Interpolation Problem for Multidimensional Stationary Processes with Missing Observations

Abstract

The problem of the mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional continuous time stationary stochastic process is considered. Estimates are based on observations of the process with an additive stationary stochastic noise process at points which do not belong to some finite intervals of a real line. The problem is investigated in the case of spectral certainty, where the spectral densities of the processes are exactly known. Formulas for calculating the mean-square errors and spectral characteristics of the optimal linear estimates of functionals are proposed under the condition of spectral certainty. The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the processes are not known exactly while some sets of admissible spectral densities of the processes are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible spectral densities.

Minimax interpolation of stochastic processes with stationary increments from observations with noise

“…The values of the mean square errors and the spectral characteristics of the optimal estimate of the functional A N T ξ depending on the unobserved values of a stochastic process ξ(t) which determine a generated stationary stochastic dth increment sequence ξ (d) j with the spectral density matrix f (λ) based on observations of the process ξ(t) + η(t) at points R \ [0; (N + 1)T ] can be calculated by formulas (21), (24), (25) respectively, under the condition that spectral densities f (λ) and g(λ) of stochastic sequences ξ j and η j are exactly known.…”

Filtering of multidimensional stationary sequences with missing observations

“…Let the function F (λ) + G 0 (λ) satisfy the minimality condition (1). The spectral density G 0 (λ) is the least favorable in the classes D U V k , k = 1, 4, for the optimal linear filtering of the functional A ξ if it satisfies relations (36), (39), (42), (45), respectively, and the pair (F (λ), G 0 (λ)) is a solution of the optimization problem (19). The minimax-robust spectral characteristic of the optimal estimate of the functional A ξ is determined by formula (11).…”

“…For the first pair 39), ( 42), (45), respectively, and the pair (F (λ), G 0 (λ)) is a solution of the optimization problem (19). The minimax-robust spectral characteristic of the optimal estimate of the functional A ξ is determined by formula (11).…”

Mykhailo Moklyachuk – to the 75th anniversary of his birth

“…The monograph [11] investigates the problems of optimal estimation by the mean-squared criterion of linear functions constructed on the basis of unknown values of stationary random processes. The estimates are based on observations of processes with an additional stationary noise process.…”