Abstract
Total least squares estimation based on Gauss–Newton method in nonlinear errors-in-variables (NEIV) model will encounter the problems of convergence, correctness and accuracy of solution related to the selected initial parameter values. In this contribution, a new total least squares estimator is introduced to solve NEIV model. This method is an extension of the homotopy nonlinear weighted least square (HNWLS) method, which is used in the nonlinear Gauss–Markov model where only the dependent variables contain random errors. The new estimator is called homotopy nonlinear weighted total least squares (HNWTLS), because it adopts weighted total least squares adjustment criterion and homotopy method to estimate nonlinear model parameters. The homotopy function of HNWTLS is constructed by using the normal equation of weighted total least squares adjustment criterion. By taking the error vector of independent variables as a parameter vector, the NEIV model is transformed into a classical nonlinear adjustment model. Then, according to the conclusion of HNWLS, the calculation formula of HNWTLS is derived, and the corresponding calculation algorithm is developed accordingly, where the standard Euler prediction and Newton correction method are introduced into it to tracks the homotopy curves. Finally, three examples to demonstrate the advantage and efficiency of HNWTLS estimator are given and some conclusions are drawn.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allgower EL, Georg K (1990) Numerical continuation methods: an introduction. Springer, Heidelberg
Bates D, Watts D (1980) Relative curvature measures of nonlinearity. J Roy Stat Soc: Ser B (Methodol) 42:1–16. https://doi.org/10.1111/j.2517-6161.1980.tb01094.x
Choi S, Harney D, Book N (1996) A robust path tracking algorithm for homotopy continuation. Comput Chem Eng 20:647–655. https://doi.org/10.1016/0098-1354(95)00199-9
Chow V (1959) Open-channel hydraulics. McGraw-Hill, New York
Chow S, Mallet-Paret J, Yorke J (1978) Finding zeroes of maps: homotopy methods that are constructive with probability one. Math Comput 32:887–889. https://doi.org/10.1090/S0025-5718-1978-0492046-9
Fang X (2011) Weighted total least squares solutions for applications in geodesy. Gottfried Wilhelm Leibniz Universität Hannover, Hannover
Felus Y (2002) New methods for spatial statistics in geographic information systems. The Ohio State University, Columbus
Felus Y, Schaffrin B (2005) A total least-squares approach in two stages for semivariogram modeling of aeromagnetic data. GIS Spatial Anal 2:215–220
Golub G, Van L (1980) An analysis of the total least-squares problem. SIAM J Numer Anal 17:883–893. https://doi.org/10.1137/0717073
Hu Z, Hua X, Li Z et al (2008a) Nonlinear surveying model parameters estimation based on homotopy arithmetic. Geomat Inf Sci Wuhan Univ 9:930–933. https://CNKI:SUN:WHCH.0.2008-09-012
Hu Z, Hua X, Li H (2008) Non-linear coordinate transformation method based on homotopy arithmetic. Eng Surv Mapp 17:24–28
Hu C, Chen Y, Peng Y (2014) On weighted total least squares adjustment for solving the nonlinear problems. J Geod Sci 4:49–56. https://doi.org/10.2478/jogs-2014-0007
Hui X, Li X, Chen Y (2007) A new homotopic path-tracking algorithm based on MATLAB. J Northeast Univ (Nat Sci) 28:620–622
Jazaeri S, Amiri-Simkooei A (2015) Weighted total least squares for solving non-linear problem: GNSS point positioning. Surv Rev 47:265–271. https://doi.org/10.1179/1752270614Y.0000000132
Jazaeri S, Amiri-Simkooei A, Sharifi M (2013) Iterative algorithm for weighted total least squares adjustment. Surv Rev 46:19–27. https://doi.org/10.1179/1752270613Y.0000000052
Jegen M, Everett M, Schultz A (2001) Using homotopy to invert geophysical data. Geophysics 66:1749–1760. https://doi.org/10.1190/1.1487117
Jukič D, Scitovski R, Späth H (1999) Partial linearization of one class of the nonlinear total least squares problem by using the inverse model function. Computing 62:163–178. https://doi.org/10.1007/s006070050019
Kim B, Lee T, Ouarda T (2014) Total least square method applied to rating curves. Hydrol Process 28:4057–4066. https://doi.org/10.1002/hyp.9944
Kumar A (2011) Stage-discharge relationship. Encyclopedia of snow, Ice and Glaciers Encyclopedia of Earth Sciences Series S, pp 1079–1081. https://doi.org/10.1007/978-90-481-2642-2_537
Lenzmann L, Lenzmann E (2007) Zur lösung des nichtlinearen gauss-markov-modells. Zeitschrift für Geodäsie, Geoinformation und Landmanagement 132:108–120
Li T, Yorke J (1980) A simple reliable numerical algorithm for following homotopy paths. In: Analysis and computation of fixed points, pp 73–91. https://doi.org/10.1016/B978-0-12-590240-3.50008-2
Manfreda S, Pizarro A, Moramarco T et al (2020) Potential advantages of flow-area rating curves compared to classic stage-discharge-relations. J Hydrol 585:124752. https://doi.org/10.1016/j.jhydrol.2020.124752
Negatu T, Zimale F, Steenhuis T (2022) Establishing stage-discharge rating curves in developing countries: Lake tana basin, Ethiopia. Hydrology 9:9010013. https://doi.org/10.3390/hydrology9010013
Neitzel F (2010) Gneralization of total least-squares on example of unweighted and weighted 2d similarity transformation. J Geodesy 84:751–762. https://doi.org/10.1007/s00190-010-0408-0
Petersen-Øverleir A (2004) Accounting for heteroscedasticity in rating curve estimates. J Hydrol 292:173–181. https://doi.org/10.1016/j.jhydrol.2003.12.024
Schaffrin B, Wieser A (2008) On weighted total least-squares adjustment for linear regression. J Geodesy 82:415–421. https://doi.org/10.1007/s00190-007-0190-9
Shen Y, Li B, Chen Y (2011) An iterative solution of weighted total least-squares adjustment. J Geodesy 85:229–238. https://doi.org/10.1007/s00190-010-0431-1
Tao Y, Gao J, Yao Y (2014) TLS algorithm for GPS height fitting based on robust estimation. Surv Rev 46:184–188. https://doi.org/10.1179/1752270613Y.0000000083
Teunissen P (1989) First and second moments of non-linear least-squares estimators. Bulletin géodésique 63:253–262. https://doi.org/10.1007/BF02520475
Teunissen P (1990) Nonlinear least squares. Manuscripta Geodaetica 15:137–150. https://espace.curtin.edu.au/handle/20.500.11937/38758
Wang X (1997) Practical criterion of nonlinear model that can or cannot be linearized. J Wuhan Technical Univ Surv Mapp 24:154–158
Wang X (2002) Nonlinear model parameter estimation theory and application. Wuhan University Publisher, Wuhan
Wang B (2017) Study on the extended theories of generalized total least squares and their applications in survey data processing. Wuhan University, Wuhan
Wang Z, Gao T (1990) An introduction to homotopy methods. Chongqing Publish House, Chongqing
Wang L, Luo X (2022) Stein Monte Carlo method for nonlinear function covariance propagation. J Geodesy Geodyn 42:1–4. https://doi.org/10.14075/j.jgg.2022.01.001
Wang C, Ning Y, Wang J et al (2020) Performance analysis of parameter estimator based on closed-form newton method for ultrawideband positioning. J Geodesy 2020:9762924. https://doi.org/10.1155/2020/9762924
Watson L, Billups S, Morgan A (1987) Algorithm 652: Hompack: a suite of codes for globally convergent homotopy algorithms. ACM Trans Math Softw (TOMS) 13:281–310. https://doi.org/10.1145/29380.214343
Wolf H (1961) Das fehlerfortpflanzungsgesetz mit gliedern ii. ordnung. J Zeitschrift Fü Vermessungswesen 3 86:86–88
Xu P, Liu J, Shi C (2012) Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis. J Geodesy 86:661–675. https://doi.org/10.1007/s00190-012-0552-9
Xu G, Xu C, Wen Y et al (2017) Source parameters of the 2016–2017 central Italy earthquake sequence from the sentinel-1, ALOS-2 and GPS data. Remote Sens 9:1182. https://doi.org/10.3390/rs9111182
Xue S, Yang Y, Dang Y (2016) Barycentre method for solving distance equations. Surv Rev 48:188–194. https://doi.org/10.1179/1752270615Y.0000000020
Xue S, Yang Y, Dang Y (2014) A closed-form of newton method for solving over-determined pseudo-distance equations. J Geodesy 88:441–448. https://doi.org/10.1007/s00190-014-0695-y
Zhang Q (2002) Nonlinear least squares adjustment theory and its application in GPS. Wuhan University, Wuhan
Zhang Q, Tao B (2004) Uniform model of nonlinear least squares adjustment based on homotopy method. Geomat Inf Sci Wuhan Univ 29:708–710
Zhao Y, Wang L, Chen X et al (2017) Parameter estimation of variogram model by nonlinear weighted total least square. Sci Surv Mapp 42:20–24. https://doi.org/10.16251/j.cnki.1009-2307.2017.01.004
Acknowledgements
This work is funded by the National Key R &D Program of China (2021YFB2600600), the National Key R &D Project of China (2021YFB2600603), and the Natural Science Foundation of Chongqing, China (CSTB2022NSCQ-MSX1527).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing fnancial interests or personal relationships that could have appeared to infuence the work reported in this paper.
Ethical approval and consent to participate
The research in this paper does not involve any human or animal experiments, and there are no ethical issues.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhang, C., Hu, C., Tang, F. et al. Homotopy nonlinear weighted total least squares adjustment. Acta Geod Geophys 59, 93–117 (2024). https://doi.org/10.1007/s40328-023-00432-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40328-023-00432-9