Regression differentiation and regression integration of finite series
https://doi.org/10.22405/2226-8383-2021-22-2-27-47
Abstract
Data analysis is a complex and multi-dimensional concept. It is explained by both the objective complexity of the data itself and the subjective nature of the expert analyzing them.
Therefore, adequate formalization of this requires a completely new apparatus, on the one hand, capable of overcoming the objective complexity of data (irregularity and inaccuracy),
and, on the other hand, the vague nature of expert judgment. The development of Discrete Mathematical Analysis (DMA) is an important step in this direction. DMA is highly expert
oriented and occupies an intermediate position in data analysis between strict mathematical methods (statistical analysis, TFA, etc.) and soft combinatorial methods (simulation modeling,
neural networks, etc.).
This paper proposes new mathematical constructions of regression derivatives and regression integrals for discrete time series defined in general on an irregular grid. An important role in their study is played by the recently created by the authors projection method for solving systems of linear algebraic equations described at the end of this paper.
The obtained constructions of regression derivatives and regression integrals have a hierarchical character in the spirit of wavelet and fractal analysis.
The results of this work define another direction in further studies, namely, the propagation of regression differentiation and regression integration in finite mathematics under scenarios of the classical one.
About the Authors
Sergei Martikovich Agayan
Geophysical Center of the Russian Academy of Sciences
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Shamil Rafekovich Bogoutdinov
Geophysical Center of the Russian Academy of Sciences, Institute of Physics of the Earth of the Russian Academy of Sciences
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Nikolaevich Mikhail Dobrovolskiy
Geophysical Center of the Russian Academy of Sciences
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Olga Vasilievna Ivanchenko
National Research Nuclear University “MEPhI”, Obninsk
Institute of Atomic Energy
Russian Federation
Russian Federation
Dmitry Alfredovich Kamaev
Research and Production Association «Typhoon»
Russian Federation
Russian Federation
doctor of technical sciences
References
1. Mikhailov V., Galdeano A., Diament M., Gvishiani A., Agayan S., Bogoutdinov Sh., Graeva E., Sailhac P. 2003, “Application of artificial intelligence for Euler solutions clustering“, Geophysics,
2. vol. 68, no. 1, pp. 168-180.
3. Zlotnicki J., LeMouel J.-L., Gvishiani A., Agayan S., Mikhailov V., Bogoutdinov Sh. Automatic fuzzy-logic recognition of anomalous activity on long geophysical records. 2005, “Application to electric signals associated with the volcanic activity of la Fournaise volcano (R?union Island)“,
4. Earth and Planetary Science Letters, vol. 234, pp. 261-278.
5. Soloviev A., Chulliat A., Bogoutdinov S., Gvishiani A., Agayan S., Peltier A., Heumez B. 2012, “Automated recognition of spikes in 1 Hz data recorded at the Easter Island magnetic
6. observatory“, Earth Planets Space, vol. 64, no. 9, pp. 743-752, DOI:10.5047/eps.2012.03.004.
7. Gvishiani A.D., Agayan S.M., Bogoutdinov Sh.R., Graeva E.M., Zlotnicki J., Bonnin J. 2008, “Recognition of anomalies from time series by fuzzy logic methods“, Russian Journal of Earth Sciences, vol. 10, ES1001, DOI:10.2205/2007ES000278.
8. Agayan S., Bogoutdinov Sh., Soloviev A., Sidorov R. 2016, “The Study of Time Series Using the DMA Methods and Geophysical Applications“, Data Science Journal, pp. 1–21, DOI:
9. 5334/dsj-2016-014
10. S.M. Agayan, Sh.R. Bogoutdinov, R.I. Krasnoperov. 2018, “Short introduction into DMA“, Russian Journal of Earth Sciences, vol. 18, no 2, DOI: 10.2205/2018ES000618
11. S.M. Agayan, Sh.R. Bogoutdinov, D.A. Kamaev, M.N. Dobrovolsky. 2019, “Stochastic trends based on fuzzy mathematics“, Chebyshevskii sbornik, vol. XX, no. 3, pp. 92-106, DOI: 10.22405/2226-8383-2019-20-3-92-106
12. A.N. Kolmogorov, S.V. Fomin. 1989, “Elements of function theory and functional analysis“, M.: Nauka, 543p.
13. Fichtenholz G.M. 1970, “Differential and Integral Calculus Course“, M.: Nauka, vol. II, 800p.
14. F.R. Gantmacher, M.G. Krein. 2002, “Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems: Revised Edition“, AMS Chelsea Publishing, vol. 345, 310pp.
15. J. Abaffy, E. Spedicato. 1989, “ABS projection algorithms : mathematical techniques for linear and nonlinear equations“, Prentice Hall, 220p.
16. S.M. Agayan, Sh.R. Bogoutdinov, A.A. Bulychev, A.A. Soloviev, I.A. Firsov. 2020, “A projection method for solving systems of linear equations: gravimetry applications“, Doklady Earth Sciences, vol. 493, part 1, pp. 530–534, DOI: 10.1134/S1028334X20070053
17. Agayan, S. M., Soloviev, A. A., Bogoutdinov, Sh. R., Nikolova, Y. I. 2019, “Regression derivatives and their application in the study of geomagnetic jerks“, Geomagnetism and aeronomy, vol. 59, no. 3, pp. 383-392, DOI: 10.1134/S0016794019030027
18. Mallat S. 2009, “A wavelet tour of signal processing“, Academic Press, 832p, DOI: 10.1016/B978-0-12-374370-1.X0001-8
19. B. Mandelbrot. 1982, “The fractal geometry of nature“, W.H. Freeman and company, 480p.
20. Feder J. 1988, “Fractals“, Springer US, 284p.
Review
For citations:
Agayan S.M., Bogoutdinov Sh.R., Dobrovolskiy N.M., Ivanchenko O.V., Kamaev D.A. Regression differentiation and regression integration of finite series. Chebyshevskii Sbornik. 2021;22(2):27-47. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-2-27-47
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