Minimax-robust filtering problem for stochastic sequences with stationary increments

Abstract

We consider a stochastic sequence ξ(m) with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. The filtering problem is solved for this type of sequences based on observations with a periodically stationary noise. When spectral densities are known and allow the canonical factorizations, we derive the mean square error and the spectral characteristics of the optimal estimate of the functional Aξ = ∞ k=0 a(k)ξ(−k). Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimate of the functional are proposed in the case where the spectral densities are not known, but some sets of admissible spectral densities are given.

Filtering of multidimensional stationary sequences with missing observations

“…This condition makes it possible to find the least favourable spectral densities in some special classes of spectral densities D (see books by Ioffe & Tihomirov, 1979, Pshenichnyj, 1971, Rockafellar, 1997. Note, that the form of the functional ∆ h 0 ; F, G is convenient for application the Lagrange method of indefinite multipliers for finding solution to the problem (21). Making use the method of Lagrange multipliers and the form of subdifferentials of the indicator functions we describe relations that determine least favourable spectral densities in some special classes of spectral densities (see books by Moklyachuk,2008, Moklyachuk & Masyutka, 2012 for additional details).…”

Filtering Problem for Functionals of Stationary Sequences

“…Define the linear operator T in the space ℓ 2 by the matrix with elements (T) l,k = s(l − k), l, k ≥ 0, where the coefficients s(k), k ≥ 0, are defined in (20). Then operators T and G in the space ℓ 2 admit the factorizations T = Υ ′ Υ and G = Φ ′ Φ.…”

“…In the case of spectral uncertainty, where the spectral densities are not exactly known while a set of admissible spectral densities is specified, formulas for determination the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of the functionals Aξ and A N ξ are proposed for some specific classes of admissible spectral densities. Such approach to filtering problems for stochastic sequence with stationary nth increments was applied in the papers by Luz and Moklyachuk [20], [21]. The method of projection in the Hilbert space is also applied to extrapolation and interpolation problems for stochastic sequences with stationary nth increments in papers by Luz and Moklyachuk [19], [22].…”

Minimax interpolation of sequences with stationary increments and cointegrated sequences

“…In the book by Moklyachuk [20], the minimax-robust estimates of the linear functionals of stationary sequences and processes are presented. See also the survey paper [18], The classical and minimax-robust problems of interpolation, extrapolation, and filtering of the functional of stochastic sequences with stationary increments are investigated in the papers by Luz and Moklyachuk [14][15][16][17]19]. Particularly, the cointegrated sequences are investigated in the papers [14,15].…”