Estimation Problems for Periodically Correlated Isotropic Random Fields

Estimation Problems for Random Fields
  • Iryna Dubovetska
  • Oleksandr Masyutka
  • Mikhail Moklyachuk
Article

Abstract

Spectral theory of isotropic random fields in Euclidean space developed by M. I. Yadrenko is exploited to find a solution to the problem of optimal linear estimation of the functional
$$ A\zeta ={\sum\limits_{t=0}^{\infty}}\,\,\,{\int_{S_n}} \,\,a(t,x)\zeta (t,x)\,m_n(dx) $$
which depends on unknown values of a periodically correlated (cyclostationary with period T) with respect to time isotropic on the sphere Sn in Euclidean space En random field ζ(t, x), t ∈ Z, x ∈ Sn. Estimates are based on observations of the field ζ(t, x) + θ(t, x) at points (t, x), t = − 1, − 2, ..., x ∈ Sn, where θ(t, x) is an uncorrelated with ζ(t, x) periodically correlated with respect to time isotropic on the sphere Sn random field. Formulas for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional are obtained. The least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functional are determined for some special classes of spectral densities.

Keywords

Random field Prediction Filtering Robust estimate Mean square error Least favourable spectral densities Minimax spectral characteristic 

AMS 2000 Subject Classifications

60G60 62M40 62M20 93E10 93E11 

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References

  1. Adshead P, Hu W (2012) Fast computation of first-order feature-bispectrum corrections. Phys Rev D 85:103531CrossRefGoogle Scholar
  2. Antoni J (2009) Cyclostationarity by examples. Mech Syst Signal Process 23:987–1036CrossRefGoogle Scholar
  3. Bartlett JG (1999) The standard cosmological model and cmb anisotropies. New Astron Rev 43:83–109CrossRefGoogle Scholar
  4. Cressie N, Wikle CK (2011) Statistics for spatio-temporal data. Wiley series in probability and statisticsGoogle Scholar
  5. Franke J (1985) Minimax robust prediction of discrete time series. Z Wahrscheinlichkeitstheor Verw Geb 68:337–364CrossRefMATHMathSciNetGoogle Scholar
  6. Gaetan C, Guyon X (2010) Spatial statistics and modeling. Springer series in statistics, vol 81. Springer Science + Business MediaGoogle Scholar
  7. Gardner WA (1994) Cyclostationarity in communications and signal processing. IEEE Press, New YorkMATHGoogle Scholar
  8. Gladyshev EG (1961) Periodically correlated random sequences. Sov Math Dokl 2:385–388MATHGoogle Scholar
  9. Grenander U (1957) A prediction problem in game theory. Ark Mat 3:371–379CrossRefMATHMathSciNetGoogle Scholar
  10. Hu W, Dodelson S (2002) Cosmic microwave background anisotropies. Ann Rev Astron Astrophys 40:171–216CrossRefGoogle Scholar
  11. Hurd HL, Miamee A (2007) Periodically correlated random sequences. Wiley, New YorkCrossRefMATHGoogle Scholar
  12. Jones PD (1994) Hemispheric surface air temperature variations: a reanalysis and an update to 1993. J Climate 7:1794–1802CrossRefGoogle Scholar
  13. Kailath T (1974) A view of three decades of linear filtering theory. IEEE Trans Inf Theory 20(2):146–181CrossRefMATHMathSciNetGoogle Scholar
  14. Kakarala R (2012) The bispectrum as a source of phase-sensitive invariants for Fourier descriptors: a group-theoretic approach. J Math Imaging Vis 44:341–353CrossRefMATHMathSciNetGoogle Scholar
  15. Kassam SA, Poor HV (1985) Robust techniques for signal processing: a survey. Proc IEEE 73(3):433–481CrossRefMATHGoogle Scholar
  16. Kogo N, Komatsu E (2006) Angular trispectrum of cmb temperature anisotropy from primordial non-Gaussianity with the full radiation transfer function. Phys Rev D73:083007–083012CrossRefGoogle Scholar
  17. Kolmogorov AN (1992) Selected works of A. N. Kolmogorov. Vol. II: probability theory and mathematical statistics. Kluwer, DordrechtGoogle Scholar
  18. Kurkin OM, Korobochkin YuB, Shatalov SA (1990) Minimax information processing. Moskva, EnergoatomizdatGoogle Scholar
  19. Kulakova VI, Nebylov AV (2008) Guaranteed estimation of signals with bounded variances of derivatives. Autom Remote Control 69(1):76–88CrossRefMATHMathSciNetGoogle Scholar
  20. Makagon A (2011) Stationary sequences associated with a periodically correlated sequence. Prob Math Stat 31(2):263–283MATHMathSciNetGoogle Scholar
  21. Marinucci D, Peccati G (2011) Random fields on the sphere. London mathematical society lecture notes series, vol 389. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  22. Moklyachuk M (1995) Extrapolation of time-homogeneous random fields that are isotropic on a sphere. I. Theory Probab Math Stat 51:137–146MathSciNetGoogle Scholar
  23. Moklyachuk M (1996) Extrapolation of time-homogeneous random fields that are isotropic on a sphere. II. Theory Probab Math Stat 53:137–148MathSciNetGoogle Scholar
  24. Moklyachuk M (2008) Robust estimates for functionals of stochastic processes. Kyiv University Publishing, KyivMATHGoogle Scholar
  25. Moklyachuk M, Masyutka O (2012) Minimax-robust estimation technique for stationary stochastic processes. LAP LAMBERT Academic Publishing, SaarbrückenMATHGoogle Scholar
  26. Moklyachuk M, Yadrenko M (1979) Linear statistical problems for homogeneous isotropic random fields on a sphere. I. Theory Probab Math Stat 18:115–124MATHGoogle Scholar
  27. Moklyachuk M, Yadrenko M (1980) Linear statistical problems for homogeneous isotropic random fields on a sphere. II. Theory Probab Math Stat 19:129–139MATHGoogle Scholar
  28. Müller C (1998) Analysis of spherical symmetries in Euclidean spaces. Springer, New YorkCrossRefMATHGoogle Scholar
  29. North GR, Cahalan RF (1981) Predictability in a solvable stochastic climate model. J Atmos Sci 38:504–513CrossRefGoogle Scholar
  30. Okamoto T, Hu W (2002) Angular trispectra of cmb temperature and polarization. Phys Rev D 66:063008CrossRefGoogle Scholar
  31. Rozanov YuA (1967) Stationary stochastic processes. Holden-Day, San FranciscoGoogle Scholar
  32. Serpedin E, Panduru F, Sari I, Giannakis GB (2005) Bibliography on cyclostationarity. Signal Process 85:2233–2303CrossRefMATHGoogle Scholar
  33. Subba Rao T, Terdik G (2006) Multivariate non-linear regression with applications. In: Bertail P, Doukhan P, Soulier P (eds) Dependence in probability and statistics. Springer, New York, pp 431–470Google Scholar
  34. Subba Rao T, Terdik G (2012) Statistical analysis of spatio-temporal models and their applications. In: Rao CR (ed) Handbook of statistics, vol 30. Elsevier B.V., pp 521–541Google Scholar
  35. Terdik G (2013) Angular spectra for non-Gaussian isotropic fields. arXiv:1302.4049v1.pdf. Accessed 17 Feb 2013
  36. Vastola KS, Poor HV (1983) An analysis of the effects of spectral uncertainty on Wiener filtering. Automatica 28:289–293CrossRefMathSciNetGoogle Scholar
  37. Wiener N (1966) Extrapolation, interpolation, and smoothing of stationary time series. With engineering applications. Cambridge, MassGoogle Scholar
  38. Yadrenko MI (1983) Spectral theory of random fields. Optimization Software Inc. Publications Division, New YorkMATHGoogle Scholar
  39. Yaglom AM (1987) Correlation theory of stationary and related random functions. Vol. I: basic results. Vol. II: supplementary notes and references. Springer, New YorkGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Iryna Dubovetska
    • 1
  • Oleksandr Masyutka
    • 2
  • Mikhail Moklyachuk
    • 1
  1. 1.Department of Probability Theory, Statistics and Actuarial MathematicsKyiv National Taras Shevchenko UniversityKyivUkraine
  2. 2.Department of Mathematics and Theoretical RadiophysicsKyiv National Taras Shevchenko UniversityKyivUkraine

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