Journal of Applied Nonlinear Dynamics
Is it Possible to Replace the Probability Distribution Function Describing a Random Process by the Prony’s Spectrum (I)
Journal of Applied Nonlinear Dynamics 1(2) (2012) 173194  DOI:10.5890/JAND.2012.05.005
Raoul R. Nigmatullin
Department of Theoretical Physics, Kazan Federal University, Kremlevskaya str. 18, 420008 Russian Federation
Download Full Text PDF
Abstract
The new law that governs by the generation of frequencies ωk (k=1,2,...,K1)for the stronglycorrelated systems (having a memory) has been found. The generalization of the present idea is based on detailed analysis of the previous results obtained in paper [1] that were devoted to new solutions of the Prony’s problem. It was turned out that many complex systems with memory generate new set of frequencies based on frequencies that have been generated in the nearest past. For justification of this relationship we collected different data that confirm this statement. We created also a special mathematical program, which selects (based on some criteria) a desired hypothesis that is chosen from other six similar ones. For all available data considered there is an optimal hypothesis that describes the distribution of frequencies that follows from the recurrence relationship including in itself the neighboring frequencies. The found hypothesis provides the optimal fit of the random smoothed sequence with high accuracy (the relative error less that 10%) including also the fit of the remnant function.The physical interpretation of this law is given also. This “unexpected” discovery found for a wide class of the stronglycorrelated systems with memory allows to replace the probability distribution function associated with some process by its Prony’s spectrum. From mathematical point of view it will help to obtain new solutions of the old Prony’s problem and replace also the Fourier spectrum containing usually the excess of artifact frequencies by the informativesignificant band of frequencies obtained from new general law that, in turn, was found for the stronglycorrelated systems.
Acknowledgments
This paper was written in the frame of the scientific research program that was accepted by Kazan Federal University for 2012 year “Dielectric spectroscopy and kinetics of complex systems”.
References

[1]  Nigmatullin, R.R., Osokin, S.I., Toboev, V.A. (2011), NAFASS: Discrete spectroscopy of random signals, Journal of Chaos, Solitons and Fractals, 44, 226. 

[2]  Sheng H., Chen Y.Q., Qiu T.S. (2012), Fractional Processes and FractionalOrder Signal Processing, (Techniques and Applications), Springer. P. 295. 

[3]  Osborne M.R. and Smyth G.K. (1991) A modified Prony algorithm for fitting functions defined by difference equations, SIAM Journal of Scientific and Statistical Computing, 12, 362. 

[4]  Kahn M., Mackisack M.S., Osborne M.R. and Smyth G.K. (1992), On the consistency of Prony’s method and related algorithms, Journal of Computational and Graphical Statistics, 1, 329. 

[5]  Mackisack M.S., Osborne M.R. and Smyth G.K. (1994), A modified Prony algorithm for estimating sinusoidal frequencies, Journal of Statistical Computation and Simulation, 49, 11. 

[6]  Osborne M.R. and Smyth G.K. (1995), A modified Prony algorithm for fitting sums of exponential functions, SIAM Journal of Scientific and Statistical Computing, 16, 119. 

[7]  Merlet J.P. (2000), Singular configurations of parallel manipulators and Grassman geometry, Int. J. Robotics Research, 19, 271. 

[8]  Smyth G. K. Employing constraints for improved frequency estimation by eigenanalysis method, (2000), Technometrics, 42, 277. 

[9]  Ciurea, M.L., Lazanu, S., Stavaracher I., Lepadatu, AM., Iancu V., Mitroi, M. R. Nigmatullin R. R. and Baleanu. C. M. (2011), Stressed induced traps in multilayed structures, J of Applied Phys., 109, 013717. 

[10]  Nigmatullin, R.R., Popov, I.I., and Baleanu D. (2011), Predictions based on the cumulative curves: Basic principles and nontrivial example, Communications in Nonlinear Science and Numerical Simulation , 16, 895. 

[11]  Nigmatullin. R.R. (2010), Universal distribution function for the stronglycorrelated fluctuations: general way for description of random sequences. Communications in Nonlinear Science and Numerical Simulation, 15, 637. 

[12]  Jonscher A.K. (1983), Dielectric relaxation in solids, Chelsea Dielectric Press, London. 