Ridge estimation method for linearized general EIV adjustment model
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摘要: 通用EIV(errors-in-variables)平差模型作为经典平差模型的一般化形式，具有同时顾及多种随机误差的优势. 在通用EIV平差模型加权总体最小二乘(WTLS)的线性化估计基础上，引入正则化准则. 正则化矩阵为单位矩阵时为岭估计，添加目标函数，通过建立拉格朗日目标函数的最小化求解，导出加权通用EIV平差模型对应的岭估计解式，给出了确定岭参数的U曲线法和L曲线法. 计算了通用EIV平差模型的线性化估计、两种岭估计及其对应的方差分量值；验证岭估计对通用EIV模型的线性化估计具有促进性，可减少迭代次数，使得参数方差分量更快趋于平稳，降低参数估计的计算量.
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关键词:
- 通用EIV(errors-in-variables)模型 /
- 总体最小二乘(TLS) /
- 线性化估计 /
- 岭估计 /
- L曲线 /
- U曲线
Abstract: As a general form of classical adjustment model, general errors-in-variables (EIV) adjustment model has the advantage of taking into account multiple random errors. Based on the linear estimation of the weighted total least squares of the general EIV adjustment model, the regularization criterion is introduced. When the regularization matrix is the unit matrix, it is called the ridge estimation. The objective function is then added. By establishing the minimization solution of the Lagrange objective function, the ridge estimation solution corresponding to the weighted general EIV adjustment model is derived. The U curve method and L curve method for determining ridge parameters are given. The linear estimation, two ridge estimations and their corresponding variance components of the general EIV adjustment model are calculated. It is validated that ridge estimation can promote the linearization estimation of general EIV model, reduce the times of iterations, make the parameter variance component more stable and reduce the calculation of parameter estimation. -
表 1 含有随机误差的模拟数据矩阵
$ {\boldsymbol{B}}_{4\times 2} $ $ {\boldsymbol{A}}_{4\times 4} $ $ {\boldsymbol{L}}_{0} $ $ \boldsymbol{e} $ 10.409 7 17.543 9 12.466 9 11.096 3 15.871 7 11.725 4 27.543 0 −1425.323 18.033 4 15.171 3 8.882 6 10.291 8 2.928 7 3.665 7 20.726 8 −852.619 18.631 3 15.855 2 12.321 0 1.108 5 6.391 9 15.808 5 20.838 9 −1142.913 15.670 5 11.878 4 3.551 0 12.104 0 3.866 5 1.256 7 25.032 5 −658.407 
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表 2 参数解估计值及其方差估计值
参数解/中误差 LTLS算法 LTLS岭估计（L 曲线法） LTLS岭估计（U 曲线法） $ {\hat{X}}_{1} $ 5.012 550 72 5.010 468 13 5.008 176 42 $ {\hat{X}}_{2} $ 9.994 963 66 9.995 139 46 9.996 834 92 $ {\sigma }_{{\hat{X}}_{1}}^{2} $ 4.272 016$ \times {10}^{-25} $ 4.109 261$ \times {10}^{-25} $ 4.001 347$ \times {10}^{-25} $ $ {\sigma }_{{\hat{X}}_{2}}^{2} $ 7.245 756$ \times {10}^{-25} $ 4.532 423$ \times {10}^{-25} $ 4.076 249$ \times {10}^{-25} $ $ {\sigma }_{{\hat{X}}_{1}{\hat{X}}_{2}} $ −5.044 943$ \times {10}^{-25} $ −4.141 588$ \times {10}^{-25} $ −3.738 164$ \times {10}^{-25} $ ${\rm {tr} }\left[D\left(\hat{\boldsymbol{X} }\right)\right]$ 1.151 777$ \times {10}^{-24} $ 8.641 684$ \times {10}^{-25} $ 8.077 596$ \times {10}^{-25} $ $ {\parallel \mathrm{\Delta }\boldsymbol{X}\parallel }^{2} $ 1.828 853$ \times {10}^{-4} $ 1.332 065$ \times {10}^{-4} $ 7.687 158$ \times {10}^{-5} $ $ \mathrm{迭}\mathrm{代}\mathrm{次}\mathrm{数} $ 11 8 8
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表 3 坐标观测值及相应权值
点号 观测数据 权值 $ {y}_{2} $ $ {x}_{2} $ $ {P}_{y} $ $ {P}_{x} $ 1 5.9 0.0 1.0 1 000.0 2 5.4 0.9 1.8 1 000.0 3 4.4 1.8 4.0 500.0 4 4.6 2.6 8.0 800.0 5 3.5 3.3 20.0 200.0 6 3.7 4.4 20.0 80.0 7 2.8 5.2 70.0 60.0 8 2.8 6.1 70.0 20.0 9 2.4 6.5 100.0 1.8 10 1.5 7.4 500.0 1.0
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表 4 参数及其方差估计值、迭代次数
参数解/
中误差LTLS算法 LTLS岭估计
（L 曲线法）LTLS岭估计
（U 曲线法）${\;\hat{\beta } }_{1}$ −0.489 907 073 −0.489 907 072 −0.489 907 072 ${\;\hat{\beta } }_{2}$ 5.527 557 906 5.527 557 907 5.527 557 907 $ {\sigma }_{{\hat{X}}_{1}}^{2} $ 0.656 376 350 0.656 376 380 0.656 376 380 $ {\sigma }_{{\hat{X}}_{2}}^{2} $ 1.821 311 510 1.821 311 500 1.821 311 500 $ \mathrm{迭}\mathrm{代}\mathrm{次}\mathrm{数} $ 18 14 13
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