Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T07:48:12.890Z Has data issue: false hasContentIssue false
  • English
  • Français

Unbiased Nonlinear Least Squares Estimations of Short-distance Equations

Published online by Cambridge University Press:  23 February 2017

Shuqiang Xue  and
Yuanxi Yang
Shuqiang Xue
Affiliation:
(School of Geological and Surveying Engineering, Chang'an University, Yanta Road, Xi'an 710054, China) (Chinese Academy of Surveying and Mapping, Beijing, 100830, China)
Yuanxi Yang*
Affiliation:
(National Key Laboratory for Geo-information Engineering, Xi'an Research Institute of Surveying and Mapping, Xi'an, 710054, China)
*
E-mail: yuanxi_yang@163.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Nonlinear least squares estimations have been widely applied in positioning. However, nonlinear least squares estimations are generally biased. As the Gauss-Newton method has been widely applied to obtain a nonlinear least squares solution, we propose an iterative procedure for obtaining unbiased estimations with this method. The characteristics of the linearization error are discussed and a systematic error source of the linearization error needs to be removed to guarantee the unbiasedness. Both the geometrical condition and the statistical condition for unbiased nonlinear least squares estimations are revealed. It is shown that for long-distance observations of high precision, or for a positioning configuration with the lowest Geometric Dilution Of Precision (GDOP), the nonlinear least squares estimations tend to be unbiased; but for short-distance cases, the bias in the nonlinear least squares solution should be estimated to obtain unbiased values by removing the bias from the nonlinear least squares solution. The proposed results are verified by the Monte Carlo method and this shows that the bias in nonlinear least squares solution of short-distance distances cannot be ignored.

Keywords

Distance equationBias Nonlinear least squaresNonlinear error propagation
Type
Research Article
Information
The Journal of Navigation , Volume 70 , Issue 4 , July 2017 , pp. 810 - 828
Copyright
Copyright © The Royal Institute of Navigation 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alcocer, A., Oliveira, P. and Pascoal, A. (2006). Underwater acoustic positioning systems based on buoys with GPS. Proceedings of the Eighth European Conference on Underwater Acoustics, 1–8.Google Scholar
Awange, J.L., Fukuda, Y., Takemoto, S., Ateya, I.L. and Grafarend, E.W. (2003). Ranging algebraically with more observations than unknowns. Earth Planets and Space, 55(7), 387–394.Google Scholar
Bates, D.M. and Watts, D.G. (1980). Relative curvature measures of nonlinearity. Journal of the Royal Statistical Society. Series B (Methodological), 42(1), 1–25.CrossRefGoogle Scholar
Box, M., 1971. Bias in nonlinear estimation. Journal of the Royal Statistical Society. Series B (Methodological), 33(2), 171–201.CrossRefGoogle Scholar
Clarke, G. (1980). Moments of the least squares estimators in a non-linear regression model. Journal of the Royal Statistical Society. Series B (Methodological), 42(2), 227–237.CrossRefGoogle Scholar
Cook, R., Tsai, C.-L. and Wei, B. (1986). Bias in nonlinear regression. Biometrika, 73(3), 615–623.Google Scholar
Dang, Y. and Xue, S. (2014). A New Newton-type Iterative Formula for Over-determined Distance Equations. In Proceedings of the Joint IAG Assembly “Earth on the Edge: Science for a Sustainable Planet(Rizos, Chris; Willis, Pascal (Eds.))”, 2011, Springer, Berlin Heidelberg, IAG Symposium, 139, 607–614.Google Scholar
Grafarend, E.W. (2006). Linear and nonlinear models: fixed effects, random effects, and mixed models. Walter de Gruyter, Berlin, New York.Google Scholar
Grafarend, E.W. and Schaffrin, B. (1993). Ausgleichungsrechnung in linearen Modellen. BI-Wissenschaftsverlag.Google Scholar
Jeudy, L.M.A. (1988). First and second moments of non-linear least-squares estimators Application to Explicit Least-square Adjustments. Bulletin Geodesique, 62(2), 113–124.Google Scholar
Leick, A. (2004). GPS satellite surveying. John Wiley & Sons.Google Scholar
Mathai, A.M. and Provost, S.B. (1992). Quadratic Forms in Random Variables. CRC Press, ISBN 978-0824786915.Google Scholar
Sirola, N. (2010). Closed-form algorithms in mobile positioning: Myths and misconceptions. Positioning Navigation and Communication (WPNC), 2010 7th Workshop on, 38–44.Google Scholar
Seemkooei, A.A. (1998). Analytical methods in optimization and design of geodetic networks. Department of Surveying Engineering, KN Toosi University of Technology, Tehran, Iran.Google Scholar
Teunissen, P.J.G. (1989). First and second moments of non-linear least-squares estimators. Journal of Geodesy, 63(3), 253–262.Google Scholar
Teunissen, P.J.G. (1990a). Nonlinear inversion of geodetic and geophysical data: Diagnosing nonlinearity. In: F.K. Brunner & C. Rizos (eds.) Developments in Four-Dimensional Geodesy, Springer Berlin Heidelberg: 241–264.CrossRefGoogle Scholar
Teunissen, P.J.G. (1990b). Nonlinear least squares. Manuscripta Geodaetica, 15(3), 137–150.Google Scholar
Teunissen, P.J.G. and Knickmeyer, E.H. (1988). Nonlinearity and Least-Squares. CISM Journal ACSGC, 42(4), 321–330.Google Scholar
Xue, S. and Yang, Y. (2014). Gauss-Jacobi combinatorial adjustment and its modification. Survey Review, 46(337), 298–304.Google Scholar
Xue, S. and Yang, Y. (2015). Positioning configurations with the lowest GDOP and their classification. Journal of Geodesy, 89(1), 49–71.Google Scholar
Xue, S., Yang, Y. and Dang, Y. (2014a). A Closed-form of Newton Iterative Formula for Nonlinear Adjustment of Distance Equations. Acta Geodaetica et Cartographica Sinica, 43(8), 771–777.Google Scholar
Xue, S., Yang, Y. and Dang, Y. (2014b). A closed-form of Newton method for solving over-determined pseudo-distance equations. Journal of Geodesy, 88(5), 441–448.CrossRefGoogle Scholar
Xue, S., Yang, Y., Dang, Y. and Chen, W. (2014c). Dynamic positioning configuration and its first-order optimization. Journal of Geodesy, 88(2), 127–143.CrossRefGoogle Scholar
Xue, S., Yang, Y. and Dang, Y. (2016). Barycentre method for solving distance equations. Survey Review, 48(348), 188–194.Google Scholar
Yang, Y. and Xu, J. (2016). GNSS receiver autonomous integrity monitoring (RAIM) algorithm based on robust estimation. Geodesy and Geodynamics, 7(2):117–123.Google Scholar
Yang, Y., Li, J., Xu, J., Tang, J. (2011). Generalised DOPs with consideration of the influence function of signal-in-space errors. Journal of Navigation, 64(S1): S3–S18.Google Scholar
Yang, Y. and Xu, T. (2003). An adaptive Kalman filter based on Sage windowing weights and variance components[J]. The Journal of Navigation, 56(02): 231–240.CrossRefGoogle Scholar
Yan, J., Tiberius, C., Bellusci, G. and Janssen, G. (2008). Feasibility of Gauss-Newton method for indoor positioning. Position, Location and Navigation Symposium, 2008 IEEE/ION. IEEE, 660–670.CrossRefGoogle Scholar