Vojislav Filipović

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In real situations the presence of outliers is unavoidable and that is why the distribution of a disturbance is non-Gaussian. A synthesis of an algorithm of identification based on the Newton-Raphson method is considered for this case. The method requires that the loss function should be twice differentiable. Huber loss function, relevant for the treatment of outliers, has just the first derivative. In order to overcome the problem, the pseudo- Huber loss function is introduced. This function behaves similarly to the Huber loss function and has derivatives of an arbitrary order. In this paper, the pseudo- Huber loss function is used for the second derivative of functional in the Newton-Raphson procedure. The main contributions of the paper are: (i) Design of a new robust recursive algorithm based on the synergy of Huber and pseudo – Huber functions; (ii) The convergence analysis.


Robust identification, Huber function, Pseudo – Huber function, convergence analysis

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F. Huang, J. Zhang, S. Zhang, “Mean – square – deviation analysis of probabilistic LMS algorithm,” Digit. Signal Process, vol.92, no.9, pp 26-35, 2019.

F.S.L.G. Duarte, R.A. Rios, E.R. Hruschka, R.F. de Mello, “Decomposing time series into deterministic and stochastic influences: A survey,” Digit. Signal Process, vol.95, no.12, 102582,

S.Y. Wang, W. Wang, L. Y. Dang, Y. X. Jiang, “Kernel least mean – square based on the Nystrom method,” Circuit, Systems and Signal Processing, vol.38, no.11, pp.3133-3151,2019. doi: 10.1007/s00034-018-1006-2.

P. Huber, E. Ronchetti, Robust Statistics, Wiley, New York, 2009.

M. Sugiyama, Introduction to Statistical Machine Learning, Morgan Kaufman, New York, 2016.

N.N.R. Suri, N. Murty, G. Athithan, Outlier Detection: Techniques and Application. A Data Mining Perspective, Springer, Berlin, 2019.

A.M. Zoubir, V. Koivunen, E. Ollila, M. Muma, Robust Statistics for Signal Processing, Cambridge University Press, Cambridge, 2018.

R.A. Maronna, R.D. Martin, V. Yohai, M. Salibian – Barrera, Robust Statistics. Theory and Methods (with R), Wiley, New York, 2019.

R. Pearson, Exploring Data in Engineering, the Science and Medicine, Oxford University Press, Oxford, 2011.

Ya. Z. Tsypkin, On the Foundations of Information and Identification Theory (in Russian), Nauka, Moscow, 1984.

V. Filipovic, “Recursive identification of block-oriented nonlinear systems in the presence of outliers”, J. of Process Control, vol.78, no. 6, pp. 1-12, 2019. doi: 10.1016/j.jprocont.2019.03.015.

J. Castro, A CTA model based on the Huber function, J. Domingo – Ferrer (Ed.): Privacy in Statistical Data Bases, Springer, Berlin, 2014.

R. Holtey, Multiple View Geometry in Computer Vision, Cambridge University Press, Cambridge, 2004.

L. Stefanski, D. Boss, “The calculus of M – estimation. The American Statistician,” vol.56, no. 1, pp.29-38,2002. doi: 10.1198/000313002753631330.

N. Stout, Almost Sure Convergence, Academic Press, New York, 1974.

L. Lai, Z. Wei, “Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems,” Annals of Statistics, vol.10, no.1, pp.151-166,1982. doi: 10.2307/2240506.

V. Filipovic, “Consistency of the robust recursive Hammerstein model identification algorithm,” J. Franklin Inst., vol.352 , no.5, pp. 1932-1943, 2015. doi: 101016//j.jfranklin.201502.005.

V. Filipovic, B. Kovacevic, “On robust AML identification algorithms,” Automatica, vol.30, no.11, pp. 1775-1778,1994. doi: 10.1016/0005-1098(94) 90081-7.

C. Desoer, M. Vidyasagar, Feedback Systems: Input - Output Properties, Academic Press, New York, 1975.



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