Extrapolation of time-homogeneous random fields that are isotropic on a sphere. II (original) (raw)
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Ukrainian Mathematical Journal, 1995
We study the problem of optimal linear estimation of the functional A_N \xi = \sum\limits_{k = 0}^{\rm N} {\int\limits_{S_n } {a(k,x)\xi (k,x)m_n (dx),} }$$ , which depends on unknown values of a random field ξ(k, x),k∃Z,x∃S n homogeneous in time and isotropic on a sphereS n, by observations of the field ξ(k,x)+η(k,x) with k∃ Z{0, 1, ...,N},x∃Sn (here, η (k, x) is a random field uncorrelated with ξ(k, x), homogeneous in time, and isotropic on a sphere Sn). We obtain formulas for calculation of the mean square error and spectral characteristic of the optimal estimate of the functionalA Nξ. The least favorable spectral densities and minimax (robust) spectral characteristics are found for optimal estimates of the functionalA Nξ.
Linear statistical problems for stationary isotropic random fields on a sphere. II
Theory of Probability and Mathematical Statistics, No. 19,1980, 1980
Linear extrapolation problem s are discussed concerning station ary isotropic random fields on a sphere w hich are observed on point sets E = X S n, E f = x): u -T « t < u, X = x k G S fc = 1, M \ E^ = {(?, x): t e N, -00 < t < at = Xp. G Sn, k = 1, m } and E^ -{(£ at): t N, t < u, x = x k^Sn > k = ~ M }, where u and T are fixed num bers, w hile E j is a subset o f R -(-oo? oo). Bibliography: 5 titles. UDC 519.21
Linear statistical problems for stationary isotropic random fields on a sphere. I
Theory of Probability and Mathematical Statistics, No. 18,1979, 1979
A b stra c t. T he sp e ctral re p re se n ta tio n o f a sta tio n a ry iso tro p ic ra n d o m field o n a sp h ere, as w ell as its co rre la tio n fu n c tio n , are describ ed ; ex am p les o f co rre la tio n fu n c tio n s o f such fields are p ro d u c e d . T he p ro b lem o f estim atin g th e regression co effic ie n ts an d th e u n k n o w n e x p e c ta tio n o f a ra n d o m field is discussed. B ib liography: 11 title s. UDC 519.21
ON ESTIMATES OF UNKNOWN VALUES OF RANDOM FIELDS OBSERVED WITH A NOISE
Theory of Probability and Mathematical Statistics, No.65, 2002.
The problem of optimal linear estimation is considered for the functional depending on unknown values of a time-homogeneous random field £(k,x), k Є Z, x € Sn, isotropic on the sphere Sn, from observations of the field £(fc, x) + ті(к, x) for k = -1, -2,..., x Є Sn, where rj(k,x) is a time-isotropic random field uncorrelated with £(k,x). Formulas for the evaluation of the mean-square error and the spectral characteristic of an optimal estimate of the functional are obtained. The least favorable spectral densities and minimax (robust) spectral characteristics of optimal estimates of the functional are calculated for different models of random fields.
A problem of minimax smoothing for homogeneous isotropic on a sphere random fields
Random Operators and Stochastic Equations, 2000
A b stra ct-T he problem of the least in a square-m ean linear estim ation for th e transform ation of a hom ogeneous isotropic on a sphere S n random field £(j, x), j £ N, x 6 S n , using observations of. £(i>®) Ti { h x) f°r j ^ 0, x € S n , where rj(j, x) is a hom ogeneous isotropic on a sphere S n random field uncorrelated w ith x), is considered. T he least favourable spectral densities and the m inim ax (robust) spectral characteristics are determ ined for some classes of spectral densities.
Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields
Keywords: isotropic random field, periodically correlated random field, robust estimate, mean square error, least favourable spectral density, minimax spectral characteristic. Abstract. The problem of optimal linear estimation of functionals depending on the unknown values of a random field ζ(t, x), which is mean-square continuous periodically correlated with respect to time argument t ∈ R and isotropic on the unit sphere S n with respect to spatial argument x ∈ S n. Estimates are based on observations of the field ζ(t, x) + θ(t, x) at points (t, x) : t < 0, x ∈ S n , where θ(t, x) is an uncorrelated with ζ(t, x) random field, which is mean-square continuous periodically correlated with respect to time argument t ∈ R and isotropic on the sphere S n with respect to spatial argument x ∈ S n. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.
Estimation problems for random fields from noisy data
Random Operators and Stochastic Equations, 2002
Sn a(t, x)ξ(t, x) m n (dx) dt of the unknown values of a random field ξ(t, x), t ∈ R 1 , x ∈ S n that is time homogeneous and isotropic on a sphere S n from observations of the field ξ(t, x)+η(t, x) for t < 0, x ∈ S n , where η(t, x) is a random field that is time homogeneous and isotropic on a sphere S n uncorrelated with ξ(t, x), is considered. Formulas are obtained for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional A ξ. The least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functionals are determined for some classes of random fields.