Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Simplified algebraic estimation for the quality control of DIA estimator

  • Original Article
  • Published:
Journal of Geodesy Aims and scope Submit manuscript

Abstract

Based on the unifying framework of the detection, identification and adaption (DIA) estimators, quality control indices are refined and formulated by taking the uncertainty of the combined estimation-testing procedure into account and performing the propagation of uncertainty. These indices are used to measure the confidence levels of the testing decisions, the reliability of the specified alternative hypothesis models, as well as the biasedness and dispersion of the estimated parameters. A simplified algebraic estimation (SAE) method is developed to calculate these quality control indices for the application of single outlier DIA. Compared to the conventional Monte Carlo simulation method, the proposed SAE method can achieve an adequate estimation accuracy and significantly higher computation efficiency. Using a GNSS single-point positioning example, the performance of the SAE method is evaluated and the quality control of the conventionally used DIA estimator is demonstrated for practical applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Data availability

Necessary data are accessible in the Numerical Example Section of the manuscript.

References

  • Alberda JE (1976) Quality control in surveying. Chart Surv 4(2):23–28

    Google Scholar 

  • Baarda W (1967) Statistical concepts in geodesy. Netherlands Geodetic Commission Publication on geodesy, Delft

    Google Scholar 

  • Baarda W (1968) A testing procedure for use in geodetic networks. Netherlands Geodetic Commission Publication On geodesy, Delft

    Google Scholar 

  • BaardaW (1976) Reliability and precision of geodetic networks. Tech. rep., VII Int Kurs fur Ingenieursmessungen hoher Prazision, Band I, T.H. Darmstadt, Inst. fur Geodaesie

  • Chen Y, Wang J (1996) Reliability measure for correlated observations. Z Vermess 121(5):211–219

    Google Scholar 

  • Förstner W (1983) Reliability and discernability of extended Gauss-Markov models. Seminar on mathematical models to outliers and systematic errors, Series A, No. 98, Deutsche Geodätische Kommision, Munich, Germany, pp. 79–103

  • Förstner W (1985) The reliability of block triangulation. Photogramm Eng Remote Sens 51(6):1137–1149

    Google Scholar 

  • Hekimoglu S, Berber M (2003) Effectiveness of robust methods in heterogeneous linear models. J Geod 76(11–12):706–713

    Article  Google Scholar 

  • Leick A, Emmons MB (1994) Quality control with reliability for large GPS networks. J Surv Eng 120(1):25–41

    Article  Google Scholar 

  • Lehmann R, Scheffler T (2011) Monte Carlo-based data snooping with application to a geodetic network. J Appl Geod 5:123–134

    Google Scholar 

  • Lehmann R (2012) Improved critical values for extreme normalized and studentized residuals in Gauss–Markov models. J Geod 86(12):1137–1146

    Article  Google Scholar 

  • Lehmann R (2013) On the formulation of the alternative hypothesis for geodetic outlier detection. J Geod 87(4):373–386

    Article  Google Scholar 

  • Lehmann R (2014) Transformation model selection by multiple hypotheses testing. J Geod 88:1117–1130

    Article  Google Scholar 

  • Lehmann R, Lösler M (2016) Multiple outlier detection: hypothesis tests versus model selection by information criteria. J Surv Eng 142(4):04016017

    Article  Google Scholar 

  • Lehmann R, Lösler M, Neitzel F (2020) Mean shift versus variance inflation approach for outlier detection: a comparative study. Mathematics 8(6):991

    Article  Google Scholar 

  • Koch KR (1999) Parameter estimation and hypothesis testing in linear models, 2nd edn. Springer, Berlin

    Book  Google Scholar 

  • Li B, Shen Y, Lou L (2011) Efficient estimation of variance and covariance components: a case study for GPS stochastic model evaluation. IEEE Trans Geosci Remote Sensing 49(1):203–210

    Article  Google Scholar 

  • Koch KR (2015) Minimal detectable outliers as measures of reliability. J Geod 89:483–490

    Article  Google Scholar 

  • Kok JJ (1984) On data snooping and multiple outlier testing. US Department of Commerce, National Oceanic and Atmospheric Administration, National Ocean Service, Charting and Geodetic Services

  • Ou J (1999) On the reliability for the situation of correlated observations. Acta Geodaetica et Cartographica Sinica, English Edition, pp 9–17

  • Pelzer H (1980) Some criteria for the reliability of networks. Deutsche Geodätische Kommission, Munchen

    Google Scholar 

  • Pope AJ (1976) The statistics of residuals and the detection of outliers. NOAA Technical Report NOS 65 NGS 1, U.S. National Geodetic Survey, MD, USA

  • Ryan S, Lachapelle G (2001) Marine positioning multiple multipath error detection. Hydrogr J 100:3–11

    Google Scholar 

  • Schaffrin B (1997) Reliability measures for correlated observations. J Surv Eng 123(3):126–137

    Article  Google Scholar 

  • Teunissen PJG (1985) Quality control in geodetic networks. In: Grafarend EW, Sanso F (eds) Optimization and design of geodetic networks. Springer, Berlin, pp 526–547

    Chapter  Google Scholar 

  • Teunissen PJG (2006) Testing theory an introduction. Series on mathematical geodesy and positioning, 2nd edn. Delft University of Technology, Delft

    Google Scholar 

  • Teunissen PJG (2018) Distributional theory for the DIA method. J Geod 92:59–80

    Article  Google Scholar 

  • Teunissen PJG, Zaminpardaz S, Tiberius C (2020) On the integrity of deformation monitoring. Geomat Nat Hazards Risk 11(1):399–413

    Article  Google Scholar 

  • Wang J, Chen Y (1999) Outlier detection and reliability measures for singular adjustment models. Geomat Res Aust 71:57–72

    Google Scholar 

  • Wieser A (2004) Reliability checking for GNSS baseline and network processing. GPS Solut 8:55–66

    Article  Google Scholar 

  • Xu P (1987) A test method for many outliers. ITC J 4:314–317

    Google Scholar 

  • Yang L, Wang J, Knight NL, Shen Y (2013) Outlier separability analysis with a multiple alternative hypotheses test. J Geod 87(6):591–604

    Article  Google Scholar 

  • Yang L, Li B, Shen Y, Rizos C (2017) An extension of internal reliability analysis regarding to separability analysis. J Surv Eng 143(3):04017002

    Article  Google Scholar 

  • Yang L, Shen Y, Li B (2019) M-estimation using unbiased median variance estimate. J Geod 93:911–925

    Article  Google Scholar 

  • Yang L, Shen Y (2020) Robust M estimation for 3D correlated vector observations based on modified bifactor weight reduction model. J Geod 94:31

    Article  Google Scholar 

  • Zaminpardaz S, Teunissen PJG (2019) DIA-datasnooping and identifiability. J Geod 93:85–101

    Article  Google Scholar 

  • Zaminpardaz S, Teunissen PJG, Tiberius C (2020) A risk evaluation method for deformation monitoring systems. J Geodesy 94(3):28

    Article  Google Scholar 

Download references

Acknowledgements

This work is sponsored by the National Natural Science Foundation of China (41731069, 41504022), the Shanghai Natural Science Foundation (20ZR1462000), and the Open Research Fund Program of LIESMARS (Grant No. 19R02).

Author information

Authors and Affiliations

  1. College of Surveying and Geo-informatics, Tongji University, Shanghai, 200092, China

    Ling Yang, Yunzhong Shen & Bofeng Li

  2. School of Civil and Environmental Engineering, University of New South Wales, Sydney, 2052, Australia

    Chris Rizos

Authors
  1. Ling Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunzhong Shen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bofeng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chris Rizos
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

LY proposed the key idea, designed the research, processed data, and wrote the paper draft; YS and BL supervised the research and revised the manuscript; CR substantively revised the manuscript.

Corresponding author

Correspondence to Yunzhong Shen.

Ethics declarations

Conflict of interest

The authors declare that the have no conflict of interest.

Appendix

Appendix

The detailed proof for Eqs. (37–39) is as follows.

Under hypothesis \({\mathcal{H}}_{i}\), there is.

$$ w_{i} \sim\{ N\} (\lambda _{i} ,1),w_{j} \sim\{ N\} (\lambda _{i} \rho _{{ji}} ,1),w_{k} \sim\{ N\} (\lambda _{i} \rho _{{jk}} ,1)t)$$
(50)

From Eq. (37), it is derived

$$E\left(\Delta {w}_{jk|i}\right)=\left\{\begin{array}{l}\frac{E\left({w}_{j}-{w}_{k}\right)}{\sqrt{2\left(1-{\rho }_{jk}\right)}}={\lambda }_{i}\frac{\left({\rho }_{ji}-{\rho }_{ki}\right)}{\sqrt{2\left(1-{\rho }_{jk}\right)}}, {\rho }_{ji}\ge 0\;\&\;{\rho }_{ki}\ge 0\\ \frac{E\left({w}_{j}+{w}_{k}\right)}{\sqrt{2\left(1+{\rho }_{jk}\right)}}={\lambda }_{i}\frac{\left({\rho }_{ji}+{\rho }_{ki}\right)}{\sqrt{2\left(1+{\rho }_{jk}\right)}}, {\rho }_{ji}\ge 0\;\&\;{\rho }_{ki}<0\\ -\frac{E\left({w}_{j}+{w}_{k}\right)}{\sqrt{2\left(1+{\rho }_{jk}\right)}}=-{\lambda }_{i}\frac{\left({\rho }_{ji}+{\rho }_{ki}\right)}{\sqrt{2\left(1+{\rho }_{jk}\right)}}, {\rho }_{ji}<0\;\&\;{\rho }_{ki}\ge 0\\ -\frac{E\left({w}_{j}-{w}_{k}\right)}{\sqrt{2\left(1-{\rho }_{jk}\right)}}=-{\lambda }_{i}\frac{\left({\rho }_{ji}-{\rho }_{ki}\right)}{\sqrt{2\left(1-{\rho }_{jk}\right)}}, {\rho }_{ji}<0\;\&\;{\rho }_{ki}<0\end{array}\right.$$
(51)

It can be simplified as

$$E\left(\Delta {w}_{jk|i}\right)=\left\{\begin{array}{l}{\lambda }_{i}\frac{\left(\left|{\rho }_{ji}\right|-\left|{\rho }_{ki}\right|\right)}{\sqrt{2\left(1-{\rho }_{jk}\right)}}, {\rho }_{ji} {\rho }_{ki}\ge 0 \\ {\lambda }_{i}\frac{\left(\left|{\rho }_{ji}\right|-\left|{\rho }_{ki}\right|\right)}{\sqrt{2\left(1+{\rho }_{jk}\right)}}, {\rho }_{ji} {\rho }_{ki}\le 0\end{array}\right.$$
(52)

Similarly, the variance is

$$D\left(\Delta {w}_{jk|i}\right)=\left\{\begin{array}{l}\frac{D\left({w}_{j}\right)+D\left({w}_{k}\right)-2{\rho }_{jk}}{2\left(1-{\rho }_{jk}\right)}=1, {\rho }_{ji}\ge 0 \;\&\;{\rho }_{ki}\ge 0\\ \frac{D\left({w}_{j}\right)+D\left({w}_{k}\right)+2{\rho }_{jk}}{2\left(1+{\rho }_{jk}\right)}=1, {\rho }_{ji}\ge 0\;\&\;{\rho }_{ki}<0\\ \frac{D\left({w}_{j}\right)+D\left({w}_{k}\right)+2{\rho }_{jk}}{2\left(1+{\rho }_{jk}\right)}=1, {\rho }_{ji}<0\;\&\;{\rho }_{ki}\ge 0\\ \frac{D\left({w}_{j}\right)+D\left({w}_{k}\right)-2{\rho }_{jk}}{2\left(1-{\rho }_{jk}\right)}=1, {\rho }_{ji}<0 \;\&\;{\rho }_{ki}<0\end{array}\right.$$
(53)

Therefore, Eqs. (38, 39) are proved.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, L., Shen, Y., Li, B. et al. Simplified algebraic estimation for the quality control of DIA estimator. J Geod 95, 14 (2021). https://doi.org/10.1007/s00190-020-01454-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00190-020-01454-9

Keywords

Search

Navigation