等式约束病态模型的正则化解及其统计性质

A Regularized Solution and Statistical Properties of Ill-Posed Problem with Equality Constraints

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。

Proper equality constraints between parameters can improve the accuracy of a least square 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 constraints. First, we established a regularization rule and developed a solution through the Lagrange multiplier method. Next, we evaluated its statistical properties to show that this solution is biased and has a smaller mean square error（MSE） than that of the constrained least square problem solution. Then we propese a rule that minimizes the MSE of the estimates to choose both a regularization factor and numerical algorithm. Finally, we use a simulated example and another control net to verify the feasibility of the algorithms and test the statistical properties of the solution.