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.     

Generalized maximum-likelihood estimation of variance–covariance components with non-informative prior

 A new estimator of variance–covariance components is presented. The proposed estimator is derived by applying the principle of maximum-likelihood estimation to the posterior probability density function for the case when no prior information is available. Received: 9 August 1999 / Accepted: 10 November 2000     

协方差阵非负约束的赫尔默特方差分量估计

在多源数据联合反演同震滑动分布过程中,各类观测数据之间相对权比的确定是当今大地测量与地球物理联合反演领域的研究热点和难点。近年来,赫尔默特方差分量估计法因其可以同时估计虚拟观测数据及实际观测数据之间的比例因子的优势,在联合反演滑动分布领域中得到了广泛应用。然而,利用HVCE法确定多源数据联合反演相对权比过程中易出现负方差现象。针对以上问题,本文提出在HVCE法确定联合反演相对权比过程中添加线性不等式约束,利用带有线性不等式约束的平差模型进行求解（LC-HVCE）,解决计算过程中出现负方差问题。为了验证LC-HVCE法的有效性,开展模拟地震试验,结果表明LC-HVCE法可以有效地解决HVCE法在确定联合反演相对权比过程中存在的不足,保证了同震滑动分布反演精度。蒙特卡洛模拟试验及拉奎拉实际地震反演结果进一步表明了LC-HVCE法在确定正则化参数与相对权比过程中较HVCE法的适用性与稳定性。     

Rigorous transformation of variance–covariance matrices of GPS-derived coordinates and velocities

With the advances in the field of GPS positioning and the global densification of permanent GPS tracking stations, it is now possible to determine at the highest level of accuracy the transformation parameters connecting various international terrestrial reference frame (ITRF) realizations. As a by-product of these refinements, not only the seven usual parameters of the similarity transformations between frames are available, but also their rates, all given at some epoch t _k . This paper introduces rigorous matrix equations to estimate variance–covariance matrices for transformed coordinates at any epoch t based on a stochastic model that takes into consideration all a priori information of the parameters involved at epoch t _k , and the coordinates and velocities at the reference frame initial epoch t ₀. The results of this investigation suggest that in order to attain maximum accuracy, the agencies determining the 14-parameter transformations between reference frames should also publish their full variance–covariance matrix. Electronic Publication     

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.     

偏差改正的Partial EIV模型方差分量估计

针对Partial EIV模型的方差分量估计中未考虑参数估值偏差所带来的影响,将Partial EIV模型视为非线性函数得到参数估值的偏差及二阶近似协方差表达式,计算得到偏差改正后的参数估值,结合方差分量估计方法,更新由参数估值影响的矩阵变量,给出了基于偏差改正的方差分量估计迭代方法。试验结果表明,参数估值及其协方差主要受参数估值偏差大小的影响,加入偏差改正能够得到更加合理的参数估值及方差分量估值,偏差改正后的方差分量估值可更加合理地评估参数估值的精度信息。     

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).     

方差分量估计前提初探

根据方差分量估计理论,即使随机模型本身已经正确,方差分量估计也会得到不同于通常意义上的最优线性无偏最小二乘估计。此外,由于方差分量估计计算工作量一般较大,因此,本文提出了利用统计检验方法来判断是否进行方差分量估计的想法,并进行了初步研究。     

GPS向量网的抗差方差分量估计

GPS向量网的整体平差中，基线向量间的精度往往存在评定标准不一的问题，如各同步区采用不同的基线解算软件，各同步区观测环境不同等。这类问题可能会造成权比的失调，并因此影响最后的平差结果。另外，由于GPS观测量的粗差出现率也较常规大地控制网的粗差出现率高，若不采用有效措施，也会影响最后的平差结果，本文针对上述情况，讨论了抗差方差分量估计，并给出了算例。     

Regularization of geopotential determination from satellite data by variance components

 Different types of present or future satellite data have to be combined by applying appropriate weighting for the determination of the gravity field of the Earth, for instance GPS observations for CHAMP with satellite to satellite tracking for the coming mission GRACE as well as gradiometer measurements for GOCE. In addition, the estimate of the geopotential has to be smoothed or regularized because of the inversion problem. It is proposed to solve these two tasks by Bayesian inference on variance components. The estimates of the variance components are computed by a stochastic estimator of the traces of matrices connected with the inverse of the matrix of normal equations, thus leading to a new method for determining variance components for large linear systems. The posterior density function for the variance components, weighting factors and regularization parameters are given in order to compute the confidence intervals for these quantities. Test computations with simulated gradiometer observations for GOCE and satellite to satellite tracking for GRACE show the validity of the approach. Received: 5 June 2001 / Accepted: 28 November 2001     

Generalized maximum-likelihood estimation of variance components with inverted gamma prior

A new method for the estimation of variance components is presented. The proposed method combines the concept of maximum-likelihood estimation with the Bayesian approach and facilitates computationally efficient introduction of prior information into the estimation process. Received: 22 April 1998 / Accepted: 19 April 1999     

方差分量估计迭代算法收敛性分析与改进

本文首先对方差分量估计中的收敛性问题进行分析,指出方差分量的收敛性和估计方程的摄动性有关。提出了改进收敛性的思想,在此基础上提出了一种改进收敛的方法,用算例证明了所提出方法的有效性。     

Estimation and modelling of the local empirical covariance function using gravity and satellite altimeter data

The estimation of a local empirical covariance function from a set of observations was done in the Faeroe Islands region. Gravity and adjusted Seasat altimeter data relative to theGPM2 spherical harmonic approximation were selected holding one value in celles of1/8°×1/4° covering the area. In order to center the observations they were transformed into a locally best fitting reference system having a semimajor axis1.8 m smaller than the one ofGRS80. The variance of the data then was273 mgal ² and0.12 m ² respectively. In the calculations both the space domain method and the frequency domain method were used. Using the space domain method the auto-covariances for gravity anomalies and geoid heights and the cross-covariances between the quantities were estimated. Furthermore an empirical error estimate was derived. Using the frequency domain method the auto-covariances of gridded gravity anomalies was estimated. The gridding procedure was found to have a considerable smoothing effect, but a deconvolution made the results of the two methods to agree. The local covariance function model was represented by a Tscherning/Rapp degree-variance model,A/((i−1)(i−2)(i+24))(R _B /R _E )²ⁱ⁺², and the error degree-variances related to the potential coefficient setGPM2. This covariance function was adjusted to fit the empirical values using an iterative least squares inversion procedure adjusting the factor A, the depth to the Bjerhammar sphere(R _E −R _B ), and a scale factor associated with the error degree-variances. Three different combinations of the empirical covariance values were used. The scale factor was not well determined from the gravity anomaly covariance values, and the depth to the Bjerhammar sphere was not well determined from geoid height covariance values only. A combination of the two types of auto-covariance values resulted in a well determined model.     

Helmert方差分量估计严密公式与简化公式等价性的证明

在分析了Helmert方差分量估计严密公式与简化公式迭代收敛结果一些统计性质的基础上,导出了关于收敛结果共同的求解方程,从理论上证明了两种结果的等价性,为使用简化公式提供了依据。     

Comprehensive study on the estimation of the variance components of traverse nets

This paper advances a new simplified formula for estimating variance components,sums up the basic law to calculate the weights of observed values and a circulation method using the increaments of weights when estimating the variance components of traverse nets,advances the charicteristic roots method to estimate the variance components of traveres nets and presents a practical method to make two real and symmetric matrices two diagonal ones.     

Simplified formulae for the BIQUE estimation of variance components in disjunctive observation groups

 General rigorous and simplified formulae are reported for the best invariant quadratic unbiased estimates of the variance–covariance components, which can be applied to all least-squares adjustments with the general linear stochastic model. Simplified procedures are given for two cases frequently recurring in geodetic applications: uncorrelated groups of correlated or uncorrelated observations, with more than one variance component in each group. Received: 19 November 1998 / Accepted: 21 March 2000     

An alternative expression for the variance factors in using Iterated Almost Unbiased Estimation

An alternative expression has been derived for the variance factors when estimating weights according to the Iterated Almost Unbiased Estimation (IAUE) technique. A variance factor can be approximately estimated by finding the ratio of the group redundancy numbers at any two successive iterations. The numerical example of the first-order leveling network of Taiwan indicates: that stabilization of the redundancy number of an observation occurs as the variance factor associated with it converges to unity; that variance factors tend to approach unity for individual groups with large redundancy; and that for those variance factors which fail to converge to unity, the redundancy numbers decrease monotonically as the iteration proceeds. Received: 16 April 1996 / Accepted: 9 June 1998     

Finite covariance functions

Because of the full covariance matrices and the computer storage limitations the number of measurements which can be handled by the collocation method simultaneously, is limited. This paper presents a method to compute covariance functions with a finite support yielding sparse covariance matrices. The theoretical background is pointed out and, for the one- and two-dimensional case, special functions are developed which can be combined with the usually used covariance functions to get a “finite covariance function”. Simulated examples to demonstrate the behaviour of different solution methods to solve these special, sparse covariance matrices supplement our investigations.