Further Results on a Robust Multivariate Time Series Analysis in Nonlinear Models with Autoregressive and t-Distributed Errors

  • Hamza AlkhatibEmail author
  • Boris Kargoll
  • Jens-André Paffenholz
Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


We investigate a time series model which can generally be explained as the additive combination of a multivariate, nonlinear regression model with multiple univariate, covariance stationary autoregressive (AR) processes whose white noise components obey independent scaled t-distributions. These distributions enable the stochastic modeling of heavy tails or outlier-afflicted observations and present the framework for a partially adaptive, robust maximum likelihood (ML) estimation of the deterministic model parameters, of the AR coefficients, of the scale parameters, and of the degrees of freedom of the underlying t-distributions. To carry out the ML estimation, we derive a generalized expectation maximization (GEM) algorithm, which takes the form of linearized, iteratively reweighted least squares. In order to derive a quality assessment of the resulting estimates, we extend this GEM algorithm by a Monte Carlo based bootstrap algorithm that enables the computation of the covariance matrix with respect to all estimated parameters. We apply the extended GEM algorithm to a multivariate global navigation satellite system (GNSS) time series, which is approximated by a three-dimensional circle while taking into account the colored measurement noise and partially heavy-tailed white noise components. The precision of the circle model fitted by the GEM algorithm is superior to that of the previous standard estimation approach.


Multivariate time series Nonlinear regression model AR process Scaled t-distribution Partially adaptive estimation Robust parameter estimation GEM algorithm Bootstrapping GNSS time series 


  1. 1.
    Huber, P.J.: Robust estimation of a location parameter. Ann. Math. Stat. 35, 73–101 (1964)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Lange, K.L., Little, R.J.A., Taylor, J.M.G.: Robust statistical modeling using the t-distribution. J. Am. Stat. Assoc. 84, 881–896 (1989)MathSciNetGoogle Scholar
  3. 3.
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. (Ser. B) 39, 1–38 (1977)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Liu, C.H.: ML estimation of the multivariate t distribution and the EM algorithm. J. Multivar. Anal. 63, 296–312 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Liu, C.H., Rubin, D.B.: The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika 81, 633–648 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Meng, X., Rubin, D.B.: Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80, 267–278 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions. Wiley, Hoboken, New Jersey (2008)CrossRefGoogle Scholar
  8. 8.
    Phillips, R.F.: Least absolute deviations estimation via the EM algorithm. Stat. Comput. 12, 281–285 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Wang, K., Xiong, S., Li, Y.: Modeling with noises for inertial sensors. In: Position Location and Navigation Symposium (PLANS) 2012, IEEE/ION, pp. 625-632 (2012)Google Scholar
  10. 10.
    Park, M., Gao, Y.: Error and performance analysis of MEMS-based inertial sensors with a low-cost GPS receiver. Sensors 8, 2240–2261 (2008)CrossRefGoogle Scholar
  11. 11.
    Schuh, W.D.: The processing of band-limited measurements; filtering techniques in the least squares context and in the presence of data gaps. Space Sci. Rev. 108, 67–78 (2003)CrossRefGoogle Scholar
  12. 12.
    Luo, X.: GPS Stochastic Modelling: Signal Quality Measures and ARMA Processes. Springer, Berlin, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    Kargoll, B., Omidalizarandi, M., Loth, I., Paffenholz, J.A., Alkhatib, H.: An iteratively reweighted least-squares approach to adaptive robust adjustment of parameters in linear regression models with autoregressive and t-distributed deviations. J. Geodesy. (2017). Scholar
  14. 14.
    Alkhatib, H., Kargoll, B., Paffenholz, J.A.: Robust multivariate time series analysis in nonlinear models with autoregressive and t-distributed errors. In: Valenzuela, O., Rojas, F., Pomares, H., Rojas, I. (eds.) Proceedings ITISE 2017—International work-conference on Time Series, vol. 1, pp. 23–36 (2017). ISBN: 978-84-17293-01-7Google Scholar
  15. 15.
    Efron, B.: Bootstrap methods: another look at the jackknife. Ann. Stat. 7(1), 1–26 (1997)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Alkhatib, H.: On Monte Carlo methods with applications to the current satellite gravity missions, Dissertation. Universitt Bonn, Bonn. Institute for Geodesy and Geoinformation (2008).
  17. 17.
    Li, H., Maddala, G.S.: Bootstrapping time series models. Economet. Rev. 15, 115–158 (1996)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Porat, B.: Digital Processing of Random Signals. Dover, Mineola/New York (1994)Google Scholar
  19. 19.
    Hargreaves, G.I.: Interval Analysis in MATLAB. Numerical Analysis Report No. 416, Manchester Centre for Computational Mathematics, The University of Manchester (2002). ISSN 1360-1725Google Scholar
  20. 20.
    Paffenholz, J.A.: Direct Geo-Referencing of 3D Point Clouds with 3D Positioning Sensors. German Geodetic Commission, Series C. Ph.D. thesis, No. 689, Munich (2012).
  21. 21.
    Wübbena, G., Bagge, A., Schmitz, M.: RTK networks based on Geo++ GNSMART–concepts, implementation, resultsGoogle Scholar
  22. 22.
    Nadarajah, N., Paffenholz, J.A., Teunissen, P.J.G.: Integrated GNSS attitude determination and positioning for direct geo-referencing. Sensors 14(7), 12715–12734 (2014). Scholar
  23. 23.
    Moon, P., Spencer, D.E.: Field Theory Handbook—Including Coordinate Systems, Differential Equations, and Their Solutions, 3rd edn. Springer, New York (1988)zbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Geodätisches Institut, Leibniz Universität HannoverHannoverGermany

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