On estimates of unknown values of random fields from noisy observations. (Ukrainian, English) Zbl 1026.62099

Moklyachuk, M. P.

Teor. Jmovirn. Mat. Stat. 65, 152-160 (2001); translation in Theory Probab. Math. Stat. 65, 171-180 (2002).

The problem of the least mean-square linear estimation of a linear functional of the form \[ A \xi =\sum_{k=0}^{\infty} \int_{S_{n}}a(k,x)\xi(k,x) m_n(dx) \] which depends on the unknown values of a random field \(\xi(k,x),\) \( k\in {\mathbf Z}\), \( x\in S_{n}\), that is time homogeneous and isotropic on a sphere \(S_{n}\), from observations of the field \(\xi(k,x)+\eta(k,x)\) for \(k<0\), \( x\in S_{n}\), where \( \eta(k,x)\) is a random field that is time homogeneous and isotropic on a sphere \(S_{n}\) and uncorrelated with \(\xi(k,x)\), is considered. Formulas are obtained for computing the value of the mean-square error and the spectral characteristics of the optimal linear estimate of the functional \(A \xi\) in the case where spectral densities of the fields \(\xi(k,x)\) and \(\eta(k,x)\) are known. The least favorable spectral densities and minimax (robust) spectral characteristics of the optimal estimates of functionals of the indicated type are determined for some classes of random fields.
For additional information see the author’s papers, Theory Probab. Math. Stat. 49, 137-146 (1994), translation from Teor. Jmovirn. Mat. Stat. 49, 193-205 (1993; Zbl 0861.60060); Theory Probab. Math. Stat. 50, 107-115 (1995), translation from Teor. Jmovirn. Mat. Stat. 50, 105-113 (1994; Zbl 0861.60062); Theory Probab. Math. Stat. 51, 137-146 (1995), translation from Teor. Jmovirn. Mat. Stat. 51, 131-139 (1994; Zbl 0939.60039); Theory Probab. Math. Stat. 53, 137-148 (1996); translation from Teor. Jmovirn. Mat. Stat. 53, 126-137 (1995; Zbl 0951.60056).

Reviewer: M.I.Yadrenko (Kyïv)

MSC:

62M40	Random fields; image analysis
62M15	Inference from stochastic processes and spectral analysis
60G35	Signal detection and filtering (aspects of stochastic processes)
60G60	Random fields
93E10	Estimation and detection in stochastic control theory

Keywords:

mean-square linear estimates; least favorable spectral density; minimax spectral characteristics

Citations:

Zbl 0861.60060; Zbl 0861.60062; Zbl 0939.60039; Zbl 0951.60056