﻿ 一种构造正则化矩阵的新方法
 文章快速检索 高级检索
 大地测量与地球动力学  2019, Vol. 39 Issue (1): 61-65  DOI: 10.14075/j.jgg.2019.01.012

### 引用本文

WU Guangming, LU Tieding, DENG Xiaoyuan, et al. A New Method of Constructing Regularized Matrix[J]. Journal of Geodesy and Geodynamics, 2019, 39(1): 61-65.

### Foundation support

National Natural Science Foundation of China, No.41374007, 41464001; Science and Technology Landing Project of Jiangxi Province, No.KJLD12077; Science and Technology Project of the Education Department of Jiangxi Province, No.GJJ13457;Natural Science Foundation of Jiangxi Province, No.2017BAB203032; National Key Research and Development Program of China, No.2016YFB0501405, 2016YFB0502601-04.

### Corresponding author

LU Tieding, PhD, professor, majors in error theory and survey adjustment, E-mail: tdlu@whu.edu.cn.

### 第一作者简介

WU Guangming, postgraduate, majors in surveying and mapping data processing, E-mail: 821345314@qq.com.

### 文章历史

1. 东华理工大学测绘工程学院, 南昌市广兰大道 418号, 330013;
2. 流域生态与地理环境监测国家测绘地理信息局重点实验室, 南昌市广兰大道 418 号, 330013;
3. 江西省数字国土重点实验室, 南昌市广兰大道418号, 330013;
4. 浙江省地理信息中心，杭州市保俶北路83号, 310012

1 岭估计原理

 $\mathit{\boldsymbol{L}} + \mathit{\boldsymbol{ \boldsymbol{\varDelta} }} = \mathit{\boldsymbol{AX}}$ (1)

 $\mathit{\boldsymbol{\hat X}} = {({\mathit{\boldsymbol{A}}^{\rm T}}\mathit{\boldsymbol{A}})^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm T}}\mathit{\boldsymbol{L}}$ (2)
 $D(\mathit{\boldsymbol{\hat X}}) = \sigma _0^2{\rm{tr}}{({\mathit{\boldsymbol{A}}^{\rm T}}\mathit{\boldsymbol{A}})^{ - 1}} = \sigma _0^2\sum\limits_{i = 1}^n {\frac{1}{{\lambda _i^2}}}$ (3)

 $\begin{array}{c} \left\| {\mathit{\boldsymbol{AX}} - \mathit{\boldsymbol{L}}} \right\|_{\rm{2}}^{\rm{2}} + \alpha \mathit{\Omega }(\mathit{\boldsymbol{X}}) = \\ \left\| {\mathit{\boldsymbol{AX}} - \mathit{\boldsymbol{L}}} \right\|_{\rm{2}}^{\rm{2}} + \alpha {\mathit{\boldsymbol{X}}^{\rm T}}\mathit{\boldsymbol{RX}} = \min \end{array}$ (4)

 ${{\mathit{\boldsymbol{\hat X}}}_1} = {({\mathit{\boldsymbol{A}}^{\rm T}}\mathit{\boldsymbol{A}} + \alpha \mathit{\boldsymbol{I}})^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm T}}\mathit{\boldsymbol{L}}$ (5)

 ${{\mathit{\boldsymbol{\hat X}}}_1} = {[\mathit{\boldsymbol{G}}(\mathit{\boldsymbol{ \boldsymbol{\varLambda} }} + \alpha \mathit{\boldsymbol{I}}){\mathit{\boldsymbol{G}}^{\rm T}}]^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm T}}\mathit{\boldsymbol{L}}$ (6)

2 正则化矩阵构造新方法

 ${\mathit{\boldsymbol{R}}_1} = \sum\limits_{i = k}^n {{\mathit{\boldsymbol{G}}_i}\mathit{\boldsymbol{G}}_i^{\rm{T}}}$ (7)

 $\mathit{\boldsymbol{M}} = {\mathit{\boldsymbol{R}}_1} + \left( {\alpha - 1} \right){\mathit{\boldsymbol{R}}_2}$ (8)

 ${\mathit{\boldsymbol{R}}_3} = \mathit{\boldsymbol{I}} + \left( {\alpha - 1} \right){\mathit{\boldsymbol{R}}_2}$ (9)

 ${{\mathit{\boldsymbol{\hat X}}}_2} = {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}} + {\mathit{\boldsymbol{R}}_3}} \right)^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}}$ (10)
3 新正则化矩阵适用性分析

 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat X}}}_2} = {{\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}} + {\mathit{\boldsymbol{R}}_3}} \right)}^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}} = }\\ {{{\left[ {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}} + \alpha \mathit{\boldsymbol{I}} + \left( {1 - \alpha } \right)\left( {\mathit{\boldsymbol{I}} - {\mathit{\boldsymbol{R}}_2}} \right)} \right]}^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}}} \end{array}$ (11)

ATA +α I = A1TA1R4=(1－α)(IR2)，根据矩阵反演公式[7]可得:

 ${{\mathit{\boldsymbol{\hat X}}}_2} = {\left( {\mathit{\boldsymbol{A}}_1^{\rm{T}}{\mathit{\boldsymbol{A}}_1} + {\mathit{\boldsymbol{R}}_4}} \right)^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}}$ (12)

 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat X}}}_2} = {{\left( {\mathit{\boldsymbol{A}}_1^{\rm{T}}{\mathit{\boldsymbol{A}}_1}} \right)}^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}} - }\\ {\left[ {{{\left( {\mathit{\boldsymbol{A}}_1^{\rm{T}}{\mathit{\boldsymbol{A}}_1}} \right)}^{ - 1}}{{\left( {{{\left( {\mathit{\boldsymbol{A}}_1^{\rm{T}}{\mathit{\boldsymbol{A}}_1}} \right)}^{ - 1}} + \mathit{\boldsymbol{R}}_4^{ - 1}} \right)}^{ - 1}}} \right]{{\left( {\mathit{\boldsymbol{A}}_1^{\rm{T}}{\mathit{\boldsymbol{A}}_1}} \right)}^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}}} \end{array}$ (13)

4.2 算例2

4.3 算例分析

5 结语

 [1] Bouhamidi A, Jbilou K, Reichel L, et al. An Extrapolated TSVD Method for Linear Discrete Ill-Posed Problems with Kronecker Structure[J]. Linear Algebra & Its Applications, 2014, 434(7): 1677-1688 (0) [2] Xu P L. Truncated SVD Methods for Discrete Linear Ill-Posed Problems[J]. Geophysical Journal of the Royal Astronomical Society, 1998, 135(2): 505-514 DOI:10.1046/j.1365-246X.1998.00652.x (0) [3] Tikhonov A N. Solution of Incorrectly Formulated Problems and the Regularization Method[J]. Soviet Math, 1963, 4: 1035-1038 (0) [4] Bell J B, Tikhonov A N, Arsenin V Y. Solutions of Ill-Posed Problems[J]. Mathematics of Computation, 1978, 32(144): 1320 DOI:10.2307/2006360 (0) [5] Hoerl A E, Kennard R W. Ridge Regression: Biased Estimation for Nonorthogonal Problems[J]. Technometrics, 1970, 12(1): 55-67 DOI:10.1080/00401706.1970.10488634 (0) [6] Hoerl A E, Kennard R W. Ridge Regression: Applications to Nonorthogonal Problems[J]. Technometrics, 1970, 12(1): 55-67 DOI:10.1080/00401706.1970.10488634 (0) [7] 林东方, 朱建军, 宋迎春. 顾及截断偏差影响的TSVD截断参数确定方法[J]. 测绘学报, 2017, 46(6): 679-688 (Lin Dongfang, Zhu Jianjun, Song Yingchun. Truncation Method for TSVD with Account of Truncation Bias[J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(6): 679-688) (0) [8] Hansen P C. Analysis of Discrete Ill-posed Problems by Means of the L-Curve[J]. SIAM Review, 1992, 34(4): 561-580 DOI:10.1137/1034115 (0) [9] Hansen P C, O' Leary D P. The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems[J]. SIAM Sci Comput, 1993, 14(6): 1487-1503 DOI:10.1137/0914086 (0) [10] 陈希孺, 王松桂. 近代回归分析——原理方法及应用[M]. 合肥: 安徽教育出版社, 1987 (Chen Xiru, Wang Songgui. Modern Regression Analysis Method and Application[M]. Hefei: Anhui Education Press, 1987) (0) [11] Golub G H, Heath M, Wahba G. Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter[J]. Technometrics, 1979, 21(2): 215-223 DOI:10.1080/00401706.1979.10489751 (0) [12] Sourbron S, Luypaert R, Schuerbeek V P, et al. Choice of the Regularization Parameter for Perfusion Quantification with MRI[J]. Physics in Medicine & Biology, 2004, 49(14): 3307 (0) [13] 崔希璋, 於宗俦, 陶本藻, 等. 广义测量平差[M]. 武汉: 武汉大学出版社, 2009 (Cui Xizhang, Yu Zongchou, Tao Benzao, et al. Generalized Surveying Adjustment[M]. Wuhan: Wuhan University Press, 2009) (0) [14] 林东方, 朱建军, 宋迎春, 等. 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, et al. Construction Method of Regularization by Singular Value Decomposition of Design Matrix[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(8): 883-889) (0) [15] 王振杰, 欧吉坤, 柳林涛. 一种解算病态问题的方法——两步解法[J]. 武汉大学学报:信息科学版, 2005, 30(9): 821-824 (Wang Zhenjie, Ou Jikun, Liu Lintao. A Method for Resolving Ill-Conditioned Problems: Two Step Solution[J]. Geomatics and Information Science of Wuhan University, 2005, 30(9): 821-824) (0) [16] 徐天河, 杨元喜. 均方误差意义下正则化解优于最小二乘解的条件[J]. 武汉大学学报:信息科学版, 2004, 29(3): 223-226 (Xu Tianhe, Yang Yuanxi. Condition of Regularization Solution Superior to LS Solution Based on MSE Principle[J]. Geomatics and Information Science of Wuhan University, 2004, 29(3): 223-226) (0) [17] 王振杰. 测量中不适定问题的正则化解法[M]. 北京: 科学出版社, 2006 (Wang Zhenjie. Research on the Regularization Solutions of Ill-Posed Problems in Geodesy[M]. Beijing: Science Press, 2006) (0) [18] 杨文采. 地球物理反演和地震层析成像[M]. 北京: 地质出版社, 1989 (Yang Wencai. Geophysical Inversion and Seismic Tomography[M]. Beijing: Geological Publishing House, 1989) (0) [19] 鲁铁定.总体最小二乘平差理论及其在测绘数据处理中的应用[D].武汉: 武汉大学, 2010 (Lu Tieding.Research on the Total Least Squares and Its Applications in Surveying Data Processing[D].Wuhan: Wuhan University, 2010) http://www.cnki.com.cn/Article/CJFDTotal-CHXB201304028.htm (0)
A New Method of Constructing Regularized Matrix
WU Guangming1     LU Tieding1,2,3     DENG Xiaoyuan1,4     QIU Dechao1
1. Faculty of Geomatics, East China University of Technology, 418 Guanglan Road, Nanchang 330013, China;
2. Key Laboratory of Watershed Ecology and Geographical Environment Monitoring, NASMG, 418 Guanglan Road, Nanchang 330013, China;
3. Key Lab for Digital Land and Resources of Jiangxi Province, 418 Guanglan Road, Nanchang 330013, China;
4. Geomatics Center of Zhejiang Province, 83 North-Baochu Road, Hangzhou 310012, China
Abstract: In parameter solving under the conditions of coefficient matrix, the rational selection of regularization parameters and regularization matrix can improve the reliability of parameter estimation. The symmetric matrix is constructed by the eigenvectors corresponding to the smaller singular values of the matrix. The diagonal matrix is constructed by the main diagonal elements of the matrix, and then a new regularization matrix is obtained by combining with the unit matrix. The experimental results show that when the regularization parameter is less than 1, the parameter estimation of this algorithm is better than the ridge estimation.
Key words: coefficient matrix; regularization matrix; singular value; mean square error; ridge estimates