International Journal of Nonlinear Analysis and Applications

Reliability analysis of Helmert model for Robust M-estimator

Document Type : Research Paper

Authors

1 Department of Finance and Banking, College of Economic and Administration, Al-Iraqia University, Baghdad, Iraq

2 Department of Business Administration, College of Economic and Administration, Al-Iraqia University, Baghdad, Iraq

Abstract

Traditional techniques of the Helmert model are known to be used for the purpose of smoothing the data. Therefore, the special Helmert robust technique is adopted in this research to reach the internal and external reliability and suitability for the data using observed points and corresponding weights, which form the objective of this study. The least-squares method poorly performed in the presence of outliers, in comparison with the use of the robust traditional techniques, to reduce the outliers’ impact. Geodetic reliability consists of two basic components: internal reliability and external reliability, which are critical measures of validating the model. Through the analyses, the internal reliability values represented by the minimum detectable bias become larger, increasing the reliability value and decreasing the probability of error. After comparing the two methods, the best method was chosen based on the final value of the probability of error. The values of the measuring parameters were modified accordingly. This method is essential to determine the reliability of the model and is bound by the effect of observations on the estimated variance of the parameters. Depending on the using various methods to estimate the reliability of the Helmert model, the results indicated that the suggested robust M method is the most accurate. And in this contribution, the minimum detectable bias was used to determine the outliers in the Gauss-Helmert model. The results showed that the robust M method has the ability to detect outliers and reduce their impact by calculating the value of the Mean Absolute Percentage Error.

Keywords

[1] Adjustment computations based on the least-squares principle. [ in German]
[2] A. Amiri-Simkooei and S. Jazaeri, Weighted total least squares formulated by standard least squares theory, J. Geodetic Sci. 2 (2012), no. 2, 113–124.
[3] J.D. Cothren, Reliability in constrained Gauss-Markov models: an analytical and differential approach with applications in photogrammetry, The Ohio State University, 2004.
[4] J.-F. Guo, J.-K. Ou, and Y.-B. Yuan, Reliability analysis for a Robust m-estimator, J. Survey. Engin. 137 (2011), no. 1, 9–13.
[5] K.-R. Koch, Robust estimations for the nonlinear Gauss Helmert model by the expectation-maximization algorithm, J. Geodesy 88 (2014), no. 3, 263–271.
[6] D. Mihajlovic and Z. Cvijetinovic, Weighted coordinate transformation formulated by standard least-squares theory, Survey Rev. 49 (2017), no. 356, 328–345.
[7] F. Neitzel, Generalization of total least-squares on example of unweighted and weighted 2d similarity transformation, J. Geodesy 84 (2010), no. 12, 751–762.
[8] F. Neitzel and B. Schaffrin, On the Gauss–Helmert model with a singular dispersion matrix where bq is of smaller rank than b, J. Comput. Appl. Math. 291 (2016), 458–467.
[9] F. Neri, G. Saitta, and S. Chiofalo, An accurate and straightforward approach to line regression analysis of error-affected experimental data, J. Phys. E: Sci. Instrum. 22 (1989), no. 4, 215.
[10] B. Schaffrin, Reliability measures for correlated observations, J. Survey. Engin. 123 (1997), no. 3, 126–137.
[11] Z. Zhou and Y. Wu, System model bias processing approach for regional coordinated states information involved filtering, Math. Prob. Engin. 2016 (2016).
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 2565-2572
  • Receive Date: 18 February 2022
  • Revise Date: 27 March 2022
  • Accept Date: 17 May 2022