加乘性混合误差模型精度评定的SUT法

doi:10.11947/j.AGCS.2022.20200514

摘要/Abstract

摘要： 已有加乘性混合误差模型参数估计方法能达到二阶精度,但精度评定方法只能达到一阶精度,若通过传统泰勒级数展开近似函数法来获取参数估值的二阶精度信息,由于加乘性混合误差模型中参数估值与观测值为一个复杂的非线性关系,必然需要复杂的求导运算。针对该问题,本文使用一种无须求导、无须了解非线性函数构成的比例无迹变换(scaled unscented transformation,SUT)法来计算参数估值的二阶精度信息。通过算例分析表明,利用SUT法求解加乘性混合误差模型能够有效避免复杂的求导运算,所求得的参数估值及其协方差阵均能达到二阶精度,从而验证了本文方法的可行性和优势。

关键词: 加乘性混合误差模型, 加权最小二乘, 非线性函数, SUT法, 精度评定

Abstract: The existing parameter estimation method of mixed additive and multiplicative random error model can achieve second-order precision, but the precision estimation method can only achieve first-order precision. If the traditional Taylor series expansion approximate function method is used to obtain the second-order precision information of parameter estimations, it will inevitably require complicated derivation operation due to the complex nonlinear relationship between parameter estimations and observations in the mixed additive and multiplicative random error model. Aiming at this problem, this paper uses the scaled unscented transformation method, which does not require derivative operation and understand the composition of nonlinear function, to obtain the second-order precision information of parameter estimations. The results of experiments show that using the SUT method to solve the mixed additive and multiplicative random error model can effectively avoid complicated derivation operation, and the obtained parameter estimations and covariance matrix can both achieve second-order precision, thus verifies the feasibility and advantages of the proposed method in this paper.

Key words: mixed additive and multiplicative random error model, weighted least squares, nonlinear function, SUT method, precision estimation

中图分类号:

P207

王乐洋, 陈涛. 加乘性混合误差模型精度评定的SUT法[J]. 测绘学报, 2022, 51(11): 2303-2316.

WANG Leyang, CHEN Tao. The SUT method for precision estimation of mixed additive and multiplicative random error model[J]. Acta Geodaetica et Cartographica Sinica, 2022, 51(11): 2303-2316.

参考文献

[1] 江勇. 最小二乘及其扩展方法在测绘中的应用[D]. 淮南: 安徽理工大学, 2016. JIANG Yong. Application of least square method and its extension method in surveying and mapping[D]. Huainan: Anhui University of Science and Technology, 2016.
[2] TEUNISSEN P J G, KNICKMEYER E H. Nonlinearity and least-squares[J]. CISM Journal ASCGC, 1988, 42(4): 321-330.
[3] 师芸, 徐培亮, 彭军还. 乘性误差模型平差理论研究进展概述[J]. 工程勘察, 2014, 42(6): 60-66. SHI Yun, XU Peiliang, PENG Junhuan. Multiplicative error models: an applications-oriented review of parameter estimation methods and statistical error analysis[J]. Geotechnical Investigation & Surveying, 2014, 42(6): 60-66.
[4] SHI Yun, XU Peiliang, PENG Junhuan. An overview of adjustment methods for mixed additive and multiplicative random error models[C]//Proceedings of 2015 International Association Ⅷ Hotine-Marussi Symposium on Mathematical Geodesy. Berlin, Germany: Springer, 2015, 142: 283-290.
[5] XU Peiliang, SHI Yun, PENG Junhuan, et al. Adjustment of geodetic measurements with mixed multiplicative and additive random errors[J]. Journal of Geodesy, 2013, 87(7): 629-643.
[6] SHI Yun, XU Peiliang. Adjustment of measurements with multiplicative random errors and trends[J]. IEEE Geoscience and Remote Sensing Letters, 2021, 18(11): 1916-1920.
[7] WANG Leyang, CHEN Tao. Virtual observation iteration solution and A-optimal design method for ill-posed mixed additive and multiplicative random error model in geodetic measurement[J]. Journal of Surveying Engineering, 2021, 147(4): 04021016.
[8] XU Peiliang, SHIMADA Seiichi. Least squares parameter estimation in multiplicative noise models[J]. Communications in Statistics-Simulation and Computation, 2000,29(1): 83-96.
[9] 师芸. 加乘性混合误差模型参数估计方法及其应用[J]. 武汉大学学报(信息科学版), 2014, 39(9): 1033-1037. SHI Yun. Least squares parameter estimation in additive/multiplicative error models for use in geodesy[J]. Geomatics and Information Science of Wuhan University, 2014, 39(9): 1033-1037.
[10] JULIER S J, UHLMANN J K, DURRANT-WHYTE H F. A new approach for filtering nonlinear systems[C]//Proceedings of 1995 American Control Conference. Seattle, WA, America: IEEE, 1995, 3: 1628-1632.
[11] JULIER S J, UHLMANN J K. New extension of the Kalman filter to nonlinear systems[C]//Proceedings of 1997 International Society for Optics and Photonics Signal Processing, Sensor Fusion, and Target Recognition VI. Orlando, USA: Society of Photo Optical Instrumentation Engineers, 1997, 3068: 182-193.
[12] WAN E A, MERWE R V D. The unscented Kalman filter[M]. New York, USA: Wiley, 2001.
[13] MERWE R V D. Sigma-point Kalman filters for probabilistic inference in dynamic state-space models[D]. Oregon: Oregon Health & Science University, 2004.
[14] GUSTAFSSON F, HENDEBY G. Some relations between extended and unscented Kalman filters[J]. IEEE Transactions on Signal Processing, 2012, 60(2): 545-555.
[15] MENEGAZ H M T, ISHIHARA J Y, BORGES G A, et al. A systematization of the unscented Kalman filter theory[J]. IEEE Transactions on Automatic Control, 2015, 60(10): 2583-2598.
[16] WANG Leyang, ZHAO Yingwen. Scaled unscented transformation for nonlinear error propagation: accuracy, sensitivity and applications[J]. Journal of Surveying Engineering, 2018, 144(1): 04017022.
[17] WANG Leyang, ZHAO Yingwen. Unscented transformation with scaled symmetric sampling strategy for precision estimation of total least squares[J]. Studia Geophysica et Geodaetica, 2017, 61(3): 385-411.
[18] 王乐洋, 丁锐, 吴璐璐. SUT法偏差改正的Partial EIV模型方差分量估计及其精度评定[J].大地测量与地球动力学, 2019, 39(7): 711-716. WANG Leyang, DING Rui, WU Lulu. Partial EIV model variance component estimation and accuracy evaluation of SUT method by deviation correction[J]. Journal of Geodesy and Geodynamics, 2019, 39(7): 711-716.
[19] WANG Leyang, ZHAO Yingwen, ZOU Chuanyi, et al.Scaled unscented transformation method and adaptive Monte Carlo method used for effects of fault parameters estimation on green function matrix in slip distribution inversion[J]. Geodesy and Geodynamics, 2020, 11(5): 328-337.
[20] WANG Leyang, DING Rui. Inversion and precision estimation of earthquake fault parameters based on scaled unscented transformation and hybrid PSO/simplex algorithm with GPS measurement data[J]. Measurement, 2020, 153: 107422.
[21] 武汉大学测绘学院测量平差学科组. 误差理论与测量平差基础[M]. 3版. 武汉: 武汉大学出版社, 2014. Surveying Adjustment Group of School of Geodesy and Geomatics, Wuhan University. Error theory and foundation of surveying adjustment[M]. 3rd ed. Wuhan: Wuhan University Press, 2014.
[22] MAGNUS J R, NEUDECKER H. Matrix differential calculus with applications in statistics and econometrics[M]. New York, USA: Wiley, 1999.
[23] 徐培亮. 非线性函数的协方差传播公式[J]. 武汉测绘科技大学学报, 1986, 11(2): 92-99. XU Peiliang. Variance-covariance propagation for a nonlinear function[J]. Geomatics and Information Science of Wuhan University, 1986, 11(2): 92-99.
[24] LUTKEPOHL H. Handbook of matrices[M]. New York, USA: Wiley, 1996.
[25] JULIER S J, UHLMANN J K. A general method for approximating nonlinear transformations of probability distributions[M]. Oxford, UK: Oxford University Press, 1996.
[26] YORK D, EVENSEN N M, MARTINEZ M L, et al. Unified equations for the slope, intercept, and standard errors of the best straight line[J]. American Journal of Physics, 2004, 72(3): 367-375.
[27] SHEN Yunzhong, LI Bofeng, CHEN Yi. An iteration solution of weighted total least-squares adjustment[J]. Journal of Geodesy, 2011, 85(4): 229-238.
[28] 王乐洋, 邹传义. 乘性误差模型参数估计及精度评定的Sterling插值方法[J]. 武汉大学学报(信息科学版), 2022, 47(2): 238-244. WANG Leyang, ZOU Chuanyi. Sterling interpolation method for parameter estimation and precision estimation in multiplicative error model[J]. Geomatics and Information Science of Wuhan University, 2022, 47(2): 238-244.
[29] 王乐洋, 陈涛, 邹传义. 病态乘性误差模型的加权最小二乘正则化迭代解法及精度评定[J]. 测绘学报, 2021, 50(5): 589-599. DOI: 10.11947/j.AGCS.2021.20200126. WANG Leyang, CHEN Tao, ZOU Chuanyi. Weighted least squares regularization iteration solution and precision estimation for ill-posed multiplicative error model[J]. Acta Geodaetica et Cartographica Sinica, 2021, 50(5): 589-599. DOI: 10.11947/j.AGCS.2021.20200126.
[30] 陈杨. 乘性误差随机模型的粗差探测[D]. 西安: 西安科技大学, 2017. CHEN Yang. Outlier detection on random model of multiplicative error[D]. Xi’an: Xi’an University of Science and Technology, 2017.
[31] BOLKAS D, FOTOPOULOS G, BRAUN A, et al. Assessing digital elevation model uncertainty using GPS survey data[J]. Journal of Surveying Engineering, 2016, 142(3): 04016001.
[32] 李志林, 朱庆, 谢潇. 数字高程模型[M]. 3版. 北京: 科学出版社, 2017. LI Zhilin, ZHU Qing, XIE Xiao. Digital elevation model[M]. 3rd ed. Beijing: Science Press, 2017.
[33] LIU Xiaoye. Airborne LiDAR for DEM generation: some critical issues[J]. Progress in Physical Geography, 2008, 32(1): 31-49.
[34] HLADIK C, ALBER M. Accuracy assessment and correction of a LiDAR-derived salt marsh digital elevation model[J]. Remote Sensing of Environment, 2012, 121(138): 224-235.
[35] AKIMA H. Algorithm 760: rectangular-grid-data surface fitting that has the accuracy of a bicubic polynomial[J]. Acm Transactions on Mathematical Software, 1996, 22(3): 357-361.
[36] ZIMMERMAN D, PAVLIK C, RUGGLES A, et al. An experimental comparison of ordinary and universal Kriging and inverse distance weighting[J]. Mathematical Geology, 1999, 31(4): 375-390.
[37] KIDNER D B. Higher-order interpolation of regular grid digital elevation models[J]. International Journal of Remote Sensing, 2003, 24(14): 2981-2987.
[38] HILL C A, HARRIS M, RIDLEY K D, et al. LiDAR frequency modulation vibrometry in the presence of speckle[J]. Applied Optics, 2003, 42(6): 1091-1100.
[39] KOBLER A, PFEIFER N, OGRINC P, et al. Repetitive interpolation: a robust algorithm for DTM generation from aerial laser scanner data in forested terrain[J]. Remote Sensing of Environment, 2007, 108(1): 9-23.
[40] LEIGH C L, KIDNER D B, THOMAS M C. The use of LiDAR in digital surface modelling: issues and errors[J]. Transactions in GIS, 2010, 13(4): 345-361.
[41] SHI Yun, XU Peiliang, PENG Junhuan, et al. Adjustment of measurements with multiplicative errors: error analysis, estimates of the variance of unit weight, and effect on volume estimation from LiDAR-type digital elevation models[J]. Sensors, 2014, 14(1): 1249-1266.
[42] 王乐洋, 许才军, 鲁铁定. 病态加权总体最小二乘平差的岭估计解法[J]. 武汉大学学报(信息科学版), 2010, 35(11): 1346-1350. WANG Leyang, XU Caijun, LU Tieding. Ridge estimation method in ill-posed weighted total least squares adjustment[J].Geomatics and Information Science of Wuhan University, 2010, 35(11): 1346-1350.
[43] WANG Leyang, CHEN Tao. Ridge estimation iterative solution of ill-posed mixed additive and multiplicative random error model with equality constraints[J]. Geodesy and Geodynamics, 2021, 12(5), 336-346.
[44] HE Yufang, ZHU Wu, LEI Yang, et al. A comparative study of ionospheric correction on SAR interferometry:a case study of L’Aquila earthquake[J]. Journal of Geodesy and Geoinformation Science, 2022, 5(1): 5-13.
[45] ZHU Wu, LEI Yang, SUN Quan. Detection, estimation and compensation of ionospheric effect on SAR interferometry using azimuth shift[J]. Journal of Geodesy and Geoinformation Science, 2022, 5(1): 14-24
[46] 王乐洋, 温贵森. 偏差改正的 Partial EIV 模型方差分量估计[J]. 测绘学报, 2019, 48(4): 412-421. DOI: 10.11947/j.AGCS.2019.20170693. WANG Leyang, WEN Guisen. Bias-corrected variance components estimation of partial EIV model[J]. Acta Geodaetica et Cartographica Sinica, 2019, 48(4): 412-421. DOI: 10.11947/j.AGCS.2019.20170693.