Minimax-robust filtering of functionals from periodically correlated random fields

Keywords: isotropic random field, periodically correlated random field, robust estimate, mean square error, least favourable spectral density, minimax spectral characteristic.Abstract. The problem of optimal linear estimation of functionals depending on the unknown values of a random field ζ(t, x), which is mean-square continuous periodically correlated with respect to time argument t ∈ R and isotropic on the unit sphere S n with respect to spatial argument x ∈ S n . Estimates are based on observations of the field ζ(t, x) + θ(t, x) at points (t, x) : t < 0, x ∈ S n , where θ(t, x) is an uncorrelated with ζ(t, x) random field, which is mean-square continuous periodically correlated with respect to time argument t ∈ R and isotropic on the sphere S n with respect to spatial argument x ∈ S n . Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case of spectral certainty where the spectral densities of the fields are exactly known. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics are proposed in the case where the spectral densities are not exactly known while a class of admissible spectral densities is given.

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“…The constrained optimization problem (20) is equivalent to the following unconstrained optimization problem…”

Section: Minimax-robust Methods Of Filteringmentioning

confidence: 99%

Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

Golichenko¹,

Masyutka

Moklyachuk

2017

BMSA

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“…can be calculated by formulas (12), (11). (5) which depends on the unknown values ξ(0), ξ(1), ξ (5) of the stochastic sequence ξ(j) from observations at points…”

Section: Theorem 21mentioning

confidence: 99%

“…Note, that the form of the functional ∆ ( h 0 ; f ) is convenient for application the Lagrange method of indefinite multipliers for finding solution to the problem (15). 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 [11,29,30] for additional details).…”

Section: Lemma 31mentioning

confidence: 99%

“…Masyutka and M.P. Moklyachuk [3], I. I. Dubovets'ka and M. P. Moklyachuk [4] - [7], I. I. Golichenko and M. P. Moklyachuk [11] investigate the interpolation, extrapolation and filtering problems for periodically correlated stochastic sequences. The paper by M. M. Luz and M. P. Moklyachuk [20] - [24] deals with the estimation problems for functionals which depend on the unknown values of stochastic sequences with stationary increments.…”

Section: Introductionmentioning

confidence: 99%

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Interpolation Problem for Stationary Sequences with Missing Observations

Moklyachuk

Sidei

2015

Stat., optim. inf. comput.

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The problem of the mean-square optimal estimation of the linear functional AsξFormulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functional are derived under the condition of spectral certainty, where the spectral density of the sequence ξ(j) is exactly known. The minimax (robust) method of estimation is applied in the case where the spectral density is not known exactly, but sets of admissible spectral densities are given. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities.

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“…In the further papers Dubovetska and investigated the minimax-robust extrapolation, interpolation and filtering problems for periodically correlated processes and sequences. See the book by Golichenko and Moklyachuk (2014) for more relative results and references. The minimax-robust extrapolation, interpolation and filtering problems for stochastic sequences and processes with nth stationary increments were solved by Luz and Moklyachuk (2012, 2015a, 2015b, 2015c, 2016a, 2016bMoklyachuk & Luz, 2013).…”

Section: Introductionmentioning

confidence: 99%

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

Luz

Moklyachuk

2016

Cogent Mathematics

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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.

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Minimax-robust filtering of functionals from periodically correlated random fields

Cited by 8 publications

References 26 publications

Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

Interpolation Problem for Stationary Sequences with Missing Observations

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

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