On minimax filtering of homogeneous random fields. (English. Russian original) Zbl 0788.62085

Moklyachuk, M. P.; Tatarinov, S. V.

Theory Probab. Math. Stat. 44, 103-111 (1992); translation from Teor. Veroyatn. Mat. Stat., Kiev 44, 105-115 (1991).

Summary: The problem of mean-square optimal linear estimation of the transform \[ A\xi= \sum^ \infty_{k,j=0} a(k,j)\xi(-k,-j) \] is considered for a homogeneous random field \(\xi(k,j)\) with density \(f(\lambda,\mu)\) from observations of the field \(\xi(k,j)+ \eta(k,j)\) for \(k\leq 0\) and \(j\leq 0\), where \(\eta(k,j)\) is a homogeneous random field with density \(g(\lambda,\mu)\) that is uncorrelated with \(\xi(k,j)\).
Formulas are obtained for computing the mean-square error and the spectral characteristic of an optimal linear estimator of the value of \(A\xi\). The least favorable spectral densities \(f_ 0(\lambda,\mu)\) and \(g_ 0(\lambda,\mu)\) are found along with the minimax (robust) spectral characteristics of an optimal estimator of \(A\xi\) for various classes \({\mathcal D}_ f\) and \({\mathcal D}_ g\) of densities.

MSC:

62M20	Inference from stochastic processes and prediction
62M40	Random fields; image analysis
60G60	Random fields
60G35	Signal detection and filtering (aspects of stochastic processes)

Keywords:

minimax filtering; mean-square optimal linear estimation; homogeneous random field; mean-square error; optimal linear estimator; least favorable spectral densities