Robust filtering of sequences with periodically stationary multiplicative seasonal increments

TL;DR

This paper develops robust filtering methods for stochastic sequences with complex, periodically stationary fractional increments, providing formulas for optimal estimates and robustness under spectral uncertainty.

Contribution

It introduces new formulas for filtering and robust estimation of sequences with periodically stationary fractional increments, addressing spectral uncertainty.

Findings

Formulas for mean square errors and spectral characteristics of optimal estimates.

Methods for determining least favorable spectral densities.

Robust filtering techniques under spectral density uncertainty.

Abstract

We study stochastic sequences $\xi(k)$ 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 $\xi(k)$ based on observations with the 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 estimates of the functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of the functionals are proposed in the case where spectral densities of sequences are not exactly known while some sets of admissible spectral densities are given.