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.

Bibliographie

[1] J. Franke, “On the robust prediction and interpolation of time series in the presence of correlated noise”, J. Time Series Analysis, 5:4 (1984), 227–244

[2] J. Franke, “Minimax robust prediction of discrete time series”, Z.Wahrsch.Verw. Gebiete., 68 (1985), 337–364

[3] J. Franke, H. V. Poor, “Minimax–robust filtering and finite–length robust predictors”, Robust and Nonlinear Time Series Analysis, Heidelberg, 1983, Lecture Notes in Statistics, 26, Springer-Verlag, 1984, 87–126

[4] J. Franke, “A general version of Breiman’s minimax filter”, Note di Matematica, 11 (1991), 157–175

[5] U. Grenander, “A prediction problem in game theory”, Ark.Mat., 3 (1957), 371–379

[6] T. Kailath, “A view of three decades of linear filtering theory”, IEEE Trans. on Inform. Theory, 20:2 (1974), 146–181

[7] S. A. Kassam, H. V. Poor, “Robust techniques for signal processing: A survey”, Proc. IEEE, 73:3 (1985), 433–481

[8] A. N. Kolmogorov, Selected works of A. N. Kolmogorov, v. II, Mathematics and Its Applications. Soviet Series, 26, Probability theory and mathematical statistics, ed. A. N. Shiryayev, Kluwer Academic Publishers, Dordrecht etc., 1992

[9] M. P. Moklyachuk, “Extrapolation of stationary sequences from observations with noise”, Theor. Probab. and Math. Stat., 57 (1998), 133–141

[10] M. P. Moklyachuk, “Robust procedures in time series analysis”, Theory Stoch. Process., 6(22):3–4 (2000), 127–147

[11] M. P. Moklyachuk, “Game theory and convex optimization methods in robust estimation problems”, Theory Stoch. Process., 7(23):1–2 (2001), 253–264

[12] M. P. Moklyachuk, “On estimates of unknown values of random fields from noisy observations”, Teor. Jmovirn. Mat. Stat., 65 (2001), 152–160

[13] M. P. Moklyachuk, “Estimation problems for random fields from noisy data”, Random Oper. and Stoch. Eq., 10 (2002), 223–232

[14] B. N. Pshenichnyi, Necessary conditions for an extremum, 2 ed., Nauka, Moscow, 1982

[15] Yu. A. Rozanov, Stationary stochastic processes, English transl. of 1st ed., Holden-Day, San Francisco, 1967, 2nd rev. ed., Nauka, Moscow, 1990

[16] K. S. Vastola, H. V. Poor, “An analysis of the effects of spectral uncertainty on Wiener filtering”, Automatica, 28 (1983), 289–293

[17] N. Wiener, Extrapolation, interpolation, and smoothing of stationary time series. With engineering applications, M. I. T. Press, Massachusetts Institute of Technology, Cambridge, Mass, 1966

[18] A. M. Yaglom, Correlation theory of stationary and related random functions, Springer Series in Statistics, v. I, Basic results, Springer-Verlag, New York etc., 1987

[19] A. M. Yaglom, Correlation theory of stationary and related random functions, Springer Series in Statistics, v. II, Supplementary notes and references, Springer-Verlag, New York etc., 1987

[20] M. I. Yadrenko, Spectral theory of random fields. Optimization Software, Inc., Springer-Verlag, New York-Heidelberg-Berlin, 1983