Statistical modelling of a 3D random field by using the Kotelnikov-Shannon decomposition

Authors:
Z. O. Vyzhva and K. V. Fedorenko

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **88** (2013).

Journal:
Theor. Probability and Math. Statist. **88** (2014), 19-34

MSC (2010):
Primary 60G60, 65C05

DOI:
https://doi.org/10.1090/S0094-9000-2014-00916-3

Published electronically:
July 24, 2014

MathSciNet review:
3112632

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Real- valued random fields , , , that are homogeneous with respect to time and homogeneous isotropic with respect to spatial variables in the plane are studied. The problem of approximation of such random fields by random fields with a bounded spectrum is considered. An analogue of the Kotelnikov-Shannon theorem for random fields with a bounded spectrum is presented. Estimates of the mean-square approximation of random fields in the space by a model constructed with the help of the spectral decomposition and interpolation Kotelnikov-Shannon formula are obtained. Some procedures for the statistical simulation of realizations of Gaussian random fields that are homogeneous with respect to time and homogeneous isotropic with respect to spatial variables in the plane are developed.

**1.**Yu. K. Belyaev,*Analytic random processes*, Teor. Veroyatnost. i Primenen**4**(1959), 437–444 (Russian, with English summary). MR**0112174****2.**Z. O. Vyzhva,*Statistical Modeling of Random Processes and Fields*, ``Obrii'', Kyiv, 2011. (Ukrainian)**3.**Z. O. Vizhva,*Statistical modeling of random fields on a plane with a uniform interpolation grid*, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki**5**(2003), 7–12 (Ukrainian, with English summary). MR**2040126****4.**I. S. Gradshteyn and I. M. Ryzhik,*Tables of Integrals, Series, and Products*, ``Nauka'', Moscow, 1971; English transl., Academic Press, New York, 1965.**5.**Yu. V. Kozachenko, A. O. Pashko, and I. V. Rozora,*Modelling of Random Processes and Fields*, ``Zadruga'', Kyiv, 2007. (Ukrainian)**6.**V. N. Nagornyĭ,*The interpolation of random processes. I, II*, Teor. Verojatnost. i Mat. Statist.**3**(1970), 93–96; ibid. No. 2 (1970), 97–104 (Russian, with English summary). MR**0292246****7.**A. Ya. Olenko,*A bound of the interpolation error in the multivariate Kotelnikov-Shannon theorem*, Visnyk Kiev Univer. Ser. Fiz.-Mat. Nauk**3**(2004), 49-54. (Ukrainian)**8.**A. Ya. Olenko,*A comparison of approximation errors in the Kotelnikov-Shannon theorem*, Visnyk Kiev Univer. Ser. Fiz.-Mat. Nauk**13**(2005), 41-45. (Ukrainian)**9.**Z. A. Piranašvili,*The problem of interpolation of random processes*, Teor. Verojatnost. i Primenen**12**(1967), 708–717 (Russian, with English summary). MR**0219125****10.**S. I. Khalikulov and Z. O. Vyzhva,*Kotelnikov-Shannon theorem for time homogeneous and isotropic fields on a sphere and the statistical modelling*, Visnyk Kiev Univer. Ser. Mat. Mekh.**6**(2001), 66-71. (Ukrainian)**11.**S. I. Khalikulov and V. M. Yadrenko,*Kotelnikov-Shannon theorem for homogeneous fields on a cylinder*, Visnyk Kiev Univer. Ser. Mat. Mekh.**5**(2000), 55-60. (Ukrainian)**12.**M. Ĭ. Yadrenko,*Spectral theory of random fields*, Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division, New York, 1983. Translated from the Russian. MR**697386****13.**Paul L. Butzer, Gerhard Schmeisser, and Rudolf L. Stens,*Shannon’s sampling theorem for bandlimited signals and their Hilbert transform, Boas-type formulae for higher order derivatives—the aliasing error involved by their extensions from bandlimited to non-bandlimited signals*, Entropy**14**(2012), no. 11, 2192–2226. MR**3000067**, https://doi.org/10.3390/e14112192**14.**Z. A. Grikh, M. J. Yadrenko, and O. M. Yadrenko,*On the approximation and statistical simulation of isotropic random fields*, Random Oper. Stochastic Equations**1**(1993), no. 1, 37–45. MR**1254174**, https://doi.org/10.1515/rose.1993.1.1.37**15.**J. R. Higgins,*Sampling Theory in Fourier and Signal Analysis*, Clarendon Press, Oxford, New York, 1996.**16.**A. Ya. Olenko and T. K. Poganī,*The least upper bound for error in the interpolation of random processes*, Teor. Ĭmovīr. Mat. Stat.**71**(2004), 133–144 (Ukrainian, with Russian summary); English transl., Theory Probab. Math. Statist.**71**(2005), 151–163. MR**2144328**, https://doi.org/10.1090/S0094-9000-05-00655-1**17.**Andriy Olenko and Tibor Pogány,*Average sampling restoration of harmonizable processes*, Comm. Statist. Theory Methods**40**(2011), no. 19-20, 3587–3598. MR**2860759**, https://doi.org/10.1080/03610926.2011.581180**18.**Kristian Seip,*Interpolation and sampling in spaces of analytic functions*, University Lecture Series, vol. 33, American Mathematical Society, Providence, RI, 2004. MR**2040080**

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Additional Information

**Z. O. Vyzhva**

Affiliation:
Department of General Mathematics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue, 4-e, Kyiv 03127, Ukraine

Email:
vsa@univ.kiev.ua

**K. V. Fedorenko**

Affiliation:
Department of General Mathematics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue, 4-e, Kyiv 03127, Ukraine

Email:
slimsmentol@mail.ru

DOI:
https://doi.org/10.1090/S0094-9000-2014-00916-3

Keywords:
Random fields,
modelling,
Kotelnikov--Shannon decomposition

Received by editor(s):
February 13, 2012

Published electronically:
July 24, 2014

Article copyright:
© Copyright 2014
American Mathematical Society