Minimax-robust forecasting of sequences with periodically stationary long memory multiple seasonal increments

Abstract

We introduce a stochastic sequence $\zeta(k)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the problem of optimal estimation of linear functionals constructed from unobserved values of the stochastic sequence $\zeta(k)$ based on its observations at points $ k<0$. 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 the sequence are not exactly known while some sets of admissible spectral densities are given.

Robust Filtering of Sequences with Periodically Stationary Multiplicative Seasonal Increments

“…In this article, we present methods of solution of the filtering problem for stochastic sequences with periodically stationary long memory multiple seasonal increments, or sequences with periodically stationary general multiplicative (GM) increments, introduced in the article by Luz and Moklyachuk [33]. These non-stationary stochastic sequences combine periodic structure of covariation functions of sequences as well as multiple seasonal factors, including the integrating one.…”

“…This article is dedicated to the robust filtering of stochastic sequences with periodically stationary long memory multiple seasonal increments, or sequences with periodically stationary general multiplicative (GM) increments, introduced by Luz and Moklyachuk [33]. In the recent years, models with multiple seasonal and periodic patterns are in scope of interest, see Dudek [8], Gould et al [14] and Reisen et al [44], Hurd and Piparas [21].…”

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

“…Here N * = N \ {0}. The explicit representation of the coefficients e γ (k) is given in [23]. Within the article, δ lp denotes Kronecker symbols, n l = n!…”

“…In this subsection, we present definition, justification and a brief review of the spectral theory of stochastic sequences with periodically stationary multiple seasonal increments, introduced in [23].…”

“…µ,s (ξ p (m)) is the GM increment of the p-th component of the vector-valued sequence ⃗ ξ(m). The following theorem describes the spectral structure of the vector-valued GM increment [16], [23].…”