M-estimator for the 3D symmetric Helmert coordinate transformation
M-estimator for the 3D symmetric Helmert coordinate transformation
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用于 3D 对称 Helmert 坐标变换的 M 估计器
DOI：
10.1007/s00190-017-1043-9
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发表时间：
2018-01-01
影响因子：
4.4
通讯作者：
Wang, Qianxin
中科院分区：
文献类型：
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作者：
Chang, Guobin;Xu, Tianhe;Wang, Qianxin
给出了三维对称Helmert坐标变换问题的M-估计。放弃了小角度旋转假设。用方向余弦矩阵或四元数表示旋转。定义3x 1乘性误差向量以表示旋转估计误差。如果异常值不大，则可以采用解析解来提供迭代的初始近似。迭代采用迭代加权最小二乘方法。在第一次迭代后的每次迭代中，使用可用的参数估计对测量方程进行线性化，使用在前一次迭代中获得的残差构造重新加权矩阵，然后计算参数估计及其方差-协方差矩阵。推导了单个伪量测对最小二乘估计量和M估计量的影响函数，从理论上证明了该方法的鲁棒性。在求解过程中，为了提高数值稳定性，对参数进行了重新标度。蒙特卡罗实验进行了检查开发的方法。不同的情况下，调查是否假设的随机模型是正确的。模拟数据与真实模型略有偏离的结果被用来显示所开发的方法的统计功效在假设的随机模型，其对假设的随机模型的偏差的鲁棒性，以及估计的方差-协方差矩阵的有效性，无论假设的随机模型是正确的或不。
The M-estimator for the 3D symmetric Helmert coordinate transformation problem is developed. Small-angle rotation assumption is abandoned. The direction cosine matrix or the quaternion is used to represent the rotation. The 3x1 multiplicative error vector is defined to represent the rotation estimation error. An analytical solution can be employed to provide the initial approximate for iteration, if the outliers are not large. The iteration is carried out using the iterative reweighted least-squares scheme. In each iteration after the first one, the measurement equation is linearized using the available parameter estimates, the reweighting matrix is constructed using the residuals obtained in the previous iteration, and then the parameter estimates with their variance-covariance matrix are calculated. The influence functions of a single pseudo-measurement on the least-squares estimator and on the M-estimator are derived to theoretically show the robustness. In the solution process, the parameter is rescaled in order to improve the numerical stability. Monte Carlo experiments are conducted to check the developed method. Different cases to investigate whether the assumed stochastic model is correct are considered. The results with the simulated data slightly deviating from the true model are used to show the developed method's statistical efficacy at the assumed stochastic model, its robustness against the deviations from the assumed stochastic model, and the validity of the estimated variance-covariance matrix no matter whether the assumed stochastic model is correct or not.