Double Truncated Singular Value Estimation Based on Signal-to ...

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E-mail Phone Title Content Verification Code LI Hao, GU Yongwei, HAN Songhui. Double Truncated Singular Value Estimation Based on Signal-to-Noise Ratio Test[J]. Geomatics and Information Science of Wuhan University, 2019, 44(2): 228-232, 239. doi: 10.13203/j.whugis20170051 Citation: LI Hao, GU Yongwei, HAN Songhui. Double Truncated Singular Value Estimation Based on Signal-to-Noise Ratio Test[J]. Geomatics and Information Science of Wuhan University , 2019, 44(2): 228-232, 239. doi: 10.13203/j.whugis20170051

Author Bio:

LI Hao, postgraduate, specializes in applied statistics. E-mail: lihao6618@163.com

Received Date: 2017-12-19

Publish Date: 2019-02-05

Abstract

A double truncated singular value estimation based on signal-to-noise ratio test is proposed by combining the measurement results of the harm degree that complex common linear exerts on parameter estimation and the truncated singular value decomposition(TSVD) estimation. Based on the signal to noise ratio(SNR) test, all the parameters are divided into two parts, according to the estimated SNR value of each parameter's least square estimation. The TSVD estimation of these two parts is truncated at different intensities. We choose a smaller truncated parameter for parameters which suffer more from complex common linear, and a bigger truncated parameter for parameters which suffer less from complex common linear, thus minimize the deviation and reduce the variance of parameter estimation effectively. This new method is applied to the simulation example of the GEO satellite orbit determination. The experimental results show that the new method is more accurate. Keywords: signal to noise ratio, complex common linear, double truncated singular value, GEO orbit determination 归庆明, 李国重.岭-压缩组合估计及其在测量平差中的应用[J].大地测量与地球动力学, 2002, 22(1):16-21 http://www.cnki.com.cn/Article/CJFDTOTAL-DKXB200201003.htm

Gui Qingming, Li Guozhong. Combining Ridge and Shrunken Estimator and Its Applications in Geodetic Adjustment[J]. Journal of Geodesy and Geodynamics, 2002, 22(1):16-21 http://www.cnki.com.cn/Article/CJFDTOTAL-DKXB200201003.htm 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 Problems in Surveying[M]. Beijing:Science Press, 2006 顾勇为, 归庆明. TSVD解算中选择截断参数的新方法[J].测绘科学技术学报, 2010, 27(3):176-179 doi: 10.3969/j.issn.1673-6338.2010.03.006

Gu Yongwei, Gui Qingming. A New Method to Select Truncated Parameter in TSVD[J].Journal of Geomatics Science and Technology, 2010, 27(3):176-179 doi: 10.3969/j.issn.1673-6338.2010.03.006 陈希孺, 王松桂.近代回归分析[M].合肥:安徽教育出版社, 1987

Chen Xiru, Wang Songgui. Advanced Regression Analysis[M].Hefei:Anhui Education Press, 1987 卢秀山.病态系统分析理论及其在测量中的应用[D].武汉: 中国科学院测量与地球物理研究所, 1999 http://www.chinasmp.com/BookStore/BookDetail.aspx?BookID=76

Lu Xiushan.Analysis Theory on Ill-Conditioned System with Application in Surveying[D].Wuhan: Institute of Geodesy and Geophysics, Chinese Academy of Sciences, 1999 http://www.chinasmp.com/BookStore/BookDetail.aspx?BookID=76 归庆明, 郭建锋, 边少锋.基于特征系统的病态性诊断[J].测绘科学, 2002, 27(2):13-15, 19 doi: 10.3771/j.issn.1009-2307.2002.02.004

Gui Qingming, Guo Jianfeng, Bian Shaofeng. Ill-Conditioning Diagnostics Based on Eigensystem[J].Science of Surveying and Mapping, 2002, 27(2):13-15, 19 doi: 10.3771/j.issn.1009-2307.2002.02.004 郭建锋.测量平差系统病态性的诊断与处理[D].郑州: 信息工程大学, 2002 http://cdmd.cnki.com.cn/Article/CDMD-90008-2002124626.htm

Guo Jianfeng.Study on Diagnostics and Processing of Ill-Conditioning Adjustment System[D]. Zhengzhou: Information Engineering University, 2002 http://cdmd.cnki.com.cn/Article/CDMD-90008-2002124626.htm Moran K M, Cicci D A. Sensitivity of Ridge-type Estimation Methods to Condition Number[J]. Applied Mathematics and Computation, 2000, 112(1):143-159 doi: 10.1016/S0096-3003(99)00095-8 归庆明, 姚绍文, 顾勇为, 等.诊断复共线性的条件指标-方差分解比法[J].测绘学报, 2006, 35(3):210-214 doi: 10.3321/j.issn:1001-1595.2006.03.003

Gui Qingming, Yao Shaowen, Gu Yongwei, et al. A New Method to Diagnose Multicollinearity Based on Condition Index and Variance Decomposition Proportion[J].Acta Geodaetica et Cartographica Sinica, 2006, 35(3):210-214 doi: 10.3321/j.issn:1001-1595.2006.03.003 张磊, 顾勇为, 归庆明, 等.基于复共线性诊断和度量的有偏估计[J].大地测量与地球动力学, 2007, 27(2):99-102 http://d.old.wanfangdata.com.cn/Periodical/dkxbydz200702019

Zhang Lei, Gu Yongwei, Gui Qingming et al. Biased Estimator Based on Diagnosis and Measure of Multicollinearity[J]. Journal of Geodesy and Geodynamics, 2007, 27(2):99-102 http://d.old.wanfangdata.com.cn/Periodical/dkxbydz200702019 顾勇为.基于复共线性诊断的正则化方法及其在大地测量中的应用[D].郑州: 信息工程大学, 2010

Gu Yongwei. Regularization Methods Based on Multicollinearity Diagnosis and Their Applications to Geodesy[D]. Zhengzhou: Information Enginee-ring University, 2010 韩松辉, 杜兰, 归庆明, 等.诊断复共线性的特征分析法及其在GEO定轨中的应用[J].测绘学报, 2013, 42(1):19-26 http://d.old.wanfangdata.com.cn/Periodical/chxb201301004

Han Songhui, Du Lan, Gui Qingming, et al. Characteristics Analysis Approach for Multicollinearity Diagnosis and Its Applications in Orbit Determination of GEO Satellites[J]. Acta Geodaetica et Cartographica Sinica, 2013, 42(1):19-26 http://d.old.wanfangdata.com.cn/Periodical/chxb201301004 郭杰, 归庆明, 郭淑妹, 等.利用复共线性诊断确定偏差矫正项的截断型岭估计[J].武汉大学学报·信息科学版, 2015, 40(6):785-789 http://ch.whu.edu.cn/CN/abstract/abstract3277.shtml

Guo Jie, Gui Qingming, Guo Shumei, et al. Truncation Ridge Estimation Based on the Biased-Corrected Theory by Multicollinearity Diagnosis[J]. Geomati-cs and Information Science of Wuhan University, 2015, 40(6):785-789 http://ch.whu.edu.cn/CN/abstract/abstract3277.shtml Belsley D A. Conditioning Diagnostics:Collinearity and Weak Data in Regression[M]. New York:Wiley, 1991 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 http://ch.whu.edu.cn/CN/abstract/abstract2283.shtml

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 http://ch.whu.edu.cn/CN/abstract/abstract2283.shtml

GEO orbit determination

Abstract: A double truncated singular value estimation based on signal-to-noise ratio test is proposed by combining the measurement results of the harm degree that complex common linear exerts on parameter estimation and the truncated singular value decomposition(TSVD) estimation. Based on the signal to noise ratio(SNR) test, all the parameters are divided into two parts, according to the estimated SNR value of each parameter's least square estimation. The TSVD estimation of these two parts is truncated at different intensities. We choose a smaller truncated parameter for parameters which suffer more from complex common linear, and a bigger truncated parameter for parameters which suffer less from complex common linear, thus minimize the deviation and reduce the variance of parameter estimation effectively. This new method is applied to the simulation example of the GEO satellite orbit determination. The experimental results show that the new method is more accurate.

LI Hao, GU Yongwei, HAN Songhui. Double Truncated Singular Value Estimation Based on Signal-to-Noise Ratio Test[J]. Geomatics and Information Science of Wuhan University, 2019, 44(2): 228-232, 239. doi: 10.13203/j.whugis20170051 Citation: LI Hao, GU Yongwei, HAN Songhui. Double Truncated Singular Value Estimation Based on Signal-to-Noise Ratio Test[J]. Geomatics and Information Science of Wuhan University , 2019, 44(2): 228-232, 239. doi: 10.13203/j.whugis20170051 如果设计矩阵存在复共线性，很小的观测误差就会造成估计结果严重偏离真值。为了得到精确、可靠的平差结果，必须削弱和克服设计矩阵复共线性对参数估计的不良影响。克服复共线性危害的前提是准确地找到设计阵中存在的复共线性关系，即进行复共线性诊断。到目前为止，国内外学者已经提出10余种复共线性诊断方法 ^{[

5

-

10

]} ，包括条件数法、方差扩大因子法、特征分析法、条件指标-方差分解比法等。许多学者在将复共线性诊断和处理相结合方面做了深入的研究 ^{[

11

-

14

]} 。文献[ 11 ]基于复共线性诊断和度量的结果, 提出了测量平差Gauss-Markov模型参数的部分岭估计。文献[ 13 ]利用特征分析法确定法矩阵中的复共线性关系个数, 以及具体存在于法矩阵的哪几列中, 据此提出了双 K 型岭估计, 并将其应用于GEO卫星定轨中。文献[ 14 ]首先深入分析了岭估计和截断奇异值(truncated singular value decomposition, TSVD)估计存在的缺陷和不足，结合对复共线性诊断、度量获得的重要信息，引入适当的滤波因子设计基于复共线性诊断的截断型岭估计, 计算其偏差并从中去掉偏差, 得到偏差矫正的截断型岭估计。

值得注意的是，虽然复共线性经常令测量平差受挫，造成参数估计不准确，并且这种危害有时是灾难性的，然而复共线性也并非总是过分伤害到了每个参数的估计，有时复共线性虽然引起了参数估计的方差膨胀，但未膨胀到严重掩盖参数真值的程度。由此看来，复共线性对参数估计的危害是潜在的，但每个参数估计受到复共线性的危害的大小却不尽相同。因此，需要对复共线性对参数估计危害大小作出合适的度量，由此可以客观评价采用最小二乘(least square, LS)估计作为测量平差模型中未知参数的估计的合理性，以及科学地判断对LS估计作出进一步修改的必要性。Belsley系统地介绍了信号-噪声比(简称信噪比)的概念和思想 ^{[

15

]} 。文献[ 12 ]针对测量平差实际，引进和发展信噪比的概念，运用该方法能够对每个参数的估计是否受到复共线性的危害及其危害的大小作出较为合理的判断。

利用文献[ 12 ]的检验法则：当 F _i ≤ F _{1,

n

-

r

,

χ

₁
²
(

γ

)} ( ω )时，认为 X _i 的信噪比估计量 F _i 较小，复共线性对相应参数估计$\hat X_{{\text{LS}}}^i$的危害比较严重，其估计效果不好；当 F _i > F _{1,

n

-

r

,

χ

₁
²
(

γ

)} ( ω )时，认为 X _i 的信噪比估计量 F _i 较大，复共线性对相应参数估计$\hat X_{{\text{LS}}}^i$的危害比较小，其估计效果较好。其中 ω 为显著性水平； F _{1,

n

-

r

,

χ

₁
²
(

γ

)} ( ω )是非中心 F 分布 F _{1,

n

-

r

,

χ

₁
²
(

γ

)} 的上侧 ω 分位点。实际应用中，判断信噪比估计量大小的阈值的选取可以根据具体情况灵活确定，不必拘泥于由显著性水平所确定的分位点，具体选取方法可参阅文献[ 12 ]。

通过计算信噪比估计量，可将待估参数分为两部分 X =[ X _a ^T X _b ^T ] ^T ，其中信噪比估计量较大的 s 个参数为 X _a =[ X ₁ X ₂ … X _s ] ^T ，它们受复共线性的危害较小；信噪比估计量较小的 t － s 个参数为 X _b =[ X _{s

+1} X _{s

+2} … X _t ] ^T ，它们受复共线性的危害较大。根据信噪比检验的结果，对信噪比估计量较小的参数截断多一些，对信噪比估计量较大的参数截断少一些。据此，下面构造基于复共线性诊断的双截断奇异值估计(double truncated singular value estimation based on signal-to-noise ratio test，DTS)：

式中， X ₀ ⁱ 为真值 X _i 的对比值，称式(10)为第 i 个参数 X _i 的TSVD估计$\hat X_{{\text{TSVD}}}^i$的信噪比估计量。当反映各个待估参数 X ₁ … X _t 估计效果差异的信噪比估计量 T ₁ … T _t 之间的差异很大时，说明总体估计质量不好。分析式(10)， t 个信噪比估计量 T _i 的差异性体现在其分子上，即 t 个${\left( {\hat X_{{\text{TSVD}}}^i-X_0^i} \right)^2}/{\text{Var}}\left( {\hat X_{{\text{TSVD}}}^i} \right)$之间差异很大，参数估计值与其标准差之间的协调性出现了问题。因为小数量级参数在被估计时，若标准差较大，则可能严重失真。无论参数的数量级如何，${\left( {\hat X_{{\text{TSVD}}}^i-X_0^i} \right)^2}/{\text{Var}}\left( {\hat X_{{\text{TSVD}}}^i} \right)$应大致相当，或差异不大时才协调。若过分失调，解算结果势必扭曲失真。为衡量 t 个参数的这种差异性，给出如下指标：

算例中，法矩阵的条件数为4.602×10 ³ ，属于病态问题。首先计算出LS估计的信噪比估计量为(7.711×10 ³ ，4.212×10 ³ ，0.027×10 ³ ，8.346×10 ³ ，0.000 1×10 ³ ，1.004×10 ³ )，可以看出, 第3和第5个参数的LS估计的信噪比估计量明显小于其他参数，由此确定第3和第5个参数受到复共线性危害较大，其余参数受到复共线性危害较小。根据步骤2)、步骤3)确定截断参数 k ₁ =5， k ₂ =6，再分别计算LS估计、岭估计、TSVD估计、DTS估计的结果。参数估值$\mathit{\boldsymbol{\hat X}}$以及$\left\| {\Delta \mathit{\boldsymbol{\hat X}}} \right\| = \left\| {\mathit{\boldsymbol{\hat X}}-\mathit{\boldsymbol{X}}} \right\|$的大小如表 2 所示。

Hoerl A E , Kennard R W . Ridge Regression:Biased Estimation for Non-orthogonal Problems[J]. Technometrics , 1970, 12(): 55-67. doi: 10.1080/00401706.1970.10488634 归庆明 , 李国重 . 岭-压缩组合估计及其在测量平差中的应用[J]. 大地测量与地球动力学, 2002, 22(1): 16-21. Gui Qingming , Li Guozhong . Combining Ridge and Shrunken Estimator and Its Applications in Geodetic Adjustment[J]. Journal of Geodesy and Geodynamics , 2002, 22(1): 16-21. 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 Problems in Surveying[M]. Beijing:Science Press, 2006 顾勇为 , 归庆明 . TSVD解算中选择截断参数的新方法[J]. 测绘科学技术学报, 2010, 27(3): 176-179. doi: 10.3969/j.issn.1673-6338.2010.03.006 Gu Yongwei , Gui Qingming . A New Method to Select Truncated Parameter in TSVD[J]. Journal of Geomatics Science and Technology , 2010, 27(3): 176-179. doi: 10.3969/j.issn.1673-6338.2010.03.006 陈希孺, 王松桂.近代回归分析[M].合肥:安徽教育出版社, 1987 Chen Xiru, Wang Songgui. Advanced Regression Analysis[M].Hefei:Anhui Education Press, 1987 卢秀山.病态系统分析理论及其在测量中的应用[D].武汉: 中国科学院测量与地球物理研究所, 1999 Lu Xiushan.Analysis Theory on Ill-Conditioned System with Application in Surveying[D].Wuhan: Institute of Geodesy and Geophysics, Chinese Academy of Sciences, 1999 归庆明 , 郭建锋 , 边少锋 . 基于特征系统的病态性诊断[J]. 测绘科学, 2002, 27(2): 13-15, 19. doi: 10.3771/j.issn.1009-2307.2002.02.004 Gui Qingming , Guo Jianfeng , Bian Shaofeng . Ill-Conditioning Diagnostics Based on Eigensystem[J]. Science of Surveying and Mapping , 2002, 27(2): 13-15, 19. doi: 10.3771/j.issn.1009-2307.2002.02.004 郭建锋.测量平差系统病态性的诊断与处理[D].郑州: 信息工程大学, 2002 Guo Jianfeng.Study on Diagnostics and Processing of Ill-Conditioning Adjustment System[D]. Zhengzhou: Information Engineering University, 2002 Moran K M , Cicci D A . Sensitivity of Ridge-type Estimation Methods to Condition Number[J]. Applied Mathematics and Computation , 2000, 112(1): 143-159. doi: 10.1016/S0096-3003(99)00095-8 归庆明 , 姚绍文 , 顾勇为 . 诊断复共线性的条件指标-方差分解比法[J]. 测绘学报, 2006, 35(3): 210-214. doi: 10.3321/j.issn:1001-1595.2006.03.003 Gui Qingming , Yao Shaowen , Gu Yongwei . A New Method to Diagnose Multicollinearity Based on Condition Index and Variance Decomposition Proportion[J]. Acta Geodaetica et Cartographica Sinica , 2006, 35(3): 210-214. doi: 10.3321/j.issn:1001-1595.2006.03.003 张磊 , 顾勇为 , 归庆明 . 基于复共线性诊断和度量的有偏估计[J]. 大地测量与地球动力学, 2007, 27(2): 99-102. Zhang Lei , Gu Yongwei , Gui Qingming . Biased Estimator Based on Diagnosis and Measure of Multicollinearity[J]. Journal of Geodesy and Geodynamics , 2007, 27(2): 99-102. 顾勇为.基于复共线性诊断的正则化方法及其在大地测量中的应用[D].郑州: 信息工程大学, 2010 Gu Yongwei. Regularization Methods Based on Multicollinearity Diagnosis and Their Applications to Geodesy[D]. Zhengzhou: Information Enginee-ring University, 2010 韩松辉 , 杜兰 , 归庆明 . 诊断复共线性的特征分析法及其在GEO定轨中的应用[J]. 测绘学报, 2013, 42(1): 19-26. Han Songhui , Du Lan , Gui Qingming . Characteristics Analysis Approach for Multicollinearity Diagnosis and Its Applications in Orbit Determination of GEO Satellites[J]. Acta Geodaetica et Cartographica Sinica , 2013, 42(1): 19-26. 郭杰 , 归庆明 , 郭淑妹 . 利用复共线性诊断确定偏差矫正项的截断型岭估计[J]. 武汉大学学报·信息科学版, 2015, 40(6): 785-789. Guo Jie , Gui Qingming , Guo Shumei . Truncation Ridge Estimation Based on the Biased-Corrected Theory by Multicollinearity Diagnosis[J]. Geomati-cs and Information Science of Wuhan University , 2015, 40(6): 785-789. Belsley D A. Conditioning Diagnostics:Collinearity and Weak Data in Regression[M]. New York:Wiley, 1991 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.

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