Further Results on a Modified EM Algorithm for Parameter Estimation in Linear Models with Time-Dependent Autoregressive and t-Distributed Errors

  • Boris KargollEmail author
  • Mohammad Omidalizarandi
  • Hamza Alkhatib
  • Wolf-Dieter Schuh
Conference paper
Part of the Contributions to Statistics book series (CONTRIB.STAT.)


In this contribution, we consider an expectation conditional maximization either (ECME) algorithm for the purpose of estimating the parameters of a linear observation model with time-dependent autoregressive (AR) errors. The degree of freedom (d.o.f.) of the underlying family of scaled t-distributions, which is used to account for outliers and heavy-tailedness of the white noise components, is adapted to the data, resulting in a self-tuning robust estimator. The time variability of the AR coefficients is described by a second linear model. We improve the estimation of the d.o.f. in a previous version of the ECME algorithm, which involves a zero search, by using an interval Newton method. We model the transient oscillations of a shaker table measured by a high-accuracy accelerometer, and we analyze various criteria for selecting a simultaneously parsimonious and realistic time-variability model.


Linear model Time-dependent AR process Scaled t-distribution Self-tuning robust estimator EM algorithm Vibration analysis 



The presented application of the PCB Piezotronics accelerometer within the vibration analysis experiment was performed as a part of the collaborative project “Spatio-temporal monitoring of bridge structures using low cost sensors” with ALLSAT GmbH, which is funded by the German Federal Ministry for Economic Affairs and Energy (BMWi) and the Central Innovation Programme for SMEs (ZIM Kooperationsprojekt, ZF4081803DB6). In addition, the authors acknowledge the Institute of Concrete Construction (Leibniz Universität Hannover) for providing the shaker table and the reference accelerometer.


  1. 1.
    Kuhlmann, H.: Importance of autocorrelation for parameter estimation in regression models. In: 10th FIG International Symposium on Deformation Measurements, pp. 354–361. FIG (2001)Google Scholar
  2. 2.
    Brockmann, J.M., Kargoll, B., Krasbutter, I., Schuh, W.D., Wermuth M.: GOCE data analysis: from calibrated measurements to the global Earth gravity field. In: Flechtner, F., Gruber, T., Gntner A., Mandea, M., Rothacher, M., Schne, T., Wickert, J. (eds.) Advanced Technologies in Earth Sciences System Earth via Geodetic-Geophysical Space Techniques, pp. 213–229. Springer, Berlin (2010)Google Scholar
  3. 3.
    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. Geod. (2017). Scholar
  4. 4.
    Krasbutter, I., Brockmann, J.M., Kargoll, B., Schuh, W.D., Goiginger, H., Pail, R.: Refinement of the stochastic model of GOCE scientific data in a long time series. In: Ouwehand, L. (ed.) 4th International GOCE User Workshop. ESA Publication SP-696, ESA/ESTEC (2011). ISBN: 978-92-9092-260-5Google Scholar
  5. 5.
    Kargoll, B., Omidalizarandi, M., Alkhatib, H., Schuh, W.D.: A modified EM algorithm for parameter estimation in linear models with time-dependent 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. 2, pp. 1132–1145 (2017). ISBN 978-84-17293-01-7Google Scholar
  6. 6.
    Subba Rao, T.: The fitting of non-stationary time-series models with time-dependent parameters. J. Roy. Stat. Soc. B Met. 32(2), 312–322 (1970)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Azrak, R., Mélard, G.: AR models with time-dependent coefficients—a comparison between several approaches. Technical Report 0642, IAP Statistics Network, Interuniversity Attraction Pole (2006)Google Scholar
  8. 8.
    Dahlhaus, R.: Fitting time series models to nonstationary processes. Ann. Statist. 25(1), 1–37 (1997). Scholar
  9. 9.
    Rudoy, D., Quatieri, T.F., Wolfe, P.J.: Time-varying autoregressions in speech: detection theory and applications. IEEE Trans. Audio Speech Lang. Process. 19(4), 977–989 (2011). Scholar
  10. 10.
    Schuh, W.D., Brockmann, J.M.: Refinement of the stochastic model for GOCE gravity gradients by non-stationary decorrelation filters. In: ESA Living Planet Symposium 2016, Prag, Poster 2382 (2016)Google Scholar
  11. 11.
    Tsatsanis, M.K., Giannakis, G.B.: Time-varying system identification and model validation using wavelets. IEEE Trans. Sign. Process. 41, 3512–3523 (1993). Scholar
  12. 12.
    Eom, K.B.: Analysis of acoustic signatures from moving vehicles using time-varying autoregressive models. Multidim. Sys. Sign. Process. 10(4), 357–378 (1999). Scholar
  13. 13.
    Härmä, A., Juntunen, M., Kaipio, J.P.: Time-varying autoregressive modeling of audio and speech signals. In: 10th European Signal Processing Conference. IEEE (2000)Google Scholar
  14. 14.
    Grenier, Y.: Time-dependent ARMA modeling of nonstationary signals. IEEE Trans. Acoust. Speech Sign. Process. 31(4), 899–911 (1983)CrossRefGoogle Scholar
  15. 15.
    Pierce, D.A.: Least squares estimation in the regression model with autoregressive-moving average errors. Biometrika 58(2), 299–312 (1971). Scholar
  16. 16.
    McDonald, J.B., Newey, W.K.: Partially adaptive estimation of regression models via the generalized t distribution. Econom. Theor. 4(3), 428–457 (1988)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Parzen, E.: A density-quantile function perspective on robust estimation. In: Launer, L., Wilkinson, G.N. (eds.) Robustness in Statistics, pp. 237–258. Academic Press, New York (1979). Scholar
  18. 18.
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. B Met. 39(1), 1–38 (1977)MathSciNetzbMATHGoogle Scholar
  19. 19.
    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). Scholar
  20. 20.
    Liu, C.H., Rubin, D.B.: ML estimation of the t distribution using EM and its extensions. ECM ECME. Stat. Sin. 5, 19–39 (1995)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hamilton, J.D.: Time Series Analysis. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar
  22. 22.
    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(1), 67–78 (2003). Scholar
  23. 23.
    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
  24. 24.
    Neitzel, F., Niemeier, W., Weisbrich, S., Lehmann, M.: Investigation of low-cost accelerometer, terrestrial laser scanner and ground-based radar interferometer for vibration monitoring of bridges. In: 6th European Workshop on Structural Health Monitoring (2012).
  25. 25.
    Schlittgen, R., Streitberg, B.H.J.: Zeitreihenanalyse, 9th edn. R. Oldenbourg Verlag, Munich (2001)zbMATHGoogle Scholar
  26. 26.
    Tary, J.B., Herrera, R.H., van der Baan, M.: Time-varying autoregressive model for spectral analysis of microseismic experiments and long-period volcanic events. Geophys. J. Int. 196(1), 600–611 (2014). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Boris Kargoll
    • 1
    Email author
  • Mohammad Omidalizarandi
    • 1
  • Hamza Alkhatib
    • 1
  • Wolf-Dieter Schuh
    • 2
  1. 1.Leibniz Universität Hannover, Geodätisches InstitutHannoverGermany
  2. 2.Rheinische Friedrich-Wilhelms-Universität BonnInstitut für Geodäsie und GeoinformationBonnGermany

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