Robust Kalman filter for rank deficient observation models

A robust Kalman filter is derived for rank deficient observation models. The datum for the Kalman filter is introduced at the zero epoch by the choice of a generalized inverse. The robust filter is obtained by Bayesian statistics and by applying a robust M-estimate. Outliers are not only looked for in the observations but also in the updated parameters. The ability of the robust Kalman filter to detect outliers is demonstrated by an example. Received: 8 November 1996 / Accepted: 11 February 1998     

抗差加权非等时距GM(1,1)模型在大型建筑物沉降预测中的应用

针对沉降观测中不等间隔和观测值可能含有粗差的情况,提出抗差加权非等时距GM(1,1)模型,实验表明,当观测值中不含粗差时,抗差加权非等时距GM(1,1)模型和常规GM(1,1)模型一样具有良好的预测效果;当观测值中含有粗差时,具备良好抵抗粗差的能力,具有非常好的预测效果。     

Robust estimator for correlated observations based on bifactor equivalent weights

 A new robust parameter estimator for the adjustment of correlated observations is developed based on a `bifactor reduction' model of weight elements. A shrinking factor for weight elements is proposed. The new equivalent weight matrix composed by the bifactor weight elements preserves the symmetry and keeps the original correlation coefficients unchanged. The new parameter estimator with its error influence function is derived. The robustness and efficiency of the new robust estimator is demonstrated with a simulated example and some conclusions are drawn. Received: 5 March 2001 / Accepted: 17 January 2002     

相关稳健估计及其在测量数据

在对现有的基于等价权的稳健估计的弊端进行分析的基础上,直接从方差膨胀模型出发,建立起一套基于调整观测量方差-协方差阵的相关稳健估计方法,并通过算例证明其估计思想、模型的正确性和可行性.     

基于方差分量估计的不同截止高度角下的组合单点定位

BDS/GPS/GLONASS组合系统定位时,由于系统间卫星测距精度的差异性,需要合理确定卫星间权比,Helmert方差分量估计常被用于确定不同类观测值间权比;而当观测值含有粗差时,Helmert方差分量估计定位结果容易被粗差污染或收敛失真,出现大的偏差。文中基于Helmert方差分量估计,引入等价权因子IGGIII函数,建立抗差Helmert方差分量估计权函数模型,对比分析其在低截止高度角10°、15°和20°下,在BDS/GPS/GLONASS组合系统定位中的应用及定位精度,并讨论分析在高截止高度角30°和40°下,组合系统和单系统BDS的定位精度。实验结果表明:当观测值无明显粗差时,Helmert方差分量估计和抗差分量估计的定位精度相当,略低于高度角权函数的定位结果,点位精度RMS优于2.5m;含粗差时,抗差解定位精度最高;当截止高度角为30°时,BDS单系统定位精度RMS优于5m,而组合系统RMS接近3m;为40°时,组合系统平面精度RMS优于2m,三维精度RMS优于6m,而单系统不能定位。     

基于等价方差-协方差的粗差探测敏感度分析及实现

选权迭代法在面对独立观测量中的粗差时能够表现出良好的探测效果,但由于其只是一种基于独立观测值的稳健估计法,没有考虑到观测值之间的相关性[1]。而现有的等价权函数虽然都满足稳健估计的要求,但由于所构造的等价权阵不对称,使得最后平差结果严重偏离实际情况。本文介绍在传统稳健估计的基础上,在定权时充分考虑相关观测值之间相关性的不变性,构造对称的方差—协方差阵不断扩大,并通过VB进行编程及实例分析,发现该方法对粗差的敏感度非常强,探测精度很高。     

Effect of heteroscedasticity and heterogeneousness on outlier detection for geodetic networks

The robustness of an outlier detection method strongly depends on the weights of observations, i.e., the type of the stochastic model applied (homoscedasticity, heteroscedasticity and heterogeneousness). In this paper, we have investigated how the reliability of the robust methods and tests for outliers changes depending on the weights of the observations in geodetic networks. Furthermore, the contribution of the directions and distances to horizontal control network with regard to reliability are investigated separately. The concept of a breakdown point is used as a global measure of robustness against outliers. The mean success rate (MSR) is found to be a practical tool for confirming the breakdown point. Many different “good” data samples are generated for each network and then deliberately contaminated using a Monte-Carlo simulation. Six robust methods and Baarda’s test are applied to the corrupted samples and the degree of corruption is varied. The performance of each method is measured using both local and global MSRs. Our research shows: (1) The MSRs of Baarda’s test change depending on the strength of the heteroscedasticity, but do not change for trilateration and leveling networks, (2) the global MSRs of robust methods do not differ considerably from the local ones     

Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness

The findings of this paper are summarized as follows: (1) We propose a sign-constrained robust estimation method, which can tolerate 50% of data contamination and meanwhile achieve high, least-squares-comparable efficiency. Since the objective function is identical with least squares, the method may also be called sign-constrained robust least squares. An iterative version of the method has been implemented and shown to be capable of resisting against more than 50% of contamination. As a by-product, a robust estimate of scale parameter can also be obtained. Unlike the least median of squares method and repeated medians, which use a least possible number of data to derive the solution, the sign-constrained robust least squares method attempts to employ a maximum possible number of good data to derive the robust solution, and thus will not be affected by partial near multi-collinearity among part of the data or if some of the data are clustered together; (2) although M-estimates have been reported to have a breakdown point of 1/(t+1), we have shown that the weights of observations can readily deteriorate such results and bring the breakdown point of M-estimates of Huber’s type to zero. The same zero breakdown point of the L ₁-norm method is also derived, again due to the weights of observations; (3) by assuming a prior distribution for the signs of outliers, we have developed the concept of subjective breakdown point, which may be thought of as an extension of stochastic breakdown by Donoho and Huber but can be important in explaining real-life problems in Earth Sciences and image reconstruction; and finally, (4) We have shown that the least median of squares method can still break down with a single outlier, even if no highly concentrated good data nor highly concentrated outliers exist. An erratum to this article is available at .     

精密单点定位不等精度观测值的RKF研究

推导了精密单点定位含有粗差观测数据的M-LS滤波原理,对等价权阵采用三段降权函数实现抗差。从新息和残差的协方差关系出发,利用对粗差敏感的残差标准差作为抗差因子。通过迭代减弱卫星间载波残差及其抗差因子的相关性。针对载波和伪距观测值不等观测精度和不相关性,采用双抗差因子实现静态抗差卡尔曼滤波（robust Kalman filtering,RKF）。采用标准卡尔曼滤波、基于新息RKF、基于残差的增益矩阵双抗差因子RKF、基于残差的等价权阵双抗差因子RKF等4种模型,分别对一组实测数据解算分析。结果表明,基于新息RKF对精度较高的载波粗差不敏感;基于残差的增益矩阵RKF对载波较小的粗差抗差效果较差,且发生粗差历元时刻的状态参数与真值偏差较大;而基于残差构造的等价权阵双抗差因子RKF可以非常精确和高效地实现抗差,单个卫星粗差对测站位置参数影响小于1 mm。     

我国近海平均海面及其变化的研究

建立了计算平均海面及其变化的动态抗差模型，并把它与计算平均海面的平均值法、抗差法和动态模型法作了实测数据的计算和比较，表明动态抗差模型不仅能顾及海面动态变化反应，而且能削弱海面异常变化的影响，其结果更稳定可靠，优于其他方法。最后应用动态抗差模型，计算了我国42个验潮站的平均海面及其变化，结果表明，从50年代到70提供，我国近海的海面平均以0．621mm/a的速率上升。     

抗差Tikhonov正则化方法及其应用

提出了抗差Tikhonov正则化方法，并给出了3种常用的计算正则化参数的抗差估计方法，即抗差L-曲线法、抗差广义交叉检核法和抗差广义不符原理。计算结果表明，抗差Tikhonov正则化方法不仅能克服方程病态，而且能有效地控制观测异常影响。     

Adjustment of leveling network by information spread estimation

In practical parameter estimation, we have always chosen either Least Squares Estimation(LSE) or Robust Estimation. Since the distribution of observations is unknown, to select a correct estimation method is very difficult. It is well known that if observations include gross errors, the result of LSE will be badly containinated. On the other hand, if observations do not include any gross errors, the result of robust estimation is not as good as that of LSE. To solve this problem, Wang (1999) developed an estimation method called Information Spread Estimation (ISE) based on the information spread principle. The ISE is a very good method for estimating one parameter which is very robust. However, most of instances in surveying data processing are multi-parameters' estimation, owing to the inherent restrictions of ISE, it can not be applied to the surveying data processing directly. To apply the good method to the field of surveying data processing widely, the author has done the research deeply. This paper applies ISE successfully to the adjustment of leveling network by using the specialties of leveling.     

基于卫星双向时间频率传递进行钟差预报的方法研究

用多项式拟合、谱分析、改进的AR模型三种方法对由卫星双向时间频率传递得出的钟差时间序列进行了拟舍和预报分析。为了抵制钟差时间序列中异常值的影响，引入了“抗差等价权”，利用卫星双向时间频率传递得到的1d的钟差，按不问采样率、不同时间跨度进行计算分析。结果表明，抗差估计的预报精度明显高于最小二乘估计；平滑值的预报精度高于采样值；由于钟差时间序列中有明显的周期变化，多项式进行钟差预报的精度不可靠；用谱分析进行钟差预报的精度不高，但可以发现钟差时间序列中的主要周期变化；改进的AR模型预报精度最高，预报6h钟差的RMs在1ns左右。     

Numerical prediction methods for clock deviation based on two-way satellite time and frequency transfer data

Three functional models, polynomial, spectral analysis, and modified AR model, are studied and compared in fitting and predicting clock deviation based on the data sequence derived from two-way satellite time and frequency transfer. A robust equivalent weight is applied, which controls the significant influence of outlying observations. Some conclusions show that the prediction precision of robust estimation is better than that of LS. The prediction precision calculated from smoothed observations is higher than that calculated from sampling observations. As a count of the obvious period variations in the clock deviation sequence, the predicted values of polynomial model are implausible. The prediction precision of spectral analysis model is very low, but the principal periods can be determined. The prediction RMS of 6-hour extrapolation interval is 1 ns or so, when modified AR model is used.     

支持向量回归辅助的GPS/INS组合导航抗差自适应算法

卡尔曼滤波残差分量受到观测信息误差和动力学模型误差的双重影响,由于GPS/INS松耦合导航系统中观测值个数少于状态参数个数,导致异常检测时难以正确区分误差来源,提出一种支持向量回归辅助的组合导航抗差自适应算法。该算法克服了组合系统观测信息无冗余情况下异常检测的局限性,基于遗传算法参数寻优构建回归模型,预测次优观测值,结合整体异常检验法自主选择抗差或自适应滤波,进而调整观测值或动力学模型对导航解的贡献,进行导航预报。最后利用车载实测数据进行验证,结果表明：该算法能够对存在的异常故障智能判定,减弱观测值异常和动力学模型误差影响,保证组合导航精度,提高导航解可靠性。     

一种三步抗差方案的设计

抗差估计的关键是选择恰当的极值函数ρ（．）或ψ（．）函数。本文针对抗差估计涉及的几个重要问题，提出一种新的抗差方案的构想。新的抗差方案试图实现如下三阶段目标；１）选取抗差性能很强的初值，包括定位参数和单位权中误差的初值。２）综合抗差，即要求能实现系数阵空间和观测空间同时抗差，能解决相关观测抗差估计问题，实现定位参数与单位权中误差的联合抗差。３）剔除粗差，给出有关估值的准确统计特性。     

On the use of iterative re-weighting least-squares and outlier detection for empirically modelling rates of vertical displacement

The proper identification and removal of outliers in the combination of rates of vertical displacements derived from GPS, tide gauges/satellite altimetry, and GRACE observations is presented. Outlier detection is a necessary pre-screening procedure in order to ensure reliable estimates of stochastic properties of the observations in the combined least-squares adjustment (via rescaling of covariance matrices) and to ensure that the final vertical motion model is not corrupted and/or distorted by erroneous data. Results from this study indicate that typical data snooping methods are inadequate in dealing with these heterogeneous data sets and their stochastic properties. Using simulated vertical displacement rates, it is demonstrated that a large variety of outliers (random scattered and adjacent, as well as jointly influential) can be dealt with if an iterative re-weighting least-squares adjustment is combined with a robust median estimator. Moreover, robust estimators are efficient in areas weakly constrained by the data, where even high quality observations may appear to be erroneous if their estimates are largely influenced by outliers. Four combined models for the vertical motion in the region of the Great Lakes are presented. The computed vertical displacements vary between  − 2 mm/year (subsidence) along the southern shores and 3 mm/year (uplift) along the northern shores. The derived models provide reliable empirical constraints and error bounds for postglacial rebound models in the region.     

Gross Error Compensation for Gravity Field Analysis Based on Kinematic Orbit Data

This paper aims at a comparative study of several measures to compensate for gross errors in kinematic orbit data. It starts with a simulation study on the influence of a single outlier in the orbit data on the gravity field solution. It is shown that even a single outlier can degrade the resulting gravity field solution considerably. To compensate for outliers, two different strategies are investigated: wavelet filters, which detect and eliminate gross errors, and robust estimators, which due to an iterative downweighting gradually ignore those observations that lead to large residuals. Both methods are applied in the scope of the analysis of a 2-year kinematic CHAMP (challenging minisatellite payload) orbit data set. In various real data studies, robust estimators outperform wavelet filters in terms of resolution of the derived gravity field solution. This superior performance is at the cost of computational load, as robust estimators are implemented iteratively and require the solution of large sets of linear equations several times.     

抗差贝叶斯估计及应用

当未知参数具有先验期望和方差，且观测值与未知参数先验值均服从正态分布时，最小二乘贝叶斯估计将给出参数的最优解。然而当观测值和参数先验值的实际分布有悖于正态假设时，经典贝叶斯估计使估值偏高。本文基于常用的M估计原理，对三种类型的误差模式，导出了M－LS、LS－M和M－M三种抗差贝叶斯估计解式和影响函数；讨论了相应的计算方法；给出了参数验后方差表达式。     

基于卫星双向时间频率传递进行钟差预报的几种方法

Three functional models, polynomial, spectral analysis, and modified AR model, are studied and compared in fitting and predicting clock deviation based on the data sequence derived from two-way satellite time and frequency transfer. A robust equivalent weight is applied, which controls the significant influence of outlying observations. Some conclusions show that the prediction precision of robust estimation is better than that of LS. The prediction precision calculated from smoothed observations is higher than that calculated from sampling observations. As a count of the obvious period variations in the clock deviation sequence, the predicted values of polynomial model are implausible. The prediction precision of spectral analysis model is very low, but the principal periods can be determined. The prediction RMS of 6-hour extrapolation interval is Ins or so, when modified AR model is used.