正则化的奇异值分解参数构造法

doi:10.11947/j.AGCS.2016.20150134

测绘学报 ›› 2016, Vol. 45 ›› Issue (8): 883-889.doi: 10.11947/j.AGCS.2016.20150134

• 大地测量学与导航 • 下一篇

正则化的奇异值分解参数构造法

林东方, 朱建军, 宋迎春, 何永红

中南大学地球科学与信息物理学院, 湖南长沙 410083

收稿日期:2015-03-12 修回日期:2016-06-06 出版日期:2016-08-20 发布日期:2016-08-31
作者简介:林东方(1986-),男,博士研究生,研究方向为测量平差数据处理及应用。E-mail:lindongfang223@163.com
基金资助:
国家自然科学基金（415300321；41474008）

Construction Method of Regularization by Singular Value Decomposition of Design Matrix

LIN Dongfang, ZHU Jianjun, SONG Yingchun, HE Yonghong

School of Geosciences and Info-physics, Central South University, Changsha 410083, China

Received:2015-03-12 Revised:2016-06-06 Online:2016-08-20 Published:2016-08-31
Supported by:
The National Natural Science Foundation of China (Nos. 415300321;41474008)

摘要/Abstract

摘要： Tikhonov正则化法引入正则化参数和稳定泛函来改善矩阵的病态性。稳定泛函表示为参数的二范约束时，正则化矩阵为单位阵的正则化法即为岭估计法。通过对岭估计的方差与偏差进行分析可知，岭估计改善矩阵病态性的同时也过度地引入了偏差，降低了解的可靠性，对较大奇异值的修正不能有效地减小估计的方差，却引入了偏差，而对较小奇异值的修正可有效地减小估计的方差。因此，选择较小奇异值特征向量构造正则化矩阵，调节各奇异值的修正，可有效减小参数估计的方差，减少偏差的引入，得到更为可靠的参数估计。通过试验证明了该方法的有效性。

关键词: 正则化法, 岭估计, 正则化矩阵, 奇异值, 特征向量

Abstract: Tikhonov regularization introduces regularization parameter and stable functional to improve the ill-condition. When the stable functional expressed as two-norm constraint, the regularization method is the same as ridge estimation. The analysis of the variance and bias of the ridge estimation shows that ridge estimation improved the ill-condition but introduced more bias. The estimation reliability is lowered. We get that correct the larger singular values cannot decrease the variance effectively but introduced more bias, correcting the smaller singular values can decrease the variance effectively. We choose the eigenvectors of the smaller singular values to construct the regularization matrix. It can adjust the correction of the singular values, decrease the variance and biases and finally get a more reliable estimation.

Key words: regularization solution, ridge estimation, regularization matrix, singular value, eigenvectors

中图分类号:

P207

林东方, 朱建军, 宋迎春, 何永红. 正则化的奇异值分解参数构造法[J]. 测绘学报, 2016, 45(8): 883-889.

LIN Dongfang, ZHU Jianjun, SONG Yingchun, HE Yonghong. Construction Method of Regularization by Singular Value Decomposition of Design Matrix[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(8): 883-889.

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正则化的奇异值分解参数构造法

Construction Method of Regularization by Singular Value Decomposition of Design Matrix

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