Linearization Estimation Algorithm for Universal EIV Adjustment Model
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摘要: 通用EIV模型将EIV模型扩展到了最一般化的形式，其加权整体最小二乘算法同时顾及了观测向量、观测向量的系数矩阵和参数的系数矩阵中的随机误差。本文将非线性的通用EIV函数模型展开，并将二阶项纳入模型的常数项，从而将通用EIV模型表示为线性形式的高斯-赫尔默特模型，推导出了通用EIV模型的线性化整体最小二乘算法和近似精度估计公式。通过模拟数据和实例进行了评估和分析，该算法与通用EIV模型的WTLS算法估计结果一致，验证了该算法的正确性和可行性。当模型含大量估计量时，通用EIV模型的线性化估计算法显著提升了计算效率，收敛速度更快。Abstract: The universal EIV model extends the EIV model to the most general form, and the weighted total least squares (WTLS) algorithm is proposed to take into account the random errors in observation vector, observation vector coefficient matrix and parameter coefficient matrix. In this paper, the nonlinear universal EIV function model is expanded, and the second-order term is included into the constant term of the model, so the universal EIV model is represented as Gauss-Helmert model in linear form, and the Linearized total least squares algorithm and approximate precision estimation formula of the universal EIV model are derived. Through the simulation data and examples, this algorithm is consistent with the estimation results of the WTLS algorithm of the universal EIV model, which verifies the correctness and feasibility of this algorithm. When the model contains a large number of estimators, the linearized estimation algorithm of the universal EIV model significantly improves the computational efficiency and converges faster.
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图 1 1 000组实验数据的LTLS参数解与参数真值偏差统计图
Figure 1. Statistical Graph of Deviations Between Truth Values and Parameter Values Soluted by LTLS in 1 000 Experiments
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图 2 1 000组实验数据的WTLS参数解与参数真值偏差统计图
Figure 2. Statistical Graph of Deviations Between Truth Values and Parameter Values Soluted by WTLS in 1 000 Experiments
表 1 参数解及其方差估计值
Table 1 Parameter Values and Mean Square Deviations
算法 参数解 中误差 $ \hat {\boldsymbol{X}} _1 $ $ \hat {\boldsymbol{X}} _2 $ $ σ _{\hat{\boldsymbol{X}}_1} $ $ σ_ {\hat{\boldsymbol{X}}_2} $ LTLS算法 5.012 551 9.994 964 0.039 9 0.051 9 WTLS算法 5.012 551 9.994 964 0.039 9 0.051 9 
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表 2 1000组实验的参数解均值和协因数阵
Table 2 Average Parameter Values and Co⁃variance Matrix in 1 000 Experiments
算法 参数解均值 协因数阵 $ {\rm{avg}} ( \hat{\boldsymbol{X}}_1) $ $ {\rm{avg}} ( \hat{\boldsymbol{X}}_2) $ $ {\boldsymbol{Q }}( \hat{\boldsymbol{X}} ) $ $ \bar{ \boldsymbol{Q }}( \hat{\boldsymbol{X}} ) $ LTLS算法 5.009 427 9.998 257 0.004 3 −0.005 1 0.004 2 −0.004 9 −0.005 1 0.007 3 −0.004 9 0.007 2 WTLS算法 5.009 427 9.998 257 0.004 3 −0.005 1 0.004 2 −0.004 9 −0.005 1 0.007 3 −0.004 9 0.007 2
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表 3 LTLS算法和WTLS算法计算效率的比较
Table 3 Comparison of Computational Efficiency Between LTLS Algorithm and WTLS Algorithm
待估量数量 NLTLS NWTLS tLTLS tWTLS 减少比例/% 10 5.18 4.5 0.224 ms 0.191 ms − 100 5.04 6.36 0.681 ms 0.724 ms 5.9 1 000 5 6.75 0.033 s 0.044 s 25.0 10 000 5 7.5 2.681 s 3.909 s 31.4
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表 4 距离观测值及其中误差
Table 4 Distance Observations and Standard Deviations
统计项 l1/mm l2/mm l3/mm l4/mm l5/mm l6/mm y1/m y2/m 观测值 14.1 16.6 6.1 7.1 22.1 26.3 10.0 8.0 中误差 0.10 0.10 0.10 0.10 0.10 0.10 0.05 0.05
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表 5 点P1和点P2的坐标估值/m
Table 5 Coordinate Estimates of P1 and P2 /m
算法 坐标估值 $ \hat x_1 $ $ \hat x_2 $ $ \hat x_3 $ $ \hat x_4 $ LTLS算法 6.995 056 5 49.715 632 6.981 465 5 41.968 315 9 WTLS算法 6.995 056 5 49.715 632 6.981 465 5 41.968 315 9
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表 6 距离观测值估值
Table 6 Estimation of Distance Observations
算法 $ \hat l_1 $/mm $ \hat l_2 $/mm $ \hat l_3 $/mm $ \hat l_4 $/mm $ \hat l_5 $5/mm $ \hat l_6 $/mm $ \hat y_1 $/m $ \hat y_2 $/m LTLS算法 14.070 1 16.635 1 6.032 4 7.178 4 22.137 7 26.256 7 9.994 1 8.006 8 WTLS算法 14.070 1 16.635 1 6.032 4 7.178 4 22.137 7 26.256 7 9.994 1 8.006 8
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