Prediction problem for random fields on groups

Mikhail Moklyachuk

Teoriâ slučajnyh processov, Tome 13 (2007) no. 4, pp. 148-162

Résumé

The problem considered is the problem of optimal linear estimation of the functional $A\xi=\sum^\infty_{j=0}\int_Ga(g, j)\xi(g, j)dg$ which depends on the unknown values of a homogeneous random field $\xi(g, j)$ on the group $G\times{\mathbb Z}$ from observations of the field $\xi(g, j)+\eta(g, j)$ for $(g, j)\in G\times\{-1, -2, \ldots\},$ where $\eta(g, j)$ is an uncorrelated with $\xi(g, j)$ homogeneous random field $\xi(g, j)$ on the group $G\times{\mathbb Z}.$ Formulas are proposed for calculation the mean square error and spectral characteristics of the optimal linear estimate in the case where spectral densities of the fields are known. The least favorable spectral densities and the minimax spectral characteristics of the optimal estimate of the functional are found for some classes of spectral densities.

Détail

Keywords: Random field, prediction, filtering, robust estimate, observations with noise, mean square error, least favorable spectral densities, minimax spectral characteristic.