顾及设计矩阵误差的AR模型新解法

doi:10.11947/j.AGCS.2017.20170004

测绘学报 ›› 2017, Vol. 46 ›› Issue (11): 1795-1801.doi: 10.11947/j.AGCS.2017.20170004

• 大地测量学与导航 • 下一篇

顾及设计矩阵误差的AR模型新解法

姚宜斌^1,2,3, 熊朝晖¹, 张豹¹, 张良¹, 孔建⁴

1. 武汉大学测绘学院, 湖北武汉 430079;
2. 武汉大学地球空间环境与大地测量教育部重点实验室, 湖北武汉 430079;
3. 地球空间信息技术协同创新中心, 湖北武汉 430079;
4. 武汉大学中国南极测绘研究中心, 湖北武汉 430079

收稿日期:2017-01-03 修回日期:2017-08-18 出版日期:2017-11-20 发布日期:2017-12-05
通讯作者: 熊朝晖 E-mail:cehui_xiong@whu.edu.cn
作者简介:姚宜斌(1976-),男,教授,主要从事测量数据处理理论与方法、GNSS空间环境学研究。E-mail:ybyao@sgg.whu.edu.cn
基金资助:
国家自然科学基金（41274022；41574028）；湖北省杰出青年科学基金（2015CFA036）

A New Method to Solving AR Model Parameters Considering Random Errors of Design Matrix

YAO Yibin^1,2,3, XIONG Zhaohui¹, ZHANG Bao¹, ZHANG Liang¹, KONG Jian⁴

1. School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China;
2. Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University, Wuhan 430079, China;
3. Collaborative Innovation Center for Geospatial Technology, Wuhan 430079, China;
4. Chinese Antarctic Center of Surveying and Mapping, Wuhan 430079, China

Received:2017-01-03 Revised:2017-08-18 Online:2017-11-20 Published:2017-12-05
Supported by:
The General Program of National Natural Science Foundation of China (Nos. 41274022; 41574028); Natural Science Foundation for Distinguished Young Scholars of Hubei Province of China (No. 2015CFA036)

摘要/Abstract

摘要：

在自回归模型求解中，设计矩阵和观测值均存在误差，传统的最小二乘法不能很好地解决这一问题。本文提出了一种顾及设计矩阵误差的AR模型新解法，通过引入虚拟观测值，使观测向量与设计矩阵不仅同源而且带误差的元素个数相同，然后通过对观测方程进行等价变换巧妙实现了在最小二乘框架下求解自回归问题。利用模拟数据及实测数据分别对新算法进行了内符合精度检验，并利用实测数据对新算法进行外符合精度检验，结果表明新算法得到的结果显著优于奇异值分解（singular value decomposition，SVD）解法及传统最小二乘解法，验证了算法的精度和有效性。

关键词: AR模型, 设计矩阵误差, 整体最小二乘, 虚拟观测值, 奇异值分解

Abstract:

The ordinary least square method could not solve the problem that the error exist both in design matrix and observation vector while compute parameter values of AR model. In this article, a new method is proposed which consider the random errors of design matrix. The source of design matrix and observation vector is same and the amount of parameters contain error can be equal by introducing virtual observation. Then, this problem could be solved under the framework of normal least square by equivalence transformation of observation equation. The result of this new method is superior to SVD method and normal least square method by simulation date and observation data which verify the feasibility and effectiveness of this method.

Key words: AR model, design matrix error, TLS, virtual observations, SVD method

中图分类号:

P228

姚宜斌, 熊朝晖, 张豹, 张良, 孔建. 顾及设计矩阵误差的AR模型新解法[J]. 测绘学报, 2017, 46(11): 1795-1801.

YAO Yibin, XIONG Zhaohui, ZHANG Bao, ZHANG Liang, KONG Jian. A New Method to Solving AR Model Parameters Considering Random Errors of Design Matrix[J]. Acta Geodaetica et Cartographica Sinica, 2017, 46(11): 1795-1801.

参考文献

[1] ADCOCK R J.Note on the Method of Least Squares[J].Analyst,1877,4(6):183-184.
[2] GOLUB G H,VAN LOAN C F.An Analysis of the Total Least Squares Problem[J].SIAM Journal on Numerical Analysis,1980,17(6):883-893.
[3] VAN HUFFEL S,ZHA Hongyuan.An Efficient Total Least Squares Algorithm Based on a Rank-revealing Two-sided Orthogonal Decomposition[J].Numerical Algorithms,1993,4(1):101-133.
[4] 姚宜斌,孔建.顾及设计矩阵随机误差的最小二乘组合新解法[J].武汉大学学报(信息科学版),2014,39(9):1028-1032. YAO Yibin,KONG Jian.A New Combined LS Method Considering Random Errors of Design Matrix[J].Geomatics and Information Science of Wuhan University,2014,39(9):1028-1032.
[5] 曾文宪,方兴,刘经南,等.附有不等式约束的加权整体最小二乘算法[J].测绘学报,2014,43(10):1013-1018.DOI:10.13485/j.cnki.11-2089.2014.0173. ZENG Wenxian,FANG Xing,LIU Jingnan,et al.Weighted Total Least Squares Algorithm with Inequality Constraints[J].Acta Geodaetica et Cartographica Sinica,2014,43(10):1013-1018.DOI:10.13485/j.cnki.11-2089.2014.0173.
[6] SCHAFFRIN B,WIESER A.On Weighted Total Least-squares Adjustment for Linear Regression[J].Journal of Geodesy,2008,82(7):415-421.
[7] SCHUERMANS M,MARKOVSKY I,VAN HUFFEL S.An Adapted Version of the Element-wise Weighted Total Least Squares Method for Applications in Chemometrics[J].Chemometrics and Intelligent Laboratory Systems,2007,85(1):40-46.
[8] VAN HUFFEL S,VANDEWALLE J.Analysis and Properties of the Generalized Total Least Squares Problem AX≈B When Some or All Columns in A are Subject to Error[J].SIAM Journal on Matrix Analysis and Applications,1989,10(3):294-315.
[9] XU Peiliang,LIU Jingnan,SHI Chuang.Total Least Squares Adjustment in Partial Errors-in-variables Models:Algorithm and Statistical Analysis[J].Journal of Geodesy,2012,86(8):661-675.
[10] FANG X.Weighted Total Least Squares Solutions for Applications in Geodesy[D].Hannover,Germany:Leibniz University,2011.
[11] 方兴,曾文宪,刘经南,等.三维坐标转换的通用整体最小二乘算法[J].测绘学报,2014,43(11):1139-1143.DOI:10.13485/j.cnki.11-2089.2014.0193. FANG Xing,ZENG Wenxian,LIU Jingnan,et al.A General Total Least Squares Algorithm for Three-dimensional Coordinate Transformations[J].Acta Geodaetica et Cartographica Sinica,2014,43(11):1139-1143.DOI:10.13485/j.cnki.11-2089.2014.0193.
[12] 赵俊,归庆明.部分变量误差模型的整体抗差最小二乘估计[J].测绘学报,2016,45(5):552-559.DOI:10.11947/j.AGCS.2016.20150374. ZHAO Jun,GUI Qingming.Total Robustified Least Squares Estimation in Partial Errors-in-variables Model[J].Acta Geodaetica et Cartographica Sinica,2016,45(5):552-559.DOI:10.11947/j.AGCS.2016.20150374.
[13] 王乐洋,余航,陈晓勇.Partial EIV模型的解法[J].测绘学报,2016,45(1):22-29.DOI:10.11947/j.AGCS.2016.20140560. WANG Leyang,YU Hang,CHEN Xiaoyong.An Algorithm for Partial EIV Model[J].Acta Geodaetica et Cartographica Sinica,2016,45(1):22-29.DOI:10.11947/j.AGCS.2016.20140560.
[14] 陶叶青,高井祥,姚一飞.基于中位数法的抗差总体最小二乘估计[J].测绘学报,2016,45(3):297-301.DOI:10.11947/j.AGCS.2016.20150234. TAO Yeqing,GAO Jingxiang,YAO Yifei.Solution for Robust Total Least Squares Estimation Based on Median Method[J].Acta Geodaetica et Cartographica Sinica,2016,45(3):297-301.DOI:10.11947/j.AGCS.2016.20150234.
[15] 吴富梅,杨元喜.基于高阶AR模型的陀螺随机漂移模型[J].测绘学报,2007,36(4):389-394. WU Fumei,YANG Yuanxi.Gyroscope Random Drift Model Based on the Higher-order AR Model[J].Acta Geodaetica et Cartographica Sinica,2007,36(4):389-394.
[16] 潘国荣,刘大杰.顾及邻近点变形因素项的动态模型辨识及预测[J].测绘学报,2001,30(1):32-35. PAN Guorong,LIU Dajie.Dynamic Modeling Identification and Predication in Consideration of the Adjacent Point Deformation[J].Acta Geodaetica et Cartographica Sinica,2001,30(1):32-35.
[17] 杨元喜,崔先强.动态定位有色噪声影响函数——以一阶AR模型为例[J].测绘学报,2003,32(1):6-10. YANG Yuanxi,CUI Xianqiang.Influence Functions of Colored Noises on Kinematic Positioning:Taking the AR Model of First Class as an Example[J].Acta Geodaetica et Cartographica Sinica,2003,32(1):6-10.
[18] 叶志伟,尹晖,张守建.AR模型谱在超导重力数据信号检测中的分析研究[J].武汉大学学报(信息科学版),2007,32(6):536-539. YE Zhiwei,YIN Hui,ZHANG Shoujian.Using AR Model Spectrum Algorithms to Detect Superconducting Gravimetric Signals[J].Geomatics and Information Science of Wuhan University,2007,32(6):536-539.
[19] 张昊,王琪洁,朱建军,等.对钱德勒参数进行时变修正的CLS+AR模型在极移预测中的应用[J].武汉大学学报(信息科学版),2012,37(3):286-289. ZHANG Hao,WANG Qijie,ZHU Jianjun,et al.Application of CLS+AR Model Polar Motion to Prediction Based on Time-varying Parameters Correction of Chandler Wobble[J].Geomatics and Information Science of Wuhan University,2012,37(3):286-289.
[20] 王乐洋,许才军,鲁铁定.边长变化反演应变参数的总体最小二乘方法[J].武汉大学学报(信息科学版),2010,35(2):181-184. WANG Leyang,XU Caijun,LU Tieding.Inversion of Strain Parameter Using Distance Changes Based on Total Least Squares[J].Geomatics and Information Science of Wuhan University,2010,35(2):181-184.
[21] 魏二虎,殷志祥,李广文,等.虚拟观测值法在三维坐标转换中的应用研究[J].武汉大学学报(信息科学版),2014,39(2):152-156. WEI Erhu,YIN Zhixiang,LI Guangwen,et al.On 3D Coordinate Transformations with Virtual Observation Method[J].Geomatics and Information Science of Wuhan University,2014,39(2):152-156.
[22] 姚宜斌,黄书华,孔建,等.空间直线拟合的整体最小二乘算法[J].武汉大学学报(信息科学版),2014,39(5):571-574. YAO Yibin,HUANG Shuhua,KONG Jian,et al.Total Least Squares Algorithm for Fitting Spatial Straight Lines[J].Geomatics and Information Science of Wuhan University,2014,39(5):571-574.
[23] CRYER J D,CHAN K S.时间序列分析及应用[M].潘红宇,译.北京:机械工业出版社,2011. CRYER J D,CHAN K S.Time Series Analysis with Applications in R[M].PAN Hongyu,tran.Beijing:China Machine Press,2011.
[24] 姚宜斌,黄书华,陈家君.求解自回归模型参数的整体最小二乘新方法[J].武汉大学学报(信息科学版),2014,39(12):1463-1466. YAO Yibin,HUANG Shuhua,CHEN Jiajun.A New Method of TLS to Solving the Autoregressive Model Parameter[J].Geomatics and Information Science of Wuhan University,2014,39(12):1463-1466.
[25] 王新洲,陶本藻,邱卫宁,等.高等测量平差[M].北京:测绘出版社,2013. WANG Xinzhou,TAO Benzao,QIU Weining,et al.Advanced Surveying Adjustment[M].Beijing:Mapping Publishing Company,2013.

顾及设计矩阵误差的AR模型新解法

A New Method to Solving AR Model Parameters Considering Random Errors of Design Matrix

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	胡川, 方兴, 赵立都. 融合正交几何信息的非线性等式约束整体最小二乘平差及迭代算法[J]. 测绘学报, 2020, 49(7): 816-823.
[2]	邓兴升, 彭思淳, 游扬声. 加权整体最小二乘EIO模型与算法[J]. 测绘学报, 2019, 48(7): 926-930.
[3]	王乐洋, 余航, 邹传义, 鲁铁定. 加权整体最小二乘EIV模型与算法——与“加权整体最小二乘EIO模型与算法”一文的讨论[J]. 测绘学报, 2019, 48(7): 931-937.
[4]	韩松辉, 张国超, 张宁, 朱建青. AR模型中AO类异常值探测及其在GPS卫星钟差预报中的应用[J]. 测绘学报, 2019, 48(10): 1225-1235.
[5]	余航, 王坚, 王乐洋, 宁一鹏, 刘志平. 动态EIV模型及其总体卡尔曼滤波方法[J]. 测绘学报, 2018, 47(4): 480-489.
[6]	董杨, 范大昭, 纪松, 雷蓉. 主成分分析的匹配点对提纯方法[J]. 测绘学报, 2017, 46(2): 228-236.
[7]	谢东海, 钟若飞, 吴俣, 符晗, 黄小川, 孙振兴. 球面全景影像相对定向与精度验证[J]. 测绘学报, 2017, 46(11): 1822-1829.
[8]	曾文宪, 方兴, 刘经南, 姚宜斌. 通用EIV平差模型及其加权整体最小二乘估计[J]. 测绘学报, 2016, 45(8): 890-894.
[9]	赵俊, 归庆明. 部分变量误差模型的整体抗差最小二乘估计[J]. 测绘学报, 2016, 45(5): 552-559.
[10]	方兴, 曾文宪, 刘经南, 姚宜斌, 王勇. 基于非线性高斯-赫尔默特模型的混合整体最小二乘估计[J]. 测绘学报, 2016, 45(3): 291-296.
[11]	王彬, 李建成, 高井祥, 刘超. 抗差加权整体最小二乘模型的牛顿-高斯算法[J]. 测绘学报, 2015, 44(6): 602-608.
[12]	宋迎春, 谢雪梅, 陈晓林. 不确定性平差模型的平差准则与解算方法[J]. 测绘学报, 2015, 44(2): 135-141.
[13]	王斐吴晓莉周田李宇翔. 不同Klobuchar模型参数的性能比较[J]. 测绘学报, 2014, 43(11): 1151-1157.
[14]	方兴曾文宪刘经南姚宜斌. 三维坐标转换的通用整体最小二乘算法[J]. 测绘学报, 2014, 43(11): 1139-1143.
[15]	曾文宪方兴刘经南. 附有不等式约束的加权整体最小二乘算法[J]. 测绘学报, 2014, 43(10): 1013-1018.