Statistical Linearization and Consistent Measures of Dependence: A Unified Approach

Kirill Chernyshov

Abstract

The paper presents a unified approach to the statistical linearization of input/output mapping of non-linear discrete-time stochastic systems driven with white-noise Gaussian process. The approach is concerned with a possibility of applying any consistent measures of dependence (that is those measures of dependence of a pair of random values, which vanish if and only if these random values are stochastically independent) in statistical linearization problems and oriented to the elimination of drawbacks concerned with applying correlation and dispersion (based on the correlation ratio) measures of dependence, based on linearized representations of their input/output models.

References

  1. Gebelein, H., 1941. “Das statistische Problem der Korrelation als Variations- und Eigenwertproblem und sein Zusammenhang mit der Ausgleichungsrechnung”, Zeitschrift für Angewandte Mathematik und Mechanik, vol. 21, no. 6, pp. 364-379.
  2. Kotz, S., Balakrishnan, N., and N.L. Johnson, 2000. Continuous Multivariate Distributions. Volume 1. Models and Applications / Second Edition, Wiley, New York, 752 p.
  3. Nadarajah, S. and K. Zografos, 2003. “Formulas for Rényi information and related measures for univariate distributions”, Information Sciences, vol. 155, no. 1, pp. 119-138.
  4. Nadarajah, S. and K. Zografos, 2005a. “Expressions for Rényi and Shannon entropies for bivariate distributions”, Information Sciences, vol. 170, no. 2-4, pp. 173-189..
  5. Principe, J., Xu, D., and J. Fisher, 2000. “Information Theoretic Learning”, In: Unsupervised Adaptive Filtering / Haykin (Ed.). Wiley, New York, vol. 1, pp. 265-319.
  6. Rajbman, N.S., 1981. “Extensions to nonlinear and minimax approaches”, Trends and Progress in System Identification, ed. P. Eykhoff, Pergamon Press, Oxford, pp. 185-237.
  7. Rényi, A., 1959. “On measures of dependence”, Acta Math. Hung., vol. 10, no 3-4, pp. 441-451.
  8. Rényi, A., 1961. “On measures of information and entropy”, in: Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability (June 20- July 30, 1960). University of California Press, Berkeley, California, vol. 1, pp. 547-561.
  9. Rényi, A., 1976a. “Some Fundamental Questions of Information Theory”, Selected Papers of Alfred Rényi, Akademiai Kiado, Budapest, vol. 2, pp. 526-552.
  10. Rényi, A., 1976b. “On Measures of Entropy and Information”, Selected Papers of Alfréd Renyi, Akademiai Kiado, Budapest, vol. 2, pp. 565-580.
  11. Roberts, J.B. and P.D. Spanos, 2003. Random Vibration and Statistical Linearization, Dover, New York, 464 p.
  12. Sarmanov, O.V and E.K. Zakharov, 1960. “Measures of dependence between random variables and spectra of stochastic kernels and matrices”, Matematicheskiy Sbornik, vol. 52(94), pp. 953-990. (in Russian).
  13. Sarmanov, O.V., 1963a. “Investigation of stationary Markov processes by the method of eigenfunction expansion”, Sel. Trans. Math. Statist. Probability, vol. 4, pp. 245-269.
  14. Sarmanov, O.V., 1963b. “The maximum correlation coefficient (nonsymmetric case)”, Sel. Trans. Math. Statist. Probability, vol. 4, pp. 207-210.
  15. Sarmanov, O.V., 1967. “Remarks on uncorrelated Gaussian dependent random variables”, Theory Probab. Appl., vol. 12, issue 1, pp. 124-126.
  16. Socha. L., 2008. Linearization Methods for Stochastic Dynamic Systems, Lect. Notes Phys. 730, Springer, Berlin, Heidelberg, 383 p.
  17. Zografos, K. and S. Nadarajah, 2005b. “Expressions for Rényi and Shannon entropies for multivariate distributions”, Statistics & Probability Letters, vol. 71, no. 1, pp. 71-84.
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Paper Citation


in Harvard Style

Chernyshov K. (2015). Statistical Linearization and Consistent Measures of Dependence: A Unified Approach . In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO, ISBN 978-989-758-122-9, pages 524-532. DOI: 10.5220/0005534805240532


in Bibtex Style

@conference{icinco15,
author={Kirill Chernyshov},
title={Statistical Linearization and Consistent Measures of Dependence: A Unified Approach},
booktitle={Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},
year={2015},
pages={524-532},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005534805240532},
isbn={978-989-758-122-9},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,
TI - Statistical Linearization and Consistent Measures of Dependence: A Unified Approach
SN - 978-989-758-122-9
AU - Chernyshov K.
PY - 2015
SP - 524
EP - 532
DO - 10.5220/0005534805240532