基于矩阵分解的可分离非线性最小二乘问题求解方法及其应用

doi:10.11947/j.AGCS.2022.20200502

测绘学报 ›› 2022, Vol. 51 ›› Issue (11): 2317-2327.doi: 10.11947/j.AGCS.2022.20200502

基于矩阵分解的可分离非线性最小二乘问题求解方法及其应用

王路遥, 刘国林, 王凤云, 王珂, 韩宇

山东科技大学测绘与空间信息学院, 山东青岛 266590

收稿日期:2020-10-30 修回日期:2022-02-20 发布日期:2022-11-30
通讯作者: 刘国林 E-mail:gliu@sdust.edu.cn
作者简介:王路遥(1994—),男,博士生,研究领域为现代测量数据处理。 E-mail: 541355164@qq.com
基金资助:
国家自然科学基金(42074009);山东省自然科学基金(ZR2020MD043;ZR2020MD044)

The method and application for solving separable nonlinear least squares based on matrix decomposition

WANG Luyao, LIU Guolin, WANG Fengyun, WANG Ke, HAN Yu

College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China

Received:2020-10-30 Revised:2022-02-20 Published:2022-11-30
Supported by:
The National Natural Science Foundation of China (No. 42074009); Shandong Natural Science Foundation (Nos. ZR2020MD043; ZR2020MD044)

摘要/Abstract

摘要： 针对可分离非线性函数模型的特殊结构,本文使用变量投影法(VP)将线性参数与非线性参数分离开来,并分别与矩阵的满秩分解、QR分解、奇异值分解和施密特正交化相结合,对两类参数分别求解,缩短了计算机解算方程组的运算时间,使算法更加高效,同时也使得具有一定病态程度的方程组在解算过程中保持相对较好的稳定性。本文利用Mackey-Glass时间序列拟合试验和空间直角坐标转换参数解算试验对比分析了基于不同矩阵分解方法的算法优劣性。试验结果表明,基于矩阵分解的改进变量投影法具有高效的运算效率与稳定的解算过程,也适用于解算空间直角坐标转换参数问题。

关键词: 可分离非线性最小二乘, 变量投影, 矩阵分解, Mackey-Glass时间序列, 空间直角坐标转换

Abstract: For the special structure of separable function models, the variable projection (VP) method is used to separate the linear and the nonlinear parameters in this paper, and respectively combined with the full-rank decomposition, QR decomposition, singular value decomposition and Gram-Schmidt orthogon-alization to calculate the parameters, which shorten the calculation time of solving equations by computer, and enable the algorithm more efficient and equations with a certain ill-conditioning maintain the stability in the process of solving. The superiority-inferiority of the algorithms based on different matrix decomposition methods is analyzed by Mackey-Glass time series fitting experiment and parameters calculation for spatial rectangular coordinate transformation. The experimental results show that the improved VP algorithms based on matrix decomposition are highly efficient and stable, and suitable to solve the parameters of spatial rectangular coordinate transformation models.

Key words: separable nonlinear least squares, variable projection, matrix decomposition, Mackey-Glass time series, spatial rectangular coordinates transformation

中图分类号:

P207

王路遥, 刘国林, 王凤云, 王珂, 韩宇. 基于矩阵分解的可分离非线性最小二乘问题求解方法及其应用[J]. 测绘学报, 2022, 51(11): 2317-2327.

WANG Luyao, LIU Guolin, WANG Fengyun, WANG Ke, HAN Yu. The method and application for solving separable nonlinear least squares based on matrix decomposition[J]. Acta Geodaetica et Cartographica Sinica, 2022, 51(11): 2317-2327.

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基于矩阵分解的可分离非线性最小二乘问题求解方法及其应用

The method and application for solving separable nonlinear least squares based on matrix decomposition

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