On interpolation problem for vector-valued stochastic sequences

Abstract

The classical Kolmogorov method of linear interpolation, extrapolation and filtering of weakly stationary stochastic sequences [1], [2 ] may be employed under the condition that spectral densities of stochastic sequences are known. In practice, however, the problem of estimation of unknown values of stochastic sequence arises where the spectral density is not known exactly. To solve the problem, the parametric or nonparametric estimate of the unknown spectral density is found. Then the classical method is applied provided that the estimate of the density is the true one. This procedure can result in a significant increasing of the value of the error as Vastola and Poor have demonstrated with the help of some examples [3]. For this reason it is necessary to search the estimate of the unknown value of the stochastic sequence that has the least value of the error for all densities from a certain class of spectral densities. Such an approach to the problem of interpolation, extrapolation and filtering of stationary stochastic sequences have been taken into consideration by many investigators [3] [23]. A survey of results in minimax (robust) methods of data processing can be found in the paper [9]. The paper [1 0 ] is the first one where the minimax interpolation problem for the e pollution model is investigated. The relation of the minimax interpolation problem with the problem of robust hypothesis testing is indicated in [8 ]. In papers [4] [23] the minimax interpolation, extrapolation and filtering problems are investigated with the help of the convex optimization methods.