Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Developing the Theory of Stochastic Canonic Expansions and Its Applications

  • SCIENTIFIC SCHOOLS OF THE FEDERAL RESEARCH CENTER “COMPUTER SCIENCE AND CONTROL” OF THE RUSSIAN ACADEMY OF SCIENCES, MOSCOW, THE RUSSIAN FEDERATION
  • I.N. Sinitsyn’s Scientific School
  • Published:
Pattern Recognition and Image Analysis Aims and scope Submit manuscript

Abstract

The creation of the theory of canonic expansions (CEs) is related with the names Loeve, Kolmogorov, Karhunen, and Pugachev and dates back to the 1940–1950s. The development of the theory of CEs and wavelet CEs is considered in application to the problems of the analysis, modeling, and synthesis of stochastic systems (SSs) and technologies. The direct and inverse Pugachev theorems about CEs are extended to the case of stochastic linear functionals within the framework of the correlational theory of stochastic functions (SFs). The CEs of linear and quasi-linear SFs are derived. Particular attention is paid to the problems of the equivalent regression linearization of strongly nonlinear transformations by CEs. The nonlinear regression algorithms on the basis of CEs are proposed. The theory of wavelet CEs within the specified domain of the change of the argument on the basis of Haar wavelets is developed. For stochastic elements (SEs), the direct and inverse Pugachev theorems are formulated and the correlational theory of joint CEs for two SEs is developed together with the theory of linear transformations. The solution of linear operator equations by the CEs of SEs in linear spaces with a basis is given. Special attention is focused on the CEs of SEs in Banach spaces with a basis. Some elements of the general theory of distributions for the CEs of SFs and SEs are developed. Particular attention is paid to the method based on CEs with independent components. Some new methods for the calculation of Radon–Nikodym derivatives are proposed. The considered applications of CEs and wavelet CEs to analysis, modeling, and synthesis problems are as follows: SSs and technologies, modeling, identification and recognition filtering, metrological and biometric technologies and systems, and synergic organizational technoeconomic systems (OTESs). The conclusion contains inferences and propositions for further studies. The list of references contains 43 items.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. Academician V.S. Pugachev: To His 100th Birthday, Ed. by S. V. Emel’yanov and I. N. Sinitsyn (Torus Press, Moscow, 2011).

    Google Scholar 

  2. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regionsal Conference Series in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, 1992). https://doi.org/10.1137/1.9781611970104

  3. Ya. P. Dragan, Structure and Representation of Models of Stochastic Signals (Naukova Dumka, Kiev, 1980).

    Google Scholar 

  4. I. M. Dremin, O. V. Ivanov, and V. A. Nechitaǐlo, “Wavelets and their uses,” Phys.-Usp. 44, 447–478 (2001). https://doi.org/10.1070/PU2001v044n05ABEH000918

    Article  Google Scholar 

  5. V. I. Dymkov and I. N. Sinitsyn, “Elements of concept of personal video systems,” in Systems and Means of Informatics, Annular Book of Institute of Informatics Problems, USSR Academy of Scinces (Nauka, Moscow, 1989), pp. 100–108.

  6. V. I. Dymkov and I. N. Sinitsyn, “Personal systems for processing video information,” in Fundamental Sciences to National Economy (1980), pp. 142–144.

  7. A. V. Gubin, I. N. Sinitsyn, and O. S. Ushmaev, “Metrological and biometric technologies and systems,” Istoriya Nauki Tekh., No. 7, 41–44 (2008).

  8. I. Ya. Novikov and S. B. Stechkin, “Basic wavelet theory,” Russ. Math. Surv. 53, 1159–1231 (1998). https://doi.org/10.1070/RM1998v053n06ABEH000089

    Article  Google Scholar 

  9. V. S. Pugachev, Theory of Random Functions and Its Application to Automatic Control Problems (Fizmatlit, Moscow, 1962).

    Google Scholar 

  10. V. S. Pugachev and I. N. Sinitsyn, Theory of Stochastic Systems: Textbook (Logos, Moscow, 2004).

    Google Scholar 

  11. V. S. Pugachev and I. N. Sinitsyn, Lectures on Functional Analysis and Applications (World Scientific, Singapore, 1999). https://doi.org/10.1142/3987

    Book  Google Scholar 

  12. Sergei Olegovich Novikov, Ed. by I. A. Sokolov and I. N. Sinitsyn (Inst. Probl. Informatiki Ross. Akad. Nauk, Moscow, 2013).

    Google Scholar 

  13. I. N. Sinitsyn, Canonical Representations of Random Functions and Their Applications in Problems of Computer-Aided Scientific Research (Torus Press, Moscow, 2009).

    Google Scholar 

  14. I. N. Sinitsyn, Canonical Representations of Random Functions: Theory and Applications (Torus Press, Moscow, 2023).

    Google Scholar 

  15. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and T. D. Konashenkova, “Software tools for analysis and synthesis of stochastic systems with high availability (XII),” Sist. Vysokoi Dostupnosti 17 (3), 39–58 (2021). https://doi.org/10.18127/j20729472-202103-03

  16. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and T. D. Konashenkova, “Software tools for analysis and synthesis of stochastic systems with high availability (XV),” Sist. Vysokoi Dostupnosti 18 (1), 47–61 (2022). https://doi.org/10.18127/j20729472-202201-05

  17. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and T. D. Konashenkova, “Software tools for analysis and synthesis of stochastic systems with high availability (XVI),” Sist. Vysokoi Dostupnosti 18 (2), 58–77 (2022). https://doi.org/10.18127/j20729472-202202-04

  18. I. Sinitsyn, V. Sinitsyn, E. Korepanov, and T. Konashenkova, “Bayes synthesis of linear nonstationary stochastic systems by wavelet canonical expansions,” Mathematics 10, 1517 (2022). https://doi.org/10.3390/math10091517

    Article  Google Scholar 

  19. I. N. Sinitsyn, “Stochastic systems: Theory of stochastic information technologies,” Istoriya Nauki Tekh., No. 7, 9–12 (2008).

  20. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and V. V. Belousov, “Structural methods for parameterization of distributions in stochastic systems based on canonical expansions,” in All-Russian Conf. on Spectral Methods for Information Processing in Scientific Research (Spektr 2000) (Inst. Mat. Probl. Biol. Ross. Akad. Nauk, Pushchino, Moscow oblast, 2000), pp. 150–152.

  21. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and D. F. Il’yasov, “State-of-the-art of the theory and some applications of canonical representations of random functions,” in Proc. 2nd All-Russian Conf. Dedicated to 110th Birthday of A.Ya. Khinchin: Conf. on Mathematics in the Modern World (Kaluga, 2004), pp. 43–52.

  22. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, V. V. Belousov, and E. S. Agafonov, “Symbolic mathematical software for analyzing and modeling stochastic systems based on canonical expansions of random functions,” in Proc. 8th Int. Conf. on Identification of Systems and Control Problems (2009), pp. 1059–1072.

  23. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, V. V. Belousov, E. S. Agafonov, and A. V. Kutsenko, “Mathematical support of the methods of equivalent linearization of stochastic systems based on canonical expansions of random functions,” in Proc. 9th Int. Sci.-Tech. Conf. on Cybernetics and High Technologies of the 21st Century (C&T 2008) (Sakvoe, Voronezh, 2008), pp. 723–734.

  24. I. N. Sinitsyn, I. V. Sergeev, and E. S. Agafonov, “Application of canonical representations of random functions in problems of analyzing vibration protection systems of computing machinery,” in 11th Int. Sci. Conf. on Systems of Computer Algebra and Their Applications Dedicated to 70th Birthday of Professor V.P. D’yakonov (Smolensk. Gos. Univ., Smolensk, 2010), Vol. 11, pp. 239–241.

  25. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and V. V. Belousov, “Development of software for analysis, filtration, and recognition based on canonical expansions of random functions,” in Proc. 9th Int. Conf. on Optico-Electronic Devices in Pattern Recognition, Image, and Symbol Information Processing Systems (Recognition-2010) (Kursk, 2010), pp. 28–29.

  26. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, V. V. Belousov, and I. V. Sergeev, “Development of algorithmic support for the analysis of stochastic systems based on canonical expansions of random functions,” Autom. Remote Control 72, 405–415 (2011). https://doi.org/10.1134/s0005117911020160

    Article  MathSciNet  Google Scholar 

  27. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and T. D. Konashenkova, “Software tools for analysis and synthesis of stochastic systems with high availability (XI),” Sist. Vysokoi Dostupnosti 17 (1), 25–40 (2021). https://doi.org/10.18127/j20729472-202101-03

  28. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and T. D. Konashenkova, “Software tools for analysis and synthesis of stochastic systems with high availability (XII),” Sist. Vysokoi Dostupnosti 17 (2), 26–44 (2021). https://doi.org/10.18127/j20729472-202102-03

  29. I. N. Sinitsyn, D. V. Zhukov, E. R. Korepanov, and T. D. Konashenkova, “Method of linear optimal processing of information by means of wavelet expansions,” in Computer Algebra Systems and Their Applications: Proc. 12th Int. Sci. Conf. (Izd-vo Smolensk. Gos. Univ., Smolensk, 2020), pp. 213–221. http://fizmat.smolgu.ru/images/stories/SKMP/skmp_2020.pdf

  30. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and T. D. Konashenkova, “Software tools for analysis and synthesis of stochastic systems with high availability (XIII),” Sist. Vysokoi Dostupnosti 17 (3), 39–58 (2021). https://doi.org/10.18127/j20729472-202103-03

  31. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and T. D. Konashenkova, “Software tools of wavelet optimal synthesis of nonstationary systems by complex-statistical criterion,” in Computer Algebra Systems and Their Applications (SKMP-2021): Proc. 22nd Int. Sci. Conf. (Izd-vo Smolensk. Gos. Univ., Smolensk, 2021), Vol. 22, pp. 166–175.

  32. I. N. Sinitsyn, V. I. Sinitsyn, E. R. Korepanov, and T. D. Konashenkova, “Synthesis of quality of observed multiservice network stochastic systems by the method of wavelet canonical expansions,” in The 4th Int. Sci. and Technol. Conf. on Modern Network Technologies (Moscow, 2022 (in print)).

  33. I. N. Sinitsyn and O. S. Ushmaev, “Development of metrological and biometric technologies and systems,” in Proc. 9th Int. Conf. in Pattern Recognition and Image Analysis: New Information Technologies PRIA-9-2008 (Nizhny Novgorod, 2008), Vol. 2, pp. 169–172.

  34. I. N. Sinitsyn and O. S. Ushmaev, “Software implementation of multibiometric identification in integrated analytical apps,” in Proc. Int. Sci.-Tech. Conf. on Cybernetics and High Technologies of the 21st Century (C&T 2008) (Sakvoe, Voronezh, 2008), Vol. 2, pp. 735–746.

  35. I. N. Sinitsyn and O. S. Ushmaev, “Experience of designing multifactor biometric systems,” in Proc. 8th Int. Sci.-Tech. Conf. on Cybernetics and High Technologies of the 21st Century (C&T 2007) (Sakvoe, Voronez, 2007), Vol. 1, pp. 17–28.

  36. I. N. Sinitsyn and O. S. Ushmaev, “Application of biometric technologies in high availability systems,” Sist. Vysokoi Dostupnosti 5 (1), 5–12 (2009).

    Google Scholar 

  37. I. N. Sinitsyn and O. S. Ushmaev, “Experimental study of the ratio of recognition accuracy and performance of multibiometric identification,” in Proc. 10th Int. Conf. on Pattern Recognition and Image Analysis: Novel Information Technologies (ROAI-10-2010) (St. Petersburg, 2010), pp. 221–224.

  38. I. N. Sinitsyn and A. S. Shalamov, Lecture Notes on Theory of Integrate Logistical Support Systems (Torus Press, Moscow, 2019).

    Google Scholar 

  39. I. N. Sinitsyn and A. S. Shalamov, “Optimal estimation and control in stochastic synergetic organization-engineering-economical systems: Filtration in subsystems of manufacturing and staff on the background of noise (I),” Sist. Vysokoi Dostupnosti 15 (4), 27–48 (2019).

    Google Scholar 

  40. I. N. Sinitsyn and A. S. Shalamov, “Optimal estimation and control in stochastic synergetic organization-engineering-economical systems: Filtration in subsystems of manufacturing and staff on the background of noise (II),” Sist. Vysokoi Dostupnosti 17 (1), 52–72 (2020).

    Google Scholar 

  41. I. N. Sinitsyn, A. S. Shalamov, and A. V. Kuleshov, “Nonlinear correlation modeling and reliability analysis of systems of postpurchase service of high-technology products,” in Systems and Tools of Informatics (Inst. Probl. Informatiki Ross. Akad. Nauk, Moscow, 2013), Vol. 23, pp. 80–104.

    Google Scholar 

  42. N. K. Smolentsev, Fundamentals of the Wavelet Theory: Wavelets in MATLAB (DMK Press, Moscow, 2005).

    Google Scholar 

  43. V. N. Zakharov, “Research in the area of pattern recognition and image analysis at the Federal Research Center ‘Computer Science and Control’ of the Russian Academy of Sciences and its predecessors,” Pattern Recognit. Image Anal. 33 (3) (2023).

Download references

Funding

This study was performed in the framework of research project no. 0063-2019-0008 “Theoretic-Probabilistic and Statistical Modeling Methods 2019–2023” of the Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences as part of a state assignment.

Author information

Authors and Affiliations

  1. Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119333, Moscow, Russian Federation

    I. N. Sinitsyn

Authors
  1. I. N. Sinitsyn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. N. Sinitsyn.

Ethics declarations

The author of this work declares that he has no conflicts of interest.

Additional information

Igor’ Nikolaevich Sinitsyn. Professor, Doctor of Engineering, Honored Scientist of the Russian Federation, scientist in the field of applied mathematics and control, applied mathematics and informatics, founder of the scientific school in the field of SITs and automation systems, author of more than 600 scientific papers, 60 books, and 30 inventions. He has extensive experience of work in industry and higher engineering education institutions. Currently, he is the deputy Editor-in-Chief in the journal Sistemy Vysokoi Dostupnosti, as well as a member of the editorial boards of a number of Russian and international journals, such as Informatika i Ee Primeneniya, Sistemy i Sredstva Informatiki, Pattern Recognition and Image Analysis, and Naukoemkie Tekhnologii.

Translated by E. Glushachenkova

Publisher’s Note.

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sinitsyn, I.N. Developing the Theory of Stochastic Canonic Expansions and Its Applications. Pattern Recognit. Image Anal. 33, 862–887 (2023). https://doi.org/10.1134/S1054661823040429

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1054661823040429

Keywords:

Search

Navigation