The scope of the paper is the presentation of a new method of generating numbers from a given distribution. The method uses the inverse cumulative distribution function and a method of flattening of probabilistic distributions. On the grounds of these methods, a new construction of chaotic maps was derived, which generates values from a given distribution. The analysis of the new method was conducted on the example of a newly constructed chaotic recurrences, based on the Box-Muller transformation and the quantile function of the exponential distribution. The obtained results certify that the proposed method may be successively applicable for the construction of generators of pseudo-random numbers.
REFERENCES
1.
V.M.
Anikin
, S.S.
Arkadaksky
, S.N.
Kuptsov
, A.S.
Remizov
and L.P.
Vasilenko
, Lyapunov exponent for chaotic 1D maps with uniform invariant distribution
, Bulletin of the Russian Academy of Sciences: Physics
72
(12
), 1684
–1688
(2008
).2.
M.
Berezowski
, Spatio-temporal chaos in tubular chemical reactors with the recycle of mass
, Chaos, solitons and fractals
11
(8
), 1197
–1204
(2000
).3.
M.
Berezowski
and M.
Lawnik
, Identification of fast-changing signals by means of adaptive chaotic transfor-mations
, Nonlinear Anal., Model. Control
19
(2
), 172
–177
(2014
).4.
G.E.P.
Box
and M.E.
Muller
, A Note on the Generation of Random Normal Deviates
, The Annals of Mathe-matical Statistics
29
(2
), 610
–611
(1958
).5.
G.E.P.
Box
, M.
Gwilym
and G.
Jenkins
, Time Series Analysis: Forecasting and Control
(Holden-Day
, San Francisco
, 1976
).6.
L.
Devroye
, Non-Uniform Random Variate Generation
(Springer
, New York
, 1986
).7.
S.
Grossmann
and S.
Thomae
, Invariant distributions and stationary correlation functions of one-dimensional discrete processes
, Z. Naturforsch 32
, 1353
–1363
(1977
).8.
W.
Huang
, Characterizing chaotic processes that generate uniform invariant density
, Chaos, Solitons & Frac-tals
25
(2
), 449
–460
(2005
).9.
L.
Kocarev
and S.
Lian
, Chaos-based Cryptography, Theory, Algorithms and Applications
(Springer
, Berlin-Heidelberg
, 2011
).10.
S.
Koga
, The inverse problem of Frobenius-Perron equations in 1D different systems
, Progress of Theoretical Physics
86
(5
), 991
–1002
(1991
).11.
D.
Lai
and G.
Chen
, Generating Different Statistical Distributions By The Chaotic Skew Tent Map
, Int. J. Bifurcation Chaos
10
, 1509
–1512
(2000
).12.
M.
Lawnik
, Generalized logistic map and its application in chaos based cryptography, Proceedings of the
6th International Conference on Mathematical Modeling in Physical Sciences August 28-31
, 2017 Pafos, Cyprus, accepted to print
13.
M.
Lawnik
and M.
Berezowski
, Identification of the oscillation period of chemical reactors by chaotic sampling of the conversion degree
, Chem. Process Eng.
35
(3
), 387
–393
(2014
).14.
M.
Lawnik
, The approximation of the normal distribution by means of chaotic expression
, J. Phys.: Conf. Ser.
490
(012072
), 1
–4
(2014
).C.D.
Smith
and E.F.
Jones
, Load-cycling in cubic press, in Shock Compression of Condensed Matter-2001
, AIP Con-ference Proceedings
620
, edited by M. D.
Furnish
(American Institute of Physics
, Melville, NY
, 2002
), pp. 651654
.15.
M.
Lawnik
, “Generation of numbers with the distribution close to uniform with the use of chaotic maps
”, Proceedings of the 4th International Conference on Simulation and Modeling Methodologies
, Technologies and Applications
, edited by M.S.
Obaidat
, J.
Kacprzyk
, T.
Ören
(Scitepress
, 2014
), pp. 451
–455
.16.
M.
Lawnik
, Analysis of the chaotic maps generating different statistical distributions, J. Phys.: Conf. Ser.
633
(012086
), 1
–4
(2015
).17.
18.
G.
Ljung
and G.
Box
, On a Measure of Lack of Fit in Time Series Models
, Biometrika
65
, 297
–303
(1978
).19.
D.
Pingel
, P.
Schmelcher
and F.K.
Diakonos
, Theory and examples of the inverse Frobenius-Perron problem for complete chaotic maps
, Chaos
, 9
(2
), 357
–366
(1999
).20.
Python scipy.stats module, http://docs.scipy.org/doc/scipy/reference/stats.html (last access 23.06.2017)
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