﻿ 非等间距GM(1, 1)模型的总体最小二乘算法及其病态问题
 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): 45-50  DOI: 10.14075/j.jgg.2019.01.009

### 引用本文

TAO Wuyong, HUA Xianghong, LU Tieding, et al. A Total Least Squares Algorithm for Non-Equidistant GM(1, 1) Model and Its Ill-Posed Problem[J]. Journal of Geodesy and Geodynamics, 2019, 39(1): 45-50.

### Foundation support

National Natural Science Foundation of China, No.41674005, 41374007, 41464001, 41501502; Science Research Foundation of Postdoctors Innovation and Practice Base of Wuhan Geomatics Institute, No. WGF 2016002;Science and Technology Project of the Education Department of Jiangxi Province, No. KJLD12077, GJJ13457; Natural Science Foundation of Jiangxi Province, No. 2017BAB203032; National Key Research and Development Program of China, No. 2016YFB0501405, 2016YFB0502601-04; Open Fund of Key Laboratory for Digital Land and Resources of Jiangxi Province, No. DLLJ201702; Open Fund of Key Laboratory of Precise Engineering and Industry Surveying, NASMG, No. PF2017-9.

### Corresponding author

HUA Xianghong, PhD, professor, PhD supervisor, majors in laser scanning and indoor position, E-mail:xhhua@sgg.whu.edu.cn.

### 第一作者简介

TAO Wuyong, PhD candidate, majors in laser scanning and data processing, E-mail: 781873533@qq.com.

### 文章历史

1. 武汉大学测绘学院，武汉市珞喻路129号，430079;
2. 东华理工大学测绘工程学院，南昌市广兰大道418号，330013;
3. 武汉理工大学资源与环境工程学院，武汉市珞狮路122号，430070

1 非等间距GM(1, 1)模型的病态问题 1.1 非等间距GM(1, 1)模型

 ${\mathit{\boldsymbol{X}}^{\left( 0 \right)}} = {\left[ {{x^{\left( 0 \right)}}\left( {{t_1}} \right),{x^{\left( 0 \right)}}\left( {{t_2}} \right), \cdots ,{x^{\left( 0 \right)}}\left( {{t_n}} \right)} \right]^{\rm{T}}}$ (1)

 ${\mathit{\boldsymbol{X}}^{\left( 1 \right)}} = {\left[ {{x^{\left( 1 \right)}}\left( {{t_1}} \right),{x^{\left( 1 \right)}}\left( {{t_2}} \right), \cdots ,{x^{\left( 1 \right)}}\left( {{t_n}} \right)} \right]^{\rm{T}}}$ (2)

 $\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} { - {z^{\left( 1 \right)}}\left( {{t_2}} \right)}&1\\ { - {z^{\left( 1 \right)}}\left( {{t_3}} \right)}&1\\ \vdots&\vdots \\ { - {z^{\left( 1 \right)}}\left( {{t_n}} \right)}&1 \end{array}} \right],\mathit{\boldsymbol{L}} = \left[ {\begin{array}{*{20}{c}} {{x^{\left( 0 \right)}}\left( {{t_2}} \right)}\\ {{x^{\left( 0 \right)}}\left( {{t_3}} \right)}\\ \vdots \\ {{x^{\left( 0 \right)}}\left( {{t_n}} \right)} \end{array}} \right]$ (3)

 $\mathit{\boldsymbol{\hat \xi }} = {\left[ {\begin{array}{*{20}{c}} {\hat a}&{\hat u} \end{array}} \right]^{\rm{T}}} = {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}}} \right)^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{L}}$ (4)

 ${\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {\sum\limits_{i = 2}^n {{{\left( {{z^{\left( 1 \right)}}\left( {{t_i}} \right)} \right)}^2}} }&{ - \sum\limits_{i = 2}^n {{z^{\left( 1 \right)}}\left( {{t_i}} \right)} }\\ { - \sum\limits_{i = 2}^n {{z^{\left( 1 \right)}}\left( {{t_i}} \right)} }&{n - 1} \end{array}} \right]$ (5)
1.2 病态问题

 $\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} { - {z^{\left( 1 \right)}}\left( {{t_2}} \right)}&c\\ { - {z^{\left( 1 \right)}}\left( {{t_3}} \right)}&c\\ \vdots&\vdots \\ { - {z^{\left( 1 \right)}}\left( {{t_n}} \right)}&c \end{array}} \right]$ (6)

3) 重复步骤2)，直至$\left\| {{{\mathit{\boldsymbol{\hat \xi }}}_{(i + 1)}} - {{\mathit{\boldsymbol{\hat \xi }}}_{(i)}}} \right\| \le \varepsilon$(ε为一极小值)，迭代结束；

4) 按式(20)计算单位权方差。

3 工程应用

3.1 确定c

 图 1 c值与条件数的曲线 Fig. 1 The curve graph of values of c and condition number

3.2 改变c值对计算结果的影响

 $\begin{array}{*{20}{c}} {{{\hat X}^{\left( 0 \right)}}\left( {{t_{i + 1}}} \right) = \frac{1}{{\Delta {t_{i + 1}}}}\left( {1 - {{\rm{e}}^{a\Delta {t_{i + 1}}}}} \right)}\\ {\left( {{x^{\left( 0 \right)}}\left( {{t_1}} \right) - u/a} \right){{\rm{e}}^{ - a\left( {{t_{i + 1}} - {t_1}} \right)}}} \end{array}$ (21)

3.3 总体最小二乘算法验证

1) 最小二乘(LS)；

2) 总体最小二乘(TLS)；

3) 加权总体最小二乘[17](WTLS)，系数矩阵协因数为QA=MMT，观测向量协因数阵为QL=NNT=In-1

4) 本文TLS算法。

1) LS和TLS计算结果最差，其计算得到的平均残差要大于WTLS和本文TLS，LS忽略了系数矩阵的误差，而TLS则将系数矩阵的常数项也参与了误差分配，因此采用LS和TLS求解非等间距GM(1, 1)模型都是不合理的。

2) 比较WTLS和本文TLS计算结果可以看出，本文TLS计算得到的平均残差都要小于WTLS结果，预测结果与实测数据最相符。WTLS通过协因数传播定律确定系数矩阵的协因数阵，因此确定的协因数阵顾及了系数矩阵各随机项的相关性，可以保证系数矩阵A中相同的元素有相同的改正数，但不能保证系数矩阵A与观测向量L中相同元素有相同的改正数，这与实际理论不符，因为系数矩阵A与观测向量L误差同源。而本文TLS算法则可以保证这一点，因此本文TLS更加合理。

4 结语

 [1] 陈明东, 王兰生.边坡变形破坏的灰色预报方法[C]//全国第三次工程地质大会论文选集.成都: 成都科技大学出版社, 1988 (Chen Mingdong, Wang Lansheng. The GM Model in Deformation and Failure of Slop[C]// The Selected Papers from the Third National Engineering Conference. Chengdu: Chengdu University of Science and Technology Press, 1988) http://www.wanfangdata.com.cn/details/detail.do?_type=conference&id=136231 (0) [2] 汪凡, 赵军. 基于灰色关联模型和主成分分析的上市公司绩效评价研究[J]. 商业经济, 2011(5): 110-111 (Wang Fan, Zhao Jun. The Research of Performance Evaluation of Listed Companies Based on Grey Relation Model and Principal Component Analysis[J]. Commercial Economy, 2011(5): 110-111) (0) [3] 王钟羡, 吴春笃, 史雪荣. 非等间距序列的灰色模型[J]. 数学的时间与认识, 2003, 33(10): 16-20 (Wang Zhongxian, Wu Chundu, Shi Xuerong. A Grey Model for Non-Equidistant Sequence[J]. Mathematics in Practice and Theory, 2003, 33(10): 16-20) (0) [4] 李克昭, 李志伟, 丁安民, 等. 灰线性加权非等距GM(1, 1)形变预测模型[J]. 大地测量与地球动力学报, 2016, 36(6): 513-516 (Li Kezhao, Li Zhiwei, Ding Anmin, et al. Deformation Prediction Model of Grey Line Weighted Non-Equidistant GM(1, 1)[J]. Journal of Geodesy and Geodynamics, 2016, 36(6): 513-516) (0) [5] 陈鹏宇. 非等间距GM(1, 1)模型在沉降预测中的应用探讨[J]. 大地测量与地球动力学, 2017, 37(7): 709-714 (Chen Pengyu. Discussion of the Application of Non-Equidistant GM(1, 1) Model in Subsidence Prediction[J]. Journal of Geodesy and Geodynamics, 2017, 37(7): 709-714) (0) [6] 冯健, 花向红, 王刘准. 整体最小二乘的GM(1, 1)模型在高铁中的应用研究[J]. 测绘地理信息, 2014, 39(1): 64-66 (Feng Jian, Hua Xianghong, Wang Liuzhun. The Research of Grey Model in Total Least Squares in High-Speed Rail[J]. Surveying and Mapping Geographic Information, 2014, 39(1): 64-66) (0) [7] 袁豹, 岳东杰, 李成仁. 基于总体最小二乘的改进GM(1, 1)模型及其在建筑物沉降预测中的应用[J]. 测绘工程, 2013, 22(3): 52-55 (Yuan Bao, Yue Dongjie, Li Chengren. The Improvement Grey Model Based on Total Least Squares and Its Application in Settlement Prediction of Building[J]. Surveying and Mapping Engineering, 2013, 22(3): 52-55 DOI:10.3969/j.issn.1006-7949.2013.03.014) (0) [8] 陶武勇, 鲁铁定, 吴飞. 求解GM(1, 1)模型新总体最小二乘算法[J]. 测绘科学技术学报, 2016, 33(5): 476-479 (Tao Wuyong, Lu Tieding, Wu Fei. A New Total Least Squares Algorithms for Solving Grey Model[J]. Journal of Geomatics Science and Technology, 2016, 33(5): 476-479) (0) [9] 邓聚龙. 灰理论基础[M]. 武汉: 华中科技大学出版社, 2002 (Deng Julong. Grey Theory Basis[M]. Wuhan: Huzhong University of Science and Technology Press, 2002) (0) [10] 吴正鹏, 李波, 张友萍, 等. GM(1, 1)模型的病态问题研究[J]. 中国传媒大学学报:自然科学版, 2011, 18(4): 31-34 (Wu Zhengpeng, Li Bo, Zhang Youping, et al. Study on the Morbidity Problem in Grey Model[J]. Journal of Communication University of China :Science and Technology, 2011, 18(4): 31-34 DOI:10.3969/j.issn.1673-4793.2011.04.006) (0) [11] 郑照宁, 武玉英, 包涵龄. GM模型的病态性问题[J]. 中国管理科学, 2001, 9(5): 38-44 (Zheng Zhaoning, Wu Yuying, Bao Hanling. The Morbidity Problem in GM Model[J]. Chinese Journal of Management Science, 2001, 9(5): 38-44 DOI:10.3321/j.issn:1003-207X.2001.05.006) (0) [12] Golub G H, Loan V C F. An Analysis of the Total Least Squares Problem[J]. SIAM Journal on Numerical Analysis, 1980, 17(6): 883-893 DOI:10.1137/0717073 (0) [13] 唐利民. GM(1, 1)病态问题求解的调整计量单位法[J]. 武汉大学学报:信息科学版, 2014, 39(9): 1038-1042 (Tang Limin. Adjust Measurement Unit Algorithm for Ill-Posed Problem of GM(1, 1) Model[J]. Geomatics and Information Science of Wuhan University, 2014, 39(9): 1038-1042) (0) [14] Xu P L, Liu J N, Shi C. Total Least Squares Adjustment in Partial Errors-in-Variables: Algorithm and Statisitical Analysis[J]. Journal of Geodesy, 2012, 86(8): 661-675 DOI:10.1007/s00190-012-0552-9 (0) [15] 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. Regularization of Ill-Posed Problem in Surveying[M]. Beijing: Science Press, 2006) (0) [16] 荆科, 刘业政. GM(1, 1)模型的病态问题再研究[J]. 控制与决策, 2016, 31(5): 869-874 (Jing Ke, Liu Yezheng. Morbidity Problem of GM(1, 1) Model[J]. Control and Decision, 2016, 31(5): 869-874) (0) [17] Shen Y Z, Li B F, Chen Y. An Iterative Solution of Weighted Total Least Squares Adjustment[J]. Journal of Geodesy, 2012, 86(5): 359-367 DOI:10.1007/s00190-011-0524-5 (0)
A Total Least Squares Algorithm for Non-Equidistant GM(1, 1) Model and Its Ill-Posed Problem
TAO Wuyong1     HUA Xianghong1     LU Tieding2     CHEN Xijiang3     ZHANG Wei1
1. School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China;
2. Faculty of Geomatics, East China University of Technology, 418 Guanglan Road, Nanchang 330013, China;
3. School of Resources and Environment Engineering, Wuhan University of Technology, 122 Luoshi Road, Wuhan 430070, China
Abstract: In non-equidistant GM(1, 1) model there are constant terms without error and random terms with errors in the coefficient matrix. The errors of the coefficient matrix and observation vector are from the same source; the same elements are in the coefficient matrix and observation vector. These same elements ought to have the same corrected value. Therefore, a total least squares algorithm that is suitable to solve non-equidistant GM(1, 1) model is deduced in this paper. The ill-posed problem in the non-equidistant GM(1, 1) model is taken into consideration, which has an influence on the stability of the calculated results of total least squares. The method, which is to multiply the constant column in coefficient matrix by a constant, is proposed to alleviate the ill-posed problem.
Key words: total least squares; ill-posed problem; non-equidistant GM(1, 1) model; condition number; stability