Journal of Systems Science and Complexity

, Volume 31, Issue 5, pp 1164–1185 | Cite as

Articulated Estimator Random Field and Geometrical Approach Applied in System Identification

  • Christophe CorbierEmail author


A new point of view of robust statistics based on a geometrical approach is tackled in this paper. Estimation procedures are carried out from a new robust cost function based on a chaining of elementary convex norms. This chain is randomly articulated in order to treat more efficiently natural outliers in data-set. Estimated parameters are considered as random fields and each of them, named articulated estimator random field (AERF) is a manifold or stratum of a stratified space with Riemannian geometry properties. From a high level excursion set, a probability distribution model Msta is presented and a system model validation geometric criterion (SYMOVAGEC) for system model structures Msys based on Riccian scalar curvatures is proposed. Numerical results are drawn in a context of system identification.


Articulated robust estimation estimators random field information geometry stratified space system identification 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bako L, Adaptive identification of linear systems subject to gross errors, Automatica, 2016, 67: 192–199.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Bottegal G, Aravkin A Y, Hjalmarsson H, et al., Robust EM kernel-based methods for linear system identification, Automatica, 2016, 67: 114–126.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Alessandri A and Awawdeh M, Moving-horizon estimation with guaranteed robustness for discrete-time linear and measurements subject to outliers, Automatica, 2016, 67: 85–93.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Corbier C, Badaoui M E, and Ugalde H M R, Huberian approach for reduced order arma modeling of neurodegenerative disorder signal, Signal Processing, 2015, 113: 273–284.CrossRefGoogle Scholar
  5. [5]
    Corbier C, Huberian function applied to the neurodegenerative disorder gait rhythm, Journal of Applied Statistics, 2016, 43(11): 2065–2084.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Corbier C and Romero Ugalde H M, Low-order control-oriented modeling of piezoelectric actuator using Huberian function with low threshold: Pseudolinear and neural network models, Nonlinear Dynamics, 2016, 85(2): 923–940.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Carmona J C and Alvarado V, Active noise control of a duct using robust control theory, IEEE Tran. on Control Syst. Technology, 2000, 8(6): 930–938.CrossRefGoogle Scholar
  8. [8]
    Wang H, Nie F, and Huang H, Robust distance metric learning via simultaneous L1-norm minimization and maximization, Proceedings of the 31st International Conference on Machine Learning, Beijing, 2014, 32: 1–9.Google Scholar
  9. [9]
    Ebegila M and Gokpnara F, A test statistic to choose between Liu-type and least-squares estimator based on mean square error criteria, Journal of Applied Statistics, 2012, 39(10): 2081–2096.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Canale A, L∞ estimates for variational solutions of boundary value problems in unbounded domains, Journal of Interdisciplinary Mathematics, 2008, 11(1): 127–139.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Jukic D, The Lp-norm estimation of the parameters for the JelinskiMoranda model in software reliability, International Journal of Computer Mathematics, 2012, 89(4): 467–481.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Adler R J, The Geometry of Random Fields, Classics in Applied Mathematics SIAM 62, 2010.Google Scholar
  13. [13]
    Sutton C and McCallum A, An introduction to conditional random fields, Foundations and Trends in Machine Learning, 2011, 4(4): 267–373.CrossRefzbMATHGoogle Scholar
  14. [14]
    Ferreira M A R and De Oliveira V, Bayesian reference analysis for Gaussian Markov random fields, Journal of Multivariate Analysis, 2007, 98(4): 789–812.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Francos J M and Friedlander B, Parameter estimation of two-dimensional moving average random fields, IEEE Trans. on Signal Processing, 1998, 46(8): 2157–2165.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Adler R J, Samorodnitsky G, and Taylor J E, High level excursion set geometry for non-Gaussian infinitely divisible random fields, The Annals of Probability, 2013, 41(1): 134–169.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Goresky M and MacPherson R, Stratified Morse Theory, Springer-Verlag, Berlin Heidelberg, 1988.CrossRefzbMATHGoogle Scholar
  18. [18]
    Pflaum M J, Analytic and Geometric Study of Stratified Spaces, Springer-Verlag, Berlin Heidelberg, 2001.zbMATHGoogle Scholar
  19. [19]
    Amari S, Theory of information spaces, a geometrical foundation of statistics, POST RAAG Report 106, 1980.Google Scholar
  20. [20]
    Amari S and Nagaoka H, Methods of information geometry, Translations of Mathematical Monographs, Oxford University Press, AMS, 2000, 191.Google Scholar
  21. [21]
    Amari S, Differential geometry of a parametric family of invertible linear systems-Riemannian metric, dual affine connections and divergence, Mathematical Systems Theory, 1987, 20: 53–82.CrossRefzbMATHGoogle Scholar
  22. [22]
    Greven A, Pfaffelhuber P, and Winter A, Convergence in distribution of random metric measure spaces (Λ-coalescent measure trees), Probab. Theory Relat. Fields, 2009, 145: 285–322.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Vershik A M, Random metric spaces and universality, Russian Math., 2004, 59(2): 259–295.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Tukey J W, A survey of sampling from contaminated distributions, Contributions to Probability and Statistics, Ed. by Olkin I, Stanford Univ. Press, Stanford, 1960, 448–485.Google Scholar
  25. [25]
    Huber P J and Ronchetti E M, Robust Statistics, 2nd Edition, New York, John Wiley and Sons, 2009.CrossRefzbMATHGoogle Scholar
  26. [26]
    Andrews D F, Bickel P J, Hampel F R, et al., A robust estimation of location: Survey and advances, Princeton Univ. Press, Princeton, New Jersey, 1972.zbMATHGoogle Scholar
  27. [27]
    Corbier C and Carmona J C, Extension of the tuning constant in the Huber’s function for robust modeling of piezoelectric systems, Int. J. Adapt. Control Signal Process, Published online in Wiley Online Library (, 2014, 1–16.Google Scholar
  28. [28]
    Whitney H, Tangents to an analytic variety, Annals of Mathematics, 1965, 81(3): 496–549.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Adler R J and Taylor J E, Random Fields and Geometry, Springer Monographs in Mathematics, 2007.zbMATHGoogle Scholar
  30. [30]
    Yamabe H, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., 1960, 12(1): 21–37.MathSciNetzbMATHGoogle Scholar
  31. [31]
    Corbier C and Carmona J C, Mixed Lp-estimators variety for model order reduction in control oriented system identification, Mathematical Problems in Engineering, 2014, Article ID 349070, 1–19.Google Scholar
  32. [32]
    Ljung L, System Identification: Theory for the User, Prentice Hall PTR, New York, 1999.CrossRefzbMATHGoogle Scholar
  33. [33]
    Allende H, Frery A C, and Galbiatis J, M-estimators with asymmetric influence functions: The G0A distribution case, J. of Statist. Comput. and Simul., 2006, 76(11): 941–956.CrossRefzbMATHGoogle Scholar
  34. [34]
    Romero Ugalde H M, Carmona J C, Reyes-Reyes J, et al., Balanced simplicity-accuracy neural network model families for system identification, Neural Computing and Applications, 2015, 26(1): 171–186.CrossRefGoogle Scholar
  35. [35]
    Kotz S, Kozubowski T J, and Podgorski K, Maximum likehihood estimation of asymmetric Laplace parameters, Ann. Inst. Statist. Math., 2002, 54(4): 816–826.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    Lee J Y and Nandi A K, Maximum likelihood parameter estimation of the asymmetric generalised Gaussian family of distributions, IEEE Conference in Caesarea, Higher-Order Statistics, Proceedings of the IEEE Signal Processing Workshop on, 1999, 255–258.CrossRefGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Lyon University, St-Etienne Jean Monnet University, Roanne Technology University InstituteRoanne cedexFrance

Personalised recommendations