Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences

Abstract

The problem of optimal estimation of the linear functional A = ∑ ∞ k=0 a(k) (−k) depending on the unknown values of a stochastic sequence (m) with nth stationary increments from observations of the sequence (m) + (m) at points m = 0, −1, −2, …, where (m) is a stationary sequence uncorrelated with (m), is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are derived in the case where spectral densities of stochastic sequences are exactly known and admit the canonical factorizations. In the case of spectral uncertainty, where spectral densities are not known exactly, but sets of admissible spectral densities are specified, the minimax-robust method is applied. Formulas and relations that determine the least favourable spectral densities and the minimax-robust spectral characteristics are proposed for the given sets of admissible spectral densities. The filtering problem for a class of cointegrated sequences is investigated.The crucial assumption of application of traditional methods of finding solution to the filtering problem for random processes is that spectral densities of the processes are exactly known. However, in practical situations complete information on spectral densities is impossible and the established results cannot be directly applied to practical filtering problems. This is a reason to apply the minimax-robust method of filtering and derive the minimax estimates since they minimize the maximum value of the mean-square errors for all spectral densities from a given set of admissible densities simultaneously. In this article, we deal with the problem of optimal estimation of functionals depending on the unknown values of a random process with stationary increments based on observations of the process and a noise. In the case where spectral densities of the processes are not exactly known, relations for determining least favourable spectral densities and minimax-robust spectral characteristics are proposed.

Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences

“…The values of the mean-square errors ∆(h µ (f, g); f, g) := ∆(f, g; Aξ) and ∆(h µ,N (f, g); f, g) := ∆(f, g; A N ξ) and the spectral characteristics h µ (f, g) and h µ,N (f, g) of the optimal linear estimates Aξ and A N ξ of the functionals Aξ and A N ξ which depend on the unknown values of the sequence ξ(m) based on observations of the stochastic sequence ξ(k) + η(k) can be calculated by formulas (23), (22) and (29), (27) correspondingly under the condition that spectral densities f (λ) and g(λ) of stochastic sequences ξ(m) and η(m) are exactly known. Having canonical factorizations (35) and (36) we can calculate the values of mean-square errors ∆(h µ (f, g); f, g) and spectral characteristics h µ (f, g) by formulas (42), (41) respectively.…”

“…where Z η (λ) is a random process with independent increments on [−π, π) corresponding to the spectral function [23]. The spectral density p(λ) of the sequence ζ(m) is determined by spectral densities f (λ) and g(λ) by the relation…”

“…The spectral characteristic h µ (λ) of the optimal estimate Aξ is calculated by formula (22). The value of the mean-square error ∆(f, g; Aξ) is calculated by formula (23).…”

“…∑ N k=0 a(k)ξ (−k) in the papers [21,23]. The minimax interpolation problem for the linear functional A N ξ = ∑ N k=0 a(k)ξ(k) which depends on the unknown values of the sequence ξ(k) based on observations with and without noise was investigated in papers [19,20], and for the linear functional Aξ = ∫ ∞ 0 a(t)ξ(t)dt which depends on the unknown values of a random process ξ(t) in the paper [24].…”

Filtering Problem for Functionals of Stationary Sequences

“…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].…”