均方误差意义下的正则化参数二次优化方法

doi:10.11947/j.AGCS.2020.20190148

测绘学报 ›› 2020, Vol. 49 ›› Issue (4): 443-451.doi: 10.11947/j.AGCS.2020.20190148

均方误差意义下的正则化参数二次优化方法

林东方¹, 朱建军², 付海强², 张兵²

1. 湖南科技大学地理空间信息技术国家地方联合工程实验室, 湖南湘潭 411201;
2. 中南大学地球科学与信息物理学院, 湖南长沙 410083

收稿日期:2019-04-23 修回日期:2019-10-23 发布日期:2020-04-17
通讯作者: 朱建军 E-mail:zjj@csu.edu.cn
作者简介:林东方(1986-),男,博士,讲师,研究方向为数据处理理论与方法及PolInSAR应用。E-mail:lindongfang223@163.com
基金资助:
湖南省教育厅科研项目（18C0312）；湖南省科技厅重大专项（2018GK205）；国家自然科学基金（41531068；41574006；41674012）；湖南科技大学科研项目（CXTD004）

Optimization of regularization parameter based on minimum MSE

LIN Dongfang¹, ZHU Jianjun², FU Haiqiang², ZHANG Bing²

1. National-Local Joint Engineering Laboratory of Geo-Spatial Information Technology, Hunan University of Science and Technology, Xiangtan 411201, China;
2. School of Geosciences and Info-Physics, Central South University, Changsha 410083, China

Received:2019-04-23 Revised:2019-10-23 Published:2020-04-17
Supported by:
The Research Project of Education Department of Hunan Province (No. 18C0312);The Provincial Key Research and Development Program of Hunan (No. 2018GK2015);The National Natural Science Foundation of China (Nos. 41531068;41574006;41674012);The Research Program of Hunan University of Science and Technology (No. CXTD004)

摘要/Abstract

摘要： Tikhonov正则化法是大地测量中应用最为广泛的病态问题解算方法之一。影响正则化法解算效果的重要因素是正则化参数，然而，最优正则化参数的确定一直是正则化解算的难题，如L曲线法确定的正则化参数具有稳定性好、可靠性高的优点，但存在过度平滑问题，导致正则化法对模型参数估值精度改善较小。本文从均方误差角度分析了正则化参数对模型参数估计质量的影响。基于奇异值分解技术，提出了由模型参数投影值分块计算均方误差的方法，避免了均方误差迭代计算，并基于均方误差最小准则给出了正则化参数优化方法，实现了对L曲线正则化参数的优化。数值模拟试验与PolInSAR植被高反演试验结果表明，正则化参数优化方法有效改善了正则化法解算效果，提高了模型参数估计精度。

关键词: 病态问题, 正则化方法, 正则化参数, 均方误差, L曲线

Abstract: Tikhonov regularization method is widely used in geodesy for ill-posed problems. The regularization parameter is an important factor for regularization method to solve the ill-posed problem. However, it is very difficult to determine an optimal regularization parameter. L-curve method is proposed to determine the feasible regularization parameter, which is well known to be a stable and reliable method. However, the extensive application researches show that the regularization parameter determined by L-curve method often leads to oversmoothed results. As a result, the regularization method cannot effectively improve the estimation accuracy of model parameters. Concerning this issue, this paper analyzes the effectiveness of regularization parameter on MSE (mean square error) of regularized estimation. Then, an MSE calculation method is proposed by using SVD (singular value decomposition) technology. In the method, the MSE is divided into several parts that correspond to the singular values. Therefore, the iterative calculation of MSE is avoided and the reasonable regularization parameter can be determined part to part. Using the reliable parts of MSE, the most useful regularization parameter can be determined to optimize the L-curve determined regularization parameter. Finally, the regularization parameter optimization method is proposed. Numerical example and PolInSAR vegetation inversion experiment are carried out to demonstrate the effectiveness of the regularization parameter optimization method. The results show that the regularization parameter optimization method can greatly improves the model parameter estimation of regularization method.

Key words: ill-posed problem, regularization method, regularization parameter, mean square error, L-curve

中图分类号:

P207

林东方, 朱建军, 付海强, 张兵. 均方误差意义下的正则化参数二次优化方法[J]. 测绘学报, 2020, 49(4): 443-451.

LIN Dongfang, ZHU Jianjun, FU Haiqiang, ZHANG Bing. Optimization of regularization parameter based on minimum MSE[J]. Acta Geodaetica et Cartographica Sinica, 2020, 49(4): 443-451.

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均方误差意义下的正则化参数二次优化方法

Optimization of regularization parameter based on minimum MSE

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