Interpolation Problem for Stationary Sequences with Missing Observations

Abstract

The problem of the mean-square optimal estimation of the linear functional Asξ = s−1 ∑ l=0 M l +N l+1 ∑ j=M l a(j)ξ(j), M l = l ∑ k=0 (N k + K k), N 0 = K 0 = 0, which depends on the unknown values of a stochastic stationary sequence ξ(j), j ∈ Z from observations of the sequence at points of time j ∈ Z\S, S = s−1 ∪ l=0 {M l , M l + 1,. .. , M l + N l+1 } is considered. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functional are derived under the condition of spectral certainty, where the spectral density of the sequence ξ(j) is exactly known. The minimax (robust) method of estimation is applied in the case where the spectral density is not known exactly, but sets of admissible spectral densities are given. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities.

An Adaptive Image Registration Technique to Remove Atmospheric Turbulence

“…High-quality frames are selected for further processing on the basis of sharpness, intensity similarity, and ROI size. Another technique to improve the resolution of video frames is by using bilinear interpolation [14] [15]. CGI is one of the techniques that use bilinear interpolation.…”

Interpolation problem for periodically correlated stochastic sequences with missing observations

“…In the papers by M. P. Moklyachuk and M. I. Sidei [71]- [75] results of investigations of the interpolation, extrapolation and filtering problems for stationary stochastic sequences and processes with missing observations are proposed. The results of the study of the extrapolation, interpolation and filtering problems for linear functionals constructed from unobserved values of multidimensional stochastic sequences and processes are presented in the papers by O. Yu.…”

Filtering of multidimensional stationary sequences with missing observations

“…Results of investigations of the prediction problem for stationary stochastic sequences with missing observations are presented in the papers by Bondon (2002Bondon ( , 2005, Kasahara, Pourahmadi & Inoue (2007, 2009. In papers by Moklyachuk & Sidei (2015 results of investigations of the interpolation, extrapolation and filtering problems for stationary stochastic sequences and processes with missing observations are proposed.…”