Generalized ridge estimation method for morbid weighted total least squares
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摘要: 针对加权情形下的变量误差(EIV)模型，采用广义岭估计法处理总体最小二乘平差的病态性问题. 结合最优化准则和协方差传播率推导了未知参数的改正数求解公式；根据参数估计值的均方误差最小化原理，通过求偏导数列出广义岭估计中岭参数的迭代解式，并讨论了广义岭参数的含义和作用，给出了确定岭参数的L-曲线法. 通过算例比较分析了加权最小二乘估计、总体最小二乘估计、加权最小二乘岭估计、总体最小二乘岭估计、加权最小二乘的广义岭估计和总体最小二乘广义岭估计，叙述了加权总体最小二乘的广义岭估计的优缺点.
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关键词:
- 变量误差(EIV)模型 /
- 均方误差 /
- 总体最小二乘 /
- 广义岭估计 /
- 岭参数
Abstract: For the variable error (EIV) model in the weighted case, the generalized ridge estimation method is used to deal with the morbid problem of the total least squares adjustment. Combined with optimization criterion and covariance propagation rate, the correction formula of unknown parameters is derived. Accordsing to the principle of minimizing mean square error of parameter estimation, the iterative solution of ridge parameter in generalized ridge estimation is given by solving partial derivative, and the meaning and function of generalized ridge parameter are discussed. The weighted least squares estimation, total least squares estimation, the weighted least squares ridge estimation, total least squares ridge estimation, generalized ridge estimation of the weighted least squares and generalized ridge estimation of total least squares are compared and analyzed by examples. The advantages and disadvantages of generalized ridge estimation of weighted total least squares are described. -
表 1 不同方法的解算结果
方案 $ {\hat{x}}_{1} $ $ {\hat{x}}_{2} $ $ {\hat{x}}_{3} $ $ {\hat{x}}_{4} $ $ {\hat{x}}_{5} $ 岭参数 $\parallel \Delta \hat{\boldsymbol{X} }\parallel$ 一 1.391 9 0.121 5 0.778 7 0.267 2 1.441 7 - 1.306 3 二 3.304 1 −2.803 3 0.060 0 3.587 4 2.902 7 - 6.732 2 三 1.215 7 0.372 8 0.828 0 0.598 3 1.315 7 0.299 1 0.854 7 四 1.083 6 0.625 6 0.736 4 0.789 2 1.212 3 0.150 0 0.553 4 五 0.976 5 0.800 0 0.424 2 0.962 2 1.129 6 K 0.624 8 六 0.977 4 0.800 0 0.640 1 0.891 4 1.136 0 K 0.447 6 真值 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 - - 
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表 2 方案六的协因数矩阵
参数 $ {\hat{\mathit{X}}}_{1} $ $ {\hat{\mathit{X}}}_{2} $ $ {\hat{\mathit{X}}}_{3} $ $ {\hat{\mathit{X}}}_{4} $ $ {\hat{\mathit{X}}}_{5} $ $ {\hat{\mathit{X}}}_{1} $ 2.905 8$ \times {10}^{-2} $ −4.875 2$ \times {10}^{-8} $ 2.009 7$ \times {10}^{-2} $ 1.869 0$ \times {10}^{-2} $ −7.903 2$ \times {10}^{-3} $ $ {\hat{\mathit{X}}}_{2} $ 4.718 9$ \times {10}^{-11} $ −1.395 9$ \times {10}^{-6} $ 2.260 4$ \times {10}^{-7} $ 5.107 6$ \times {10}^{-7} $ $ {\hat{\mathit{X}}}_{3} $ 1.042 2$ \times {10}^{-1} $ 6.167 1$ \times {10}^{-2} $ −2.732 8$ \times {10}^{-2} $ $ {\hat{\mathit{X}}}_{4} $ 1.391 6$ \times {10}^{-2} $ −4.221 1$ \times {10}^{-3} $ $ {\hat{\mathit{X}}}_{5} $ 3.019 1$ \times {10}^{-3} $
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