Robust Filtering of Sequences with Periodically Stationary Multiplicative Seasonal Increments

Abstract

We consider stochastic sequences with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the filtering problem for linear functionals constructed from unobserved values of a stochastic sequence of this type based on observations of the sequence with a periodically stationary noise sequence. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal filtering of the functionals. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal linear filtering of the functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.

Filtering problem for sequences with periodically stationary multi-seasonal increments with spectral densities allowing canonical factorizations

“…We proposed a solution of the filtering problem in terms of coefficients of canonical factorizations of the spectral densities of the involved stochastic sequences. The results obtained in [24] are based on the Fourier transformations of the spectral densities.…”

“…The second condition (11) is the necessary and sufficient one under which the mean square error of the optimal estimate of functional A ⃗ ξ is not equal to 0. Any linear estimate A ⃗ ξ of the functional A ⃗ ξ allows the representation [24] A…”