Filtering of multidimensional stationary sequences with missing observations

Abstract

The problem of mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence from observations of the sequence with a noise and missing observations is considered. Formulas for calculating the meansquare errors and the spectral characteristics of the optimal linear estimates of the functionals are proposed under the condition of spectral certainty, where spectral densities of the sequences are exactly known. The minimax (robust) method of estimation is applied in the case where spectral densities are not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special sets of admissible densities.

Minimax prediction of sequences with periodically stationary increments

“…, can be calculated by formulas (20), ( 21) and ( 26), ( 27) respectively, provided the spectral density f (λ) of the stochastic sequence ξ(m) is exactly known. If the spectral density f (λ) admits the canonical factorization (22), formulas (23), ( 24) and ( 28), ( 29) can be used for calculating values of the mean square errors and the spectral characteristics, respectively. However, in practical cases spectral densities of sequences usually are not exactly known.…”

“…In the case, when the spectral density f (λ) admits the canonical factorization (22), the spectral characteristic and the value of the mean square error of the optimal estimate ξ p (N) can be calculated by the formulas…”

“…The spectral characteristic h µ (λ) = {h p (λ)} T p=1 and the value of the mean square error ∆( f ; Aζ) of the optimal estimate Aζ are calculated by formulas (20) and ( 21) respectively. In the case, when the spectral density matrix f (λ) admits the canonical factorization (22), the spectral characteristic and the value of the mean square error of the optimal estimate Aξ can be calculated by formulas ( 23) and ( 24) respectively.…”

“…The value of the mean square error of the optimal estimate is calculated by the formula (32). In the case, when the spectral density f (λ) admits the canonical factorization (22), the spectral characteristic and the value of the mean square error of the optimal estimate ζ(M) can be calculated by the formulas (33), (34).…”

“…Moklyachuk and M.I. Sidei [22,28], who derive minimax estimates of stationary processes from observations with missed values. P.S.…”