Mixed Additive and Multiplicative Error Model Parameter Estimation Based on LS-VCE and M-Estimation
- 1College of Geomatics, Xi’an University of Science and Technology, Xi’an 710054, China
Abstract
Even though Global Navigation Satellite System (GNSS) observations have been proven to be affected by mixed additive and multiplicative errors, additive error models (AEMs) are still currently used for processing. Moreover, current research results on the mixed additive and multiplicative error model (MAMEM) are mostly based on simulated data without outlier and with known error variance information, lacking validation and analysis using measured data. In this paper, the IGGIII scheme from M-estimation and the least squares variance component estimation (LS-VCE) are integrated into the MAMEM, proposing a robust algorithm named MbcWLS. The accuracy of the algorithm is validated and analyzed using a linear fitting case and measured GNSS data. The linear fitting case indicates that in the absence of outlier, the accuracy of the MbcWLS algorithm suffers a slight loss. Measured by the mean root mean square error (M-RMSE) metric, the relative efficiency of MbcWLS compared to bcWLS is approximately 0.755. Under the worst-case outlier distribution scenario (characterized by high-leverage positions, consecutive clustering, unidirectional directionality, and magnitudes ranging from 5 to 10 times its own mean square error), the outlier resistance ratio of the MbcWLS algorithm is approximately 16%. Measured GNSS data indicate that when GNSS signals are affected by multipath or non-line-of-sight signals (1~500 s and 3500~5500 s), compared with the WLS algorithm (based on the AEM), both the bcWLS and MbcWLS algorithms significantly improve accuracy in the N and U components. In particular, the MbcWLS algorithm can optimize the RMSE in the U component from 11.2 m to 1.8 m, and in the N direction from 5.5 m to around 1.5 m (1~500 s). When the GNSS signal quality is excellent (1500~3000 s), the accuracy of all four algorithms is at the same level, with RMSE differences across the three components remaining largely at the centimeter level. The experimental results validate the effectiveness and practical application value of the proposed MbcWLS algorithm.
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BibTeX
@article{shi2026mixed,
title = {{Mixed Additive and Multiplicative Error Model Parameter Estimation Based on LS-VCE and M-Estimation}},
author = {Yun Shi and Guohong Li},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213926},
url = {https://www.mdpi.com/2227-7390/14/21/3926},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Mixed Additive and Multiplicative Error Model Parameter Estimation Based on LS-VCE and M-Estimation
AU - Yun Shi
AU - Guohong Li
PY - 2026
DA - 2026/10/27
JO - Mathematics
DO - 10.3390/math14213926
UR - https://www.mdpi.com/2227-7390/14/21/3926
AB - Even though Global Navigation Satellite System (GNSS) observations have been proven to be affected by mixed additive and multiplicative errors, additive error models (AEMs) are still currently used for processing. Moreover, current research results on the mixed additive and multiplicative error model (MAMEM) are mostly based on simulated data without outlier and with known error variance information, lacking validation and analysis using measured data. In this paper, the IGGIII scheme from M-estimation and the least squares variance component estimation (LS-VCE) are integrated into the MAMEM, proposing a robust algorithm named MbcWLS. The accuracy of the algorithm is validated and analyzed using a linear fitting case and measured GNSS data. The linear fitting case indicates that in the absence of outlier, the accuracy of the MbcWLS algorithm suffers a slight loss. Measured by the mean root mean square error (M-RMSE) metric, the relative efficiency of MbcWLS compared to bcWLS is approximately 0.755. Under the worst-case outlier distribution scenario (characterized by high-leverage positions, consecutive clustering, unidirectional directionality, and magnitudes ranging from 5 to 10 times its own mean square error), the outlier resistance ratio of the MbcWLS algorithm is approximately 16%. Measured GNSS data indicate that when GNSS signals are affected by multipath or non-line-of-sight signals (1~500 s and 3500~5500 s), compared with the WLS algorithm (based on the AEM), both the bcWLS and MbcWLS algorithms significantly improve accuracy in the N and U components. In particular, the MbcWLS algorithm can optimize the RMSE in the U component from 11.2 m to 1.8 m, and in the N direction from 5.5 m to around 1.5 m (1~500 s). When the GNSS signal quality is excellent (1500~3000 s), the accuracy of all four algorithms is at the same level, with RMSE differences across the three components remaining largely at the centimeter level. The experimental results validate the effectiveness and practical application value of the proposed MbcWLS algorithm.
SN - 2227-7390
ER -