方差矩阵,再进行动态导航滤波计算,能有效提高导航解的精度.文中给出了开窗估计系统误差的公式,并利用实测数据验证了该算法的可行性和实用性.计算结果表明该算法能有效地抵制系统误差对导航滤波结果的影响.     

Estimability analysis of variance and covariance components

Although variance and covariance components have been extensively investigated and a number of elegant formulae to compute them have been derived, nothing is known, without any ambiguity, about their estimability in the case of a fully unknown variance–covariance matrix. We prove that variance and covariance components in this case are not estimable, thus clarifying the ambiguity of the literature on the topic and correcting some erroneous statements in the literature. We also give a new theorem on the estimability of a linear function of variance and covariance components. Then we propose a new method to estimate the variance–covariance matrix with special structure, which can presumably be represented by, at most, r(r + 1)/2 independent parameters to guarantee its estimability in such a subspace, by directly implementing the positive definiteness of the matrix as constraint to the restricted maximum likelihood method, where r is the number of redundant measurements. Therefore, our estimates of the variance and covariance components always reconstruct a positive definite matrix and are always physically meaningful.     

移动配置法在高程异常拟合中的应用

本文将最小二乘配置理论引入到高程异常移动曲面拟合模型中(移动配置法),采用格网内插法解决了配置理论中的难点—协方差函数确定的问题。通过实验,验证了移动配置法的优越性。     

组合导航抗差自适应卡尔曼滤波

针对组合导航系统的定位精度与稳定性要求不断提高的现状,该文引入一种观测噪声协方差与抗差自适应相结合的Kalman滤波算法。利用新息向量和移动窗口协方差分析法,动态自适应修正观测噪声协方差阵;通过分析基于状态不符值、方差分量的统计量构造的自适应因子所存在的问题,提出一种由预测残差向量构造的自适应因子。仿真结果表明,该方法能够有效抑制观测异常对组合导航定位精度的影响。     

论动态自适应滤波

动态导航与定位的质量取决于对动态载体扰动和观测异常扰动的认知和控制。本文首先介绍了目前广泛使用的Sage自适应滤波，讨论了自适应滤波的残差向量、新息向量及状态参数预报值残差向量的解析关系，以及它们之间的协方差矩阵之间的关系；分析了基于新息向量、残差向量和状态参数预报值残差向量的自适应协方差估计存在的问题。对新近发展起来的抗差滤波、Sage自适应滤波及抗差自适应滤波进行了综合比较与分析，结果表明抗差自适应滤波解算理论与方法除自适应地估计载体状态预报向量的协方差矩阵外，还能自适应地估计任意历元观测量的权。计算结果证实，抗差自适应滤波不仅计算简单，而且能有效地控制观测异常和载体状态扰动异常对动态系统参数估值的影响。     

Kriging方法结合最小二乘配置在GPS高程拟合中的应用

本文利用Kriging方法结合最小二乘配置将GPS高程转换成正常高.研究了将Kriging方法中的变异函数用于计算最小二乘配置中的协方差的方法,并对一局部GPS水准网的高程作了拟合计算.通过将最小二乘配置法与平面拟合模型和多面函数拟合模型等进行比较,其外符合精度从最大的±0.0277m提高到±0.0162m.     

Lq估计的渐近方差-协方差矩阵及其特点

针对由独立同分布误差膨胀而成的独立不等精度误差,根据未知参数的M估计的Bahadur型线性表达式,本文导出了由观测量、残差向量、参数估计量和观测量平差向量组成的基本向量的Bahadur型表达式.进一步地,根据方差传播定律导出了M估计的基本向量的渐近方差-协方差矩阵,该矩阵由3个多余参数决定,第三多余参数由本文定义.对Lq范估计,分别计算了误差分别为正态分布和q范分布时的3个多余参数,以及相应的基本向量的方差协方差矩阵.对最小二乘估计,残差向量与参数估计量和观测量的平差向量统计独立,相应的协方差矩阵为零,这一性质与误差分布无关.对正态分布的Lq估计,残差向量与参数估计量和观测量平差向量的协方差不为零;而对q范分布的Lq估计,即是相应的极大似然估计,残差向量与参数估计量和观测量平差向量的协方差为零.文中所得公式和结论可用于统计分析.     

基于随机模型基准的Helmert方差分量估计

在多类观测数据联合平差中,存在某类观测值的验前协方差阵正确而其他类不正确的情况,传统Helmert方差分量估计在求解此类问题时没有对正确的验前协方差阵加以区别,从而造成正确的随机模型经估计后同样也被调整。针对上述情况,首先分析了Helmert方差分量估计迭代收敛结果的实质,然后提出了随机模型基准的概念,并推导了基于随机模型基准的Helmert方差分量估计公式。经计算表明,新公式完全可行,可以用于解决实际问题。