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Cybernetics and Systems Analysis

, Volume 55, Issue 1, pp 22–29 | Cite as

Multidimensional Scaling by Means of Pseudoinverse Operations

  • Iu. V. KrakEmail author
  • G. I. Kudin
  • A. I. Kulyas
Article
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Abstract

This article proposes a method for multidimensional information scaling based on the results of the theory of perturbation of pseudoinverse and projection matrices and solutions of systems of linear algebraic equations. An algorithm is developed for piecewise hyperplane clusterization with the verification of a given criterion of clusterization efficiency. An example of using the method for scaling characteristic features to recognize the letters of the Ukrainian sign language alphabet is given.

Keywords

scaling classification clusterization pseudoinverse matrix 
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References

  1. 1.
    M. Davison, Multidimensional Scaling [Russian translation], Finance and Statistics, Moscow (1988).Google Scholar
  2. 2.
    K. V. Vorontsov, Lectures on Algorithms of Clustering and Multidimensional Scaling [in Russian]. URL: http://www.ccas.ru/voron/download/Clustering.pdf (access date March 23, 2018).
  3. 3.
    M. Zhambyu, Hierarchical Cluster Analysis and Matching [Russian translation], Finance and Statistics, Moscow (1988).Google Scholar
  4. 4.
    N. F. Kirichenko, Yu. G. Krivonos, and N. P. Lepekha, “Optimization of the synthesis of hyperplane clusters and neurofunctional transforms in signal classification system,” Cybernetics and Systems Analysis, Vol. 44, No. 6, 832–839 (2008).MathSciNetzbMATHGoogle Scholar
  5. 5.
    N. F. Kirichenko, Yu. G. Krivonos, and N. P. Lepekha, “Synthesis of systems of neurofunctional transformations in classification problems,” Cybernetics and Systems Analysis, Vol. 43, No. 3, 353–361 (2007).MathSciNetzbMATHGoogle Scholar
  6. 6.
    N. F. Kirichenko and V. S. Donchenko, “Pseudoinverse in clustering problems,” Cybernetics and Systems Analysis, Vol. 43, No. 4, 527–541 (2007).MathSciNetzbMATHGoogle Scholar
  7. 7.
    N. F. Kirichenko and G. I. Kudin, “Analysis and synthesis of signal classification systems by perturbing pseudoinverse and projection operations,” Cybernetics and Systems Analysis, Vol. 45, No. 4, 613–622 (2009).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Iu. V. Krak, Iu. G. Kryvonos, O. V. Barmak, and A. S. Ternov, “An approach to the determination of efficient features and synthesis of an optimal band-separating classifier of dactyl elements of sign language,” Cybernetics and Systems Analysis, Vol. 52, No. 2, 173–180 (2016).MathSciNetzbMATHGoogle Scholar
  9. 9.
    I. V. Sergienko, A. N. Khimich, and M. F. Yakovlev, “Methods for obtaining reliable solutions to systems of linear algebraic equations,” Cybernetics and Systems Analysis, Vol. 47, No. 1, 62–73 (2011).MathSciNetzbMATHGoogle Scholar
  10. 10.
    A. N. Khimich and E. A. Nikolaevskaya, “Reliability analysis of computer solutions of systems of linear algebraic equations with approximate initial data,” Cybernetics and Systems Analysis, Vol. 44, No. 6, 863–874 (2008).MathSciNetzbMATHGoogle Scholar
  11. 11.
    E. A. Nikolaevskaya and A. N. Khimich, “Error estimation for a weighted minimum-norm least squares solution with positive definite weights,” Computational Mathematics and Mathematical Physics, Vol. 49, No. 3, 409–417 (2009).MathSciNetzbMATHGoogle Scholar
  12. 12.
    R. Penrose, “A generalized inverse for matrices,” in: Proc. Cambridge Philosophical Society, Vol. 51, 406–413 (1955).Google Scholar
  13. 13.
    D. Forsythe, M. Malcolm, and K. Mouler, Machine Methods of Mathematical Calculations [Russian translation], Mir, Moscow (1980).Google Scholar
  14. 14.
    A. Ben-Israel and T. N. E. Greville, Generalized Inverse: Theory and Applications, 2nd Edition, Springer, New York (2003).zbMATHGoogle Scholar
  15. 15.
    N. F. Kirichenko, “Analytical representation of perturbations of pseudoinverse matrices,” Cybernetics and Systems Analysis, Vol. 33, No. 2, 230–238 (1997).MathSciNetzbMATHGoogle Scholar
  16. 16.
    Yu. G. Krivonos, M. F. Kirichenko, Yu. V. Krak, V. S. Donchenko, and A. I. Kulyas, Analysis and Synthesis of Situations in Decision Support Systems [in Ukrainian], Naukova Dumka, Kyiv (2009).Google Scholar
  17. 17.
    N. F. Kirichenko, Yu. V. Krak, and A. A. Polishchuk, “Pseudoinverse and projection matrices in problems of synthesis of functional transformers,” Cybernetics and Systems Analysis, Vol. 40, No. 3, 407–419 (2004).MathSciNetzbMATHGoogle Scholar
  18. 18.
    Iu. G. Kryvonos, Iu. V. Krak, O. V. Barmak, and D. V. Shkilniuk, “Construction and identification of elements of sign communication,” Cybernetics and Systems Analysis, Vol. 49, No. 2, 163–172 (2013).Google Scholar
  19. 19.
    Iu. G. Kryvonos, Yu. V. Krak, Yu. V. Barchukova, and B. A. Trotsenko, “Human hand motion parametrization for dactylemes modeling,” Journal of Automation and Information Sciences, Vol. 43, Iss. 12, 1–11 (2011).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Taras Shevchenko National University of Kyiv and V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine
  2. 2.Taras Shevchenko National University of KyivKyivUkraine
  3. 3.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine

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