International Encyclopedia of Statistical Science

2011 Edition
| Editors: Miodrag Lovric

Estimation Problems for Random Fields

  • Mikhail P. Moklyachuk
Reference work entry

Estimation problems for random fields X (t),   t ∈ ℝ n (estimation of the unknown mathematical expectation, estimation of the correlationfunction, estimation of regression parameters, extrapolation, interpolation, filtering,etc) are similar to the corresponding problems for  stochastic processes (random fields of dimension 1). Complications usually are caused by the form of domain of points {tj} = D ⊂ ℝ n, where observations {X (tj) } are given, and by the dimension of the field. The complications can be overcome by considering specific domains of observations and particular classes of random fields.

Say in the domain D ⊂ ℝ n there are given observations of the random field
$$X\left (t\right ) = \sum \limits_{i=1}^{q}\theta,{g}_{ i}(t) + Y (t),$$
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References and Further Reading

  1. Grenander U (1981) Abstract inference. Wiley series in probability and mathematical statistics. Wiley, New YorkMATHGoogle Scholar
  2. Moklyachuk MP (2008) Robust estimates of functionals of stochastic processes. Vydavnycho-Poligrafichnyĭ Tsentr, Kyïvskyĭ Universytet, KyïvMATHGoogle Scholar
  3. Ramm AG (2005) Random fields estimation. World Scientific, Hackensack, NJMATHGoogle Scholar
  4. Ripley BD (1981) Spatial statistics. Wiley series in probability and mathematical statistics. Wiley, New YorkMATHGoogle Scholar
  5. Rozanov YA (1982) Markov random fields. Springer-Verlag, New YorkMATHGoogle Scholar
  6. Yadrenko MI (1983) Spectral theory of random fields. Translation series in mathematics and engineering. (Optimization Software Inc., New York) Springer-Verlag, New YorkGoogle Scholar
  7. Yaglom AM (1987) Correlation theory of stationary and related random functions, volume I and II. Springer series in statistics. Springer-Verlag, New YorkGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Mikhail P. Moklyachuk
    • 1
  1. 1.Kyiv National Taras Shevchenko UniversityKyivUkraine