引用本文: 王乐洋, 赵卫凤. 非线性加性乘性混合误差模型的参数估计方法[J]. 武汉大学学报 ( 信息科学版).
WANG Le-yang, ZHAO Wei-feng. Parameter estimation methods for nonlinear mixed additive and multiplicative random error model[J]. Geomatics and Information Science of Wuhan University.
 Citation: WANG Le-yang, ZHAO Wei-feng. Parameter estimation methods for nonlinear mixed additive and multiplicative random error model[J]. Geomatics and Information Science of Wuhan University.

## Parameter estimation methods for nonlinear mixed additive and multiplicative random error model

• 摘要: 在大地测量领域中，测量数据中不仅含有加性误差，还存在乘性误差。现有的处理加性和乘性混合误差模型的方法主要是基于未知参数和观测值是线性形式的，鲜有对未知参数和观测值是非线性形式的研究。为了扩展加性和乘性混合误差模型的参数估计方法，本文基于最小二乘原理并应用了泰勒公式展开的思想，推导了非线性加性乘性混合误差模型的最小二乘法、高斯-牛顿法、加权最小二乘法和偏差改正加权最小二乘法四种参数估计方法。通过算例的计算和对比分析可知，当模型非线性较高时，偏差改正加权最小二乘方法能够得到更好的参数估计结果，证明了该方法的有效性。该方法也更适用于处理这种非线性的加性和乘性混合误差模型的大地测量数据。

Abstract: Objectives: In the field of geodesy, with the continuous development of modern observation technology, so that the measurement data not only contain additive errors, there are also multiplicative errors, purely considering the processing of additive errors can no longer meet the requirements. Existing methods for dealing with mixed additive and multiplicative errors models are mainly based on the fact that the unknown parameters and observations are in linear form, and few studies have been conducted on the fact that the unknown parameters and observations are in nonlinear form. Methods: In order to extend the parameter estimation method of the mixed additive and multiplicative errors model, this paper determines the reasonable weight matrix of nonlinear mixed additive and multiplicative errors model based on the law of error propagation and the principle of least squares and applies the idea of Taylor's formula expansion, and derives the least squares, Gauss-Newton method, and weighted least squares of the nonlinear mixed additive and multiplicative errors model. Due to the nonlinear nature, it makes the weighted least squares solution biased, so it needs to be analyzed for its bias. The bias-corrected weighted least squares method is derived by deviation analysis and proof. Results: It can be seen through the calculation and comparative analysis of the arithmetic examples that a reasonable weighting method is conducive to improving the correctness of the model parameter estimation results, and when the model nonlinearity is high, the bias-corrected weighted least squares method can obtain better parameter estimation results. Conclusions: The feasibility and validity of four methods for parameter estimation of the nonlinear mixed additive and multiplicative errors model are demonstrated, and the bias-corrected weighted least squares method is more suitable for processing geodetic data from such nonlinear models with a mixture of mixed additive and multiplicative errors.

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