引用本文: 谢建, 朱建军. 等式约束病态模型的正则化解及其统计性质[J]. 武汉大学学报 ( 信息科学版), 2013, 38(12): 1440-1444.
XIE Jian, ZHU Jianjun. A Regularized Solution and Statistical Properties ofIll-Posed Problem with Equality Constraints[J]. Geomatics and Information Science of Wuhan University, 2013, 38(12): 1440-1444.
 Citation: XIE Jian, ZHU Jianjun. A Regularized Solution and Statistical Properties ofIll-Posed Problem with Equality Constraints[J]. Geomatics and Information Science of Wuhan University, 2013, 38(12): 1440-1444.

## A Regularized Solution and Statistical Properties ofIll-Posed Problem with Equality Constraints

• 摘要: 平差参数间合理的等式约束能显著提高最小二乘解的精度,消除自由网的秩亏问题,还能改善病态模型的解,本文由此提出了附等式约束的测量平差病态模型。建立了等式约束病态模型平差的正则化准则,由拉格朗日乘数法得到了模型的解,并给出了解的统计性质,证明它有偏,且均方误差小于约束最小二乘解。给出了确定正则化因子的均方误差极小化准则及数值解法,并以模拟的病态数值算例和测边控制网为例进行计算,验证了算法的可行性和解的优良性质。

Abstract: Proper equality constraints between parameters can improve the accuracy of a leastsquare solution,eliminate the deficiencies of free networks and improve the solution of an ill-posed problem,therefore we proposes the ill-posed adjustment model with equality con-straints.First,we established a regularization rule and developed a solution through the La-grange multiplier method.Next,we evaluated its statistical properties to show that this so-lution is biased and has a smaller mean square error(MSE)than that of the constrained leastsquare problem solution.Then we propese a rule that minimizes the MSE of the estimates tochoose both a regularization factor and numerical algorithm.Finally,we use a simulated ex-ample and another control net to verify the feasibility of the algorithms and test the statisticalproperties of the solution.

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