引用本文: 王乐洋, 韩澍豪. 不等式约束下加乘性混合误差模型的简单迭代解法[J]. 武汉大学学报 ( 信息科学版), 2024, 49(6): 996-1004.
WANG Leyang, HAN Shuhao. A Simple Iterative Solution for Mixed Additive and Multiplicative Random Error Model with Inequality Constraints[J]. Geomatics and Information Science of Wuhan University, 2024, 49(6): 996-1004.
 Citation: WANG Leyang, HAN Shuhao. A Simple Iterative Solution for Mixed Additive and Multiplicative Random Error Model with Inequality Constraints[J]. Geomatics and Information Science of Wuhan University, 2024, 49(6): 996-1004.

## A Simple Iterative Solution for Mixed Additive and Multiplicative Random Error Model with Inequality Constraints

• 摘要: 在大地测量领域中,现有的处理不等式约束的方法大多都是基于加性误差的模型,包括高斯马尔可夫模型和变量误差模型,鲜有对于加乘性混合误差模型处理方法的研究。为了拓展附有不等式约束的加乘性混合误差的方法,基于最小二乘原理并应用零权和无限权的思想,通过约束条件构建了惩罚函数,推导了在不等式约束下加乘性混合误差的一种简单迭代解法,分析了简单迭代解法在加乘性混合误差模型中的缺陷,在原有方法的基础上在惩罚项前加入了一个随迭代次数增加而增加的惩罚因子。通过算例评估分析可知,改进后的简单迭代法能够有效解决原有方法用于处理附有不等式约束的加乘性混合误差模型时不收敛的问题。通过对比其他方案可知,所提方法能够得到更好的参数估值,证明了该方法的有效性。同时,所提方法结构简单,易于实现,能够适用于大批量的数据处理。

Abstract:
Objectives With the development of modern observation techniques, the processing methods which only consider additive errors cannot meet the requirements. Most of the existing methods for dealing with inequality constraints are based on additive error models, including Gaussian-Markov model and errors-in-variables model, while the processing methods for mixed additive and multiplicative (MAM) random error models are rare.
Methods Based on the least squares principle and the ideas of zero and infinite weights, we construct a penalty function with the given inequality constraints, and derive the simple iterative method (SIM) for the estimation of MAM parameters under the inequality constraints. Then, we add a penalty factor increasing with the number of iterations before the penalty term to address the defects of the original SIM.
Results Two sets of cases show that the improved SIM can effectively solve the problem that the original method does not converge when used to deal with MAM error models with inequality constraints. The structure of improved SIM is simple and easy to implement. And it can obtain better parameter estimation compared with other schemes.
Conclusions The feasibility and effectiveness of the improved SIM for parameter estimation of MAM error models with inequality constraints are verified, and it can be applied to the processing of large batches of data.

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