Linear Modeling and Invariances

  • Igor Grabec
  • Wolfgang Sachse
Part of the Springer Series in Synergetics book series (SSSYN, volume 68)


Let us consider a natural phenomenon that can be characterized by a two- dimensional variable Z = (X, Y) and let us assume that the empirical sample points (Z 1 , Z 2 ,…, Z N ) lie close to the line described by a linear relation Ŷ = AX. In order to estimate the variable Y from a given value of X we can apply the conditional average estimator defined in the previous chapter or we can directly start with the linear relation and by minimizing the mean square error determine the coefficient A. The resulting expression A = cov(XY)/var(X), is well-known from the literature on linear statistical estimation. It can then be utilized to estimate Y from given X by the linear regression equation Ŷ = X cov(XY)/var(X).


Correlation Function Acoustic Emission Signal Ultrasonic Signal Sequential Adaptation Component Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. E. Albert, L. A. Gardner: Stochastic Approximation and Nonlinear Regression ( MIT Press, Cambridge, MA 1967 )zbMATHGoogle Scholar
  2. 2.
    S. T. Alexander: Adaptive Signal Processing ( Springer, New York 1986 )zbMATHGoogle Scholar
  3. 3.
    J. A. Anderson, E. Rosenfeld: Neurocomputing, Foundations of Research, (The MIT Press, Cambridge, MA 1988 ), Vol. 1, 2Google Scholar
  4. 4.
    J. A. Anderson, A. Pellionisz, E. Rosenfeld: Neurocomputing 2, Directions for Research ( The MIT Press, Cambridge, MA 1990 )Google Scholar
  5. 5.
    M. Bertero: “Linear Inverse and Ill-Posed Problems”, Report of Instituto Nazionale Di Fisica Nucleare, Frascati, Italy, INFN/TC-88/2Google Scholar
  6. 6.
    I. Grabec: “Application of Deconvolution in Description of Operation of Linear Systems”, Mechanical Engineering Journal, 27, 1–7 (1981)Google Scholar
  7. 7.
    I. Grabec: “Description of Operation of Noisy Linear Systems”, Mechanical Engineering Journal, 29, E 1–5 (1983)Google Scholar
  8. 8.
    I. Grabec: “Optimal Filtering of Transient AE Signals”, Proc. Ultrasonics International’85, London 1985, ( Butterworth, Guildford, UK 1985 ), pp. 219–224Google Scholar
  9. 9.
    I. Grabec: “Chaos Generated by the Cutting Process”, Physics Letters, 117, 384–386 (1986)MathSciNetCrossRefGoogle Scholar
  10. 10.
    I. Grabec, E. Elsayed: “Analysis of AE During Martensitic Transformation in the CuZnAl Alloy by the Measurement of Forces”, Physics Letters, 113, 376–378 (1986)CrossRefGoogle Scholar
  11. 11.
    I. Grabec, E. Elsayed: “Quantitative Analysis of AE During Martensitic Transformation in the CuZn Alloy”, J. Phys., D — Appl. Phys., 19, 605–614 (1986)ADSCrossRefGoogle Scholar
  12. 12.
    I. Grabec, W. Sachse: “Application of an Intelligent Signal Processing System to Acoustic Emission Analysis”, J. Acoust. Soc. Am., 85, 1226–1235 (1989)ADSCrossRefGoogle Scholar
  13. 13.
    I. Grabec, W. Sachse: “Experimental Characterization of Ultrasonic Phenomena by a Learning System”, J. Appl. Phys., 66, 3993–4000 (1989)ADSCrossRefGoogle Scholar
  14. 14.
    I. Grabec, K. Zgonc, W. Sachse: “Application of a Neural Network to Analysis of Ultrasonic Signals”, Proc. Ultrasonics International’89, Madrid 1989 ( Butterworth, Guildford, UK 1989 ), pp. 796–802Google Scholar
  15. 15.
    H. Haken: Synergetic Computers and Cognition, A Top-Down Approach to Neural Nets, Springer Series in Synergetics, Vol. 50 ( Springer, Berlin 1991 )Google Scholar
  16. 16.
    S. Haykin: Adaptive Filter Theory ( Prentice-Hall, London 1991 )zbMATHGoogle Scholar
  17. 17.
    R. Hecht-Nielsen: Neurocomputing ( Addison-Wesley, Reading, MA, 1990 )Google Scholar
  18. 18.
    D. O. Hebb: The Organization of Behavior, A Neurophysiological Theory ( Wiley, New York, 1948 )Google Scholar
  19. 19.
    T. Kohonen: Self-Organization and Associative Memory ( Springer, Berlin 1989 )Google Scholar
  20. 20.
    Y. W. Lee: Statistical Theory of Communication (J. Wiley & Sons, New York 1960 )Google Scholar
  21. 21.
    G. I. Marchuk: Methods of Numerical Mathematics ( Springer, Berlin 1975 )zbMATHGoogle Scholar
  22. 22.
    J. E. Michaels, T. E. Michaels, W. Sachse: “Application of Deconvolution to Acoustic Emission Signal Analysis”, Materials Evaluation, 39, 1032–1036 (1981)Google Scholar
  23. 23.
    S. J. Orfandis: Optimum Signal Processing: An Introduction ( Macmillan, New York 1988 )Google Scholar
  24. 24.
    Y. H. Pao: Elastic Waves and Nodestructive Testing of Materials (American Society of Mechanical Engineers, New York 1987 ), Vol. 29, pp. 107–128Google Scholar
  25. 25.
    H. Robbins, S. Monro: “A Stochastic Approximation Method”, Annals of Mathematical Statistics, 22, 400–407 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    D. E. Rumelhart, J. McClelland, and The PDP Research Group: Parallel Distributed Processing ( MIT Press, Cambridge, MA 1986 )Google Scholar
  27. 27.
    W. Sachse, S. Golan: in Elastic Waves and Non-Destructive Testing of Materials, ed. by Y. H. Pao ( AMD, ASME, New York 1978 ), Vol. 29, pp. 11–31Google Scholar
  28. 28.
    W. Sachse, N. N. Hsu: “Ultrasonic Transducers for Materials Testing and Their Characterization”, in Physical Acoustics, ed. by W. P. Mason and R. N. Thurston (Academic, New York 1979 ), Vol. 14, pp. 177–405Google Scholar
  29. 29.
    W. Sachse: “Application of Quantitative AE Methods”, in Solid Mechanics for Quantitative NDE, ed. by J. D. Achenbach, Y. Rajapakse ( Martinus Nijhoff, Dordrecht 1987 ), pp. 41–64Google Scholar
  30. 30.
    A. Tarantola: Inverse Problem Theory; Methods for Data Fitting and Model Parameter Estimation ( Elsevier, Amsterdam 1987 )zbMATHGoogle Scholar
  31. 31.
    B. Widrow, M. E. Hoff: Adaptive Switching Circuits, 1960 IRE WESCON Convention Record, 96–1054, New York, 1960Google Scholar
  32. 32.
    K. Zgonc: PhD Dissertation ( Faculty of Mechanical Engineering, University of Ljubljana 1992 )Google Scholar
  33. 33.
    K. Zgonc, I. Grabec: “A Multidimensional Optimal Deconvolution Applied to Hetero-associative Recall of Acoustic Emission Signals”, Proc. ECPD Neurocomputing, Dubrovnik, 1990, Vol. 1, No. 1, pp. 206–212Google Scholar
  34. 34.
    K. Zgonc, I. Grabec: “Multidimensional Deconvolution Applied to Acoustic Emission Analysis”, 1st Symp. Eval. Adv. Mat. by AE, Tokyo ( JSNDI, Tokyo 1990 )Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Igor Grabec
    • 1
  • Wolfgang Sachse
    • 2
  1. 1.Faculty of Mechanical EngineeringUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Theoretical and Applied MechanicsCornell UniversityIthacaUSA

Personalised recommendations