Journal of Geodesy

, Volume 92, Issue 3, pp 271–297 | Cite as

An iteratively reweighted least-squares approach to adaptive robust adjustment of parameters in linear regression models with autoregressive and t-distributed deviations

  • Boris KargollEmail author
  • Mohammad Omidalizarandi
  • Ina Loth
  • Jens-André Paffenholz
  • Hamza Alkhatib
Original Article


In this paper, we investigate a linear regression time series model of possibly outlier-afflicted observations and autocorrelated random deviations. This colored noise is represented by a covariance-stationary autoregressive (AR) process, in which the independent error components follow a scaled (Student’s) t-distribution. This error model allows for the stochastic modeling of multiple outliers and for an adaptive robust maximum likelihood (ML) estimation of the unknown regression and AR coefficients, the scale parameter, and the degree of freedom of the t-distribution. This approach is meant to be an extension of known estimators, which tend to focus only on the regression model, or on the AR error model, or on normally distributed errors. For the purpose of ML estimation, we derive an expectation conditional maximization either algorithm, which leads to an easy-to-implement version of iteratively reweighted least squares. The estimation performance of the algorithm is evaluated via Monte Carlo simulations for a Fourier as well as a spline model in connection with AR colored noise models of different orders and with three different sampling distributions generating the white noise components. We apply the algorithm to a vibration dataset recorded by a high-accuracy, single-axis accelerometer, focusing on the evaluation of the estimated AR colored noise model.


Linear regression model Autoregressive process Scaled t-distribution Adaptive robust estimation Expectation maximization (EM) algorithm Iteratively reweighted least squares 



We thank the editors and the reviewers for their constructive comments and valuable suggestions, which helped to improve this paper. 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 would like to acknowledge the Institute of Concrete Construction (Leibniz Universität Hannover) for providing the shaker table and the reference accelerometer used within this experiment.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Geodetic InstituteLeibniz Universität HannoverHannoverGermany
  2. 2.Institute of Geodesy and GeoinformationRheinische Friedrich-Wilhelms Universität BonnBonnGermany

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