病态模型改进正则化解的单位权中误差无偏估计

正则化法通过引入正则化参数对奇异值加以修正,从而改善法矩阵的病态性,然而其不加区别地对所有奇异值进行修正显然是不合理的。本文比较正则化解均方误差和最小二乘解方差的迹谱分解展开式,分析因修正奇异值导致解的均方误差变化与奇异值的关系,确定奇异值修正与否的条件,并基于残差二次型期望公式导出改进正则化解的无偏单位权中误差计算公式,最后用数值算例和病态测边网算例验证公式的正确性。

Truncated Regularized Solution to Ill-Posed Model and ItsUnbiased Estimation of Unit Weighted Variance

The regularized method improves the ill-posedness of the normal matrix by introducing the regularized parameter to correct the singular values. However, it is obviously unreasonable to correct the singular values without distinction. In this paper, we compare the trajectory decomposition expansions of the regularized mean square error and the least squares solution variance, analyze the relationship between the change of the mean square error of the solution caused by the modified singular value and the singular value, and determine the conditions for singular value correction. Based on the residual quadratic expectation formula, an error calculation formula for unbiased unit weights with improved regularization is derived. Finally, numerical examples and ill-trimmed network examples are used to verify the correctness of the formula.