Abstract
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.
Similar content being viewed by others
References
Bako L, Adaptive identification of linear systems subject to gross errors, Automatica, 2016, 67: 192–199.
Bottegal G, Aravkin A Y, Hjalmarsson H, et al., Robust EM kernel-based methods for linear system identification, Automatica, 2016, 67: 114–126.
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.
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.
Corbier C, Huberian function applied to the neurodegenerative disorder gait rhythm, Journal of Applied Statistics, 2016, 43(11): 2065–2084.
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.
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.
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.
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.
Canale A, L∞ estimates for variational solutions of boundary value problems in unbounded domains, Journal of Interdisciplinary Mathematics, 2008, 11(1): 127–139.
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.
Adler R J, The Geometry of Random Fields, Classics in Applied Mathematics SIAM 62, 2010.
Sutton C and McCallum A, An introduction to conditional random fields, Foundations and Trends in Machine Learning, 2011, 4(4): 267–373.
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.
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.
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.
Goresky M and MacPherson R, Stratified Morse Theory, Springer-Verlag, Berlin Heidelberg, 1988.
Pflaum M J, Analytic and Geometric Study of Stratified Spaces, Springer-Verlag, Berlin Heidelberg, 2001.
Amari S, Theory of information spaces, a geometrical foundation of statistics, POST RAAG Report 106, 1980.
Amari S and Nagaoka H, Methods of information geometry, Translations of Mathematical Monographs, Oxford University Press, AMS, 2000, 191.
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.
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.
Vershik A M, Random metric spaces and universality, Russian Math., 2004, 59(2): 259–295.
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.
Huber P J and Ronchetti E M, Robust Statistics, 2nd Edition, New York, John Wiley and Sons, 2009.
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.
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 (wileyonlinelibrary.com), 2014, 1–16.
Whitney H, Tangents to an analytic variety, Annals of Mathematics, 1965, 81(3): 496–549.
Adler R J and Taylor J E, Random Fields and Geometry, Springer Monographs in Mathematics, 2007.
Yamabe H, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., 1960, 12(1): 21–37.
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.
Ljung L, System Identification: Theory for the User, Prentice Hall PTR, New York, 1999.
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.
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.
Kotz S, Kozubowski T J, and Podgorski K, Maximum likehihood estimation of asymmetric Laplace parameters, Ann. Inst. Statist. Math., 2002, 54(4): 816–826.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper was recommended for publication by Editor LIU Yungang.
Rights and permissions
About this article
Cite this article
Corbier, C. Articulated Estimator Random Field and Geometrical Approach Applied in System Identification. J Syst Sci Complex 31, 1164–1185 (2018). https://doi.org/10.1007/s11424-018-6223-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-018-6223-z