非线性平差精度评定的自适应蒙特卡罗法
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摘要
针对现有的非线性平差精度评定理论中,蒙特卡罗法模拟次数的选择不具有客观性,无法对结果进行直接控制,以及没有同时考虑到平差参数估值、随机量改正数和单位权方差估值的有偏性等问题,把自适应蒙特卡罗法融入到非线性平差精度评定理论中。通过基于自适应蒙特卡罗法的估值偏差计算和参数估值协方差阵计算,设计了非线性平差精度评定一套理论完整的算法流程。基于对偶变量的思想,提出了参数估值偏差计算的对偶自适应蒙特卡罗法。直线拟合模型和椭圆拟合模型两个算例结果表明,非线性平差精度评定的自适应蒙特卡罗法能获得稳定且合理的精度评定结果,具有更强的适用性;对偶自适应蒙特卡罗法计算估值偏差的收敛速度更快,效率更高。
Among existing theories on precision estimation of nonlinear adjustment, the simulation number of Monte Carlo method generally is chosen subjectively and its result also cannot be controlled directly. Besides those, the biases of parameter estimates, corrections of observations and the estimate of variance of unit weight are not taken into consideration simultaneously. The adaptive Monte Carlo method is combined with precision estimation of nonlinear adjustment for solving problems given above in this paper. By calculating biases of estimates and covariances matrix of parameter estimates, the complete process of precision estimation based on adaptive Monte Carlo is given. With the help of the term of antithetic variates, the antithetic and adaptive Monte Carlo algorithm is proposed for biases of parameter estimates. Results from two examples of straight line fitting model and ellipse fitting model show that the adaptive Monte Carlo method in this paper can obtain the stable and reasonable effects for precision estimation of nonlinear adjustment with extensive applicability, the antithetic and adaptive Monte Carlo in this paper is better at convergence and computational efficiency for calculating biases of parameter estimates.
Among existing theories on precision estimation of nonlinear adjustment, the simulation number of Monte Carlo method generally is chosen subjectively and its result also cannot be controlled directly. Besides those, the biases of parameter estimates, corrections of observations and the estimate of variance of unit weight are not taken into consideration simultaneously. The adaptive Monte Carlo method is combined with precision estimation of nonlinear adjustment for solving problems given above in this paper. By calculating biases of estimates and covariances matrix of parameter estimates, the complete process of precision estimation based on adaptive Monte Carlo is given. With the help of the term of antithetic variates, the antithetic and adaptive Monte Carlo algorithm is proposed for biases of parameter estimates. Results from two examples of straight line fitting model and ellipse fitting model show that the adaptive Monte Carlo method in this paper can obtain the stable and reasonable effects for precision estimation of nonlinear adjustment with extensive applicability, the antithetic and adaptive Monte Carlo in this paper is better at convergence and computational efficiency for calculating biases of parameter estimates.
引文
[1] Tao Benzao. Non-linear Adjustment of Deformation Inversion Model[J]. Geomatics and Information Science of Wuhan University, 2001, 26(6): 504-508(陶本藻. 形变反演模型的非线性平差[J]. 武汉大学学报·信息科学版, 2001, 26(6): 504-508)
[2] Li Chaokui. Theory and Application of Data Processing in Space of Nonlinear Models[D]. Changsha: Central South University, 2001(李朝奎. 非线性模型空间测量数据处理理论及其应用[D]. 长沙: 中南大学, 2001)
[3] Liu Guolin. Nonlinear Least Squares and Surveying Adjustment[M]. Beijing: Surveying and Mapping Press, 2002(刘国林. 非线性最小二乘与测量平差[M]. 北京: 测绘出版社, 2002)
[4] Wang Xinzhou. The Theory and Application of Parameter Estimation of Nonlinear Model[M]. Wuhan: Wuhan University Press, 2002(王新洲. 非线性模型参数估计理论与应用[M]. 武汉: 武汉大学出版社, 2002)
[5] Zhang Songlin. The Theoretical and Application Research on Nonlinear Semiparametric Model[D]. Wuhan: Wuhan University, 2003(张松林. 非线性半参数模型最小二乘估计理论及应用研究[D]. 武汉: 武汉大学, 2003)
[6] Tang Limin. Research on the Ill-posed and Solving Methods of Nonlinear Least Squares Problem[D]. Changsha: Central South University, 2011(唐利民. 非线性最小二乘的不适定性及算法研究[D]. 长沙: 中南大学, 2011)
[7] Xue Shuqiang, Yang Yuanxi, Dang Yamin. A Closed-form of Newton Iterative Formula for Nonlinear Adjustment of Distance Equations[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(8): 771-777(薛树强, 杨元喜, 党亚民. 测距定位方程非线性平差的封闭牛顿迭代公式[J]. 测绘学报, 2014, 43(8): 771-777)
[8] Xue S, Yang Y, Dang Y. A Closed-form of Newton Method for Solving Over-Determined Pseudo-Distance Equations[J]. Journal of Geodesy, 2014, 88(5): 441-448
[9] Box M J. Bias in Nonlinear Estimation[J]. Journal of the Royal Statistical Society, Series B (Metho-dological), 1971, 33(2): 171-201
[10] Teunissen P J G, Knickmeyer E H. Nonlinearity and Least Squares[J]. CISM Journal ASCGC, 1988, 42(4): 321-330
[11] Teunissen P J G. First and Second Moments of Non-Linear Least-Squares Estimators[J]. Bulletin Géodésique, 1989, 63(3): 253-262
[12] 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
[13] Schaffrin B, Snow K. Total Least-Squares Regularization of Tykhonov Type and an Ancient Racetrack in Corinth[J]. Linear Algebra and Its Applications, 2010, 432(8): 2 061-2 076
[14] Fang X. Weighted Total Least Squares Solutions for Applications in Geodesy[D]. Hanover: Leibniz University of Hanover, 2011
[15] Shen Y, Li B, Chen Y. An Iterative Solution of Weighted Total Least-Squares Adjustment[J]. Journal of Geodesy, 2011, 85(4): 229-238
[16] Xu P, Liu J, Shi C. Total Least Squares Adjustment in Partial Errors-in-Variables Models: Algorithm and Statistical Analysis[J]. Journal of Geodesy, 2012, 86(8): 661-675
[17] 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): 1 028-1 032(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): 1 028-1 032)
[18] Tong X, Jin Y, Zhang S, et al. Bias-Corrected Weighted Total Least-Squares Adjustment of Condition Equations[J]. Journal of Surveying Engineering, 2014, 141(2): 04014013
[19] Zeng W, Liu J, Yao Y. On Partial Errors-in-Variables Models with Inequality Constraints of Parameters and Variables[J]. Journal of Geodesy, 2015, 89(2): 111-119
[20] Fang X. Weighted Total Least-Squares with Constraints: A Universal Formula for Geodetic Symmetrical Transformations[J]. Journal of Geodesy, 2015, 89(5): 459-469
[21] Schaffrin B. Adjusting the Errors-in-Variables Model: Linearized Least-Squares vs. Nonlinear Total Least-Squares[C]//VIII Hotine-Marussi Symposium on Mathematical Geodesy. Switzerland: Springer International Publishing, 2015: 301-307
[22] Fang X, Wu Y. On the Errors-in-Variables Model with Equality and Inequality Constraints for Selected Numerical Examples[J]. Acta Geodaetica et Geophysica, 2016, 51(3): 515-525
[23] Wang Leyang, Yu Hang, Chen Xiaoyong. An Algorithm for Partial EIV Model[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(1): 22-29(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)
[24] Wang Leyang, Zhao Yingwen, Chen Xiaoyong, et al. A Newton Algorithm for Multivariate Total Least Squares Problems[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(4): 411-417(王乐洋, 赵英文, 陈晓勇, 等. 多元总体最小二乘问题的牛顿解法[J]. 测绘学报, 2016, 45(4): 411-417)
[25] Zeng Wenxian, Fang Xing, Liu Jingnan, et al. Weighted Total Least Squares of Universal EIV Adjustment Model[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(8): 890-894(曾文宪, 方兴, 刘经南, 等. 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)
[26] Fang X. On Non-combinatorial Weighted Total Least Squares with Inequality Constraints[J]. Journal of Geodesy, 2014, 88(8): 805-816
[27] Amiri-Simkooei A R, Zangeneh-Nejad F, Asgari J. On the Covariance Matrix of Weighted Total Least-Squares Estimates[J]. Journal of Surveying Engineering, 2016, 142(3): 04015014
[28] JCGM 101:2008. Evaluation of Measurement Data-Supplement 1 to the “Guide to the Expression of Uncertainty in Measurement”-Propagation of Distributions Using a Monte Carlo Method[S]. Sèvres: JCGM, 2008
[29] Sneeuy N, Krumm F, Roth M. Adjustment Theory[M]. Stuttgart: University of Stuttgart, 2015
[30] Ratkowsky D A. Nonlinear Regression Modeling[M]. New York: Marcel Dekker, 1983
[31] Robert C, Casella G. Monte Carlo Statistical Methods[M]. Berlin: Springer Science & Business Media, 2013
[32] Fang X. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8): 733-749
[2] Li Chaokui. Theory and Application of Data Processing in Space of Nonlinear Models[D]. Changsha: Central South University, 2001(李朝奎. 非线性模型空间测量数据处理理论及其应用[D]. 长沙: 中南大学, 2001)
[3] Liu Guolin. Nonlinear Least Squares and Surveying Adjustment[M]. Beijing: Surveying and Mapping Press, 2002(刘国林. 非线性最小二乘与测量平差[M]. 北京: 测绘出版社, 2002)
[4] Wang Xinzhou. The Theory and Application of Parameter Estimation of Nonlinear Model[M]. Wuhan: Wuhan University Press, 2002(王新洲. 非线性模型参数估计理论与应用[M]. 武汉: 武汉大学出版社, 2002)
[5] Zhang Songlin. The Theoretical and Application Research on Nonlinear Semiparametric Model[D]. Wuhan: Wuhan University, 2003(张松林. 非线性半参数模型最小二乘估计理论及应用研究[D]. 武汉: 武汉大学, 2003)
[6] Tang Limin. Research on the Ill-posed and Solving Methods of Nonlinear Least Squares Problem[D]. Changsha: Central South University, 2011(唐利民. 非线性最小二乘的不适定性及算法研究[D]. 长沙: 中南大学, 2011)
[7] Xue Shuqiang, Yang Yuanxi, Dang Yamin. A Closed-form of Newton Iterative Formula for Nonlinear Adjustment of Distance Equations[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(8): 771-777(薛树强, 杨元喜, 党亚民. 测距定位方程非线性平差的封闭牛顿迭代公式[J]. 测绘学报, 2014, 43(8): 771-777)
[8] Xue S, Yang Y, Dang Y. A Closed-form of Newton Method for Solving Over-Determined Pseudo-Distance Equations[J]. Journal of Geodesy, 2014, 88(5): 441-448
[9] Box M J. Bias in Nonlinear Estimation[J]. Journal of the Royal Statistical Society, Series B (Metho-dological), 1971, 33(2): 171-201
[10] Teunissen P J G, Knickmeyer E H. Nonlinearity and Least Squares[J]. CISM Journal ASCGC, 1988, 42(4): 321-330
[11] Teunissen P J G. First and Second Moments of Non-Linear Least-Squares Estimators[J]. Bulletin Géodésique, 1989, 63(3): 253-262
[12] 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
[13] Schaffrin B, Snow K. Total Least-Squares Regularization of Tykhonov Type and an Ancient Racetrack in Corinth[J]. Linear Algebra and Its Applications, 2010, 432(8): 2 061-2 076
[14] Fang X. Weighted Total Least Squares Solutions for Applications in Geodesy[D]. Hanover: Leibniz University of Hanover, 2011
[15] Shen Y, Li B, Chen Y. An Iterative Solution of Weighted Total Least-Squares Adjustment[J]. Journal of Geodesy, 2011, 85(4): 229-238
[16] Xu P, Liu J, Shi C. Total Least Squares Adjustment in Partial Errors-in-Variables Models: Algorithm and Statistical Analysis[J]. Journal of Geodesy, 2012, 86(8): 661-675
[17] 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): 1 028-1 032(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): 1 028-1 032)
[18] Tong X, Jin Y, Zhang S, et al. Bias-Corrected Weighted Total Least-Squares Adjustment of Condition Equations[J]. Journal of Surveying Engineering, 2014, 141(2): 04014013
[19] Zeng W, Liu J, Yao Y. On Partial Errors-in-Variables Models with Inequality Constraints of Parameters and Variables[J]. Journal of Geodesy, 2015, 89(2): 111-119
[20] Fang X. Weighted Total Least-Squares with Constraints: A Universal Formula for Geodetic Symmetrical Transformations[J]. Journal of Geodesy, 2015, 89(5): 459-469
[21] Schaffrin B. Adjusting the Errors-in-Variables Model: Linearized Least-Squares vs. Nonlinear Total Least-Squares[C]//VIII Hotine-Marussi Symposium on Mathematical Geodesy. Switzerland: Springer International Publishing, 2015: 301-307
[22] Fang X, Wu Y. On the Errors-in-Variables Model with Equality and Inequality Constraints for Selected Numerical Examples[J]. Acta Geodaetica et Geophysica, 2016, 51(3): 515-525
[23] Wang Leyang, Yu Hang, Chen Xiaoyong. An Algorithm for Partial EIV Model[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(1): 22-29(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)
[24] Wang Leyang, Zhao Yingwen, Chen Xiaoyong, et al. A Newton Algorithm for Multivariate Total Least Squares Problems[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(4): 411-417(王乐洋, 赵英文, 陈晓勇, 等. 多元总体最小二乘问题的牛顿解法[J]. 测绘学报, 2016, 45(4): 411-417)
[25] Zeng Wenxian, Fang Xing, Liu Jingnan, et al. Weighted Total Least Squares of Universal EIV Adjustment Model[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(8): 890-894(曾文宪, 方兴, 刘经南, 等. 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)
[26] Fang X. On Non-combinatorial Weighted Total Least Squares with Inequality Constraints[J]. Journal of Geodesy, 2014, 88(8): 805-816
[27] Amiri-Simkooei A R, Zangeneh-Nejad F, Asgari J. On the Covariance Matrix of Weighted Total Least-Squares Estimates[J]. Journal of Surveying Engineering, 2016, 142(3): 04015014
[28] JCGM 101:2008. Evaluation of Measurement Data-Supplement 1 to the “Guide to the Expression of Uncertainty in Measurement”-Propagation of Distributions Using a Monte Carlo Method[S]. Sèvres: JCGM, 2008
[29] Sneeuy N, Krumm F, Roth M. Adjustment Theory[M]. Stuttgart: University of Stuttgart, 2015
[30] Ratkowsky D A. Nonlinear Regression Modeling[M]. New York: Marcel Dekker, 1983
[31] Robert C, Casella G. Monte Carlo Statistical Methods[M]. Berlin: Springer Science & Business Media, 2013
[32] Fang X. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8): 733-749