arXiv subjects

Mikhail Moklyachuk

Publications and source records attributed to Mikhail Moklyachuk.

At least 19 recordsLinked to original sources

arXiv2026

Minimax approach to the estimation problem for homogeneous random fields

The problem of the mean-square optimal estimation of the linear functionals which depend on the unknown values of a multidimensional homogeneous random field from observations of the field with noise is considered. The minimax (robust) method of estimation is applied in the case where the spectral densities of the fields are not known exactly while some 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.

math.ST↗

arXiv2025

Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations

This paper focuses on the problem of the mean square optimal estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence. Estimates are based on observations of the sequence with an additive stationary noise sequence. The aim of the paper is to develop methods of finding the optimal estimates of the functionals in the case of missing observations. The problem is investigated in the case of spectral certainty where the spectral densities of the sequences are exactly known. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of functionals are derived under the condition of spectral certainty. The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the sequences are not known exactly while sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible densities.

math.ST↗

arXiv2025

Interpolation Problem for Multidimensional Stationary Processes with Missing Observations

The problem of the mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional continuous time stationary stochastic process is considered. Estimates are based on observations of the process with an additive stationary stochastic noise process at points which do not belong to some finite intervals of a real line. The problem is investigated in the case of spectral certainty, where the spectral densities of the processes are exactly known. Formulas for calculating the mean-square errors and spectral characteristics of the optimal linear estimates of functionals are proposed under the condition of spectral certainty. The minimax (robust) method of estimation is applied in the case spectral uncertainty, where spectral densities of the processes are not known exactly while some sets of admissible spectral densities of the processes are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible spectral densities

math.ST↗

arXiv2025

Robust extrapolation problem for random processes with stationary increments

The problem of optimal estimation of linear functionals $A ξ=\int_{0}^{\infty} a(t)ξ(t)dt$ and $A_Tξ=\int_{0}^{T} a(t)ξ(t)dt$ depending on the unknown values of random process $ξ(t)$, $t\in R$, with stationary $n$th increments from observations of ttis process for $t<0$ is considered. Formulas for calculating mean square error and spectral characteristic of optimal linear estimation of the functionals are proposed in the case when spectral density is exactly known. Formulas that determine the least favorable spectral densities are proposed for given sets of admissible spectral densities.

math.ST↗

arXiv2025

On interpolation problem for multidimensional harmonizable stable sequences with noise observations

We consider the problem of optimal linear estimation of the functional $$A_N \vecξ =\sum_{j = 0}^{N} (\vec{a}(j))^{\top} \vecξ(j)$$ that depends on the unknown values $\vecξ(j),j=0,1,\dots,N,$ of a vector-valued harmonizable symmetric $α$-stable random sequence $\vecξ(j)=\left \{ ξ_ {k} (j) \right \}_{k = 1} ^ {T}$, from observations of the sequence $\vecξ(j)+\vecη(j)$ at points $j\in\mathbb Z\setminus\{0,1,\dots,N\}$. We consider the problem for mutually independent vector-valued harmonizable symmetric $α$-stable random sequences $\vecξ(j)=\left \{ ξ_ {k} (j) \right \}_{k = 1} ^ {T}$ and $\vecη(j)=\left \{ ξ_ {k} (j) \right \}_{k = 1} ^ {T}$ which have absolutely continuous spectral measures and the spectral densities $f(θ)$ and $g(θ)$ satisfying the minimality condition.

math.ST↗

arXiv2024

Minimax-robust estimation problems for stationary stochastic sequences

This survey provides an overview of optimal estimation of linear functionals which depend on the unknown values of a stationary stochastic sequence. Based on observations of the sequence without noise as well as observations of the sequence with a stationary noise, estimates could be obtained. Formulas for calculating the spectral characteristics and the mean-square errors of the optimal estimates of functionals are derived in the case of spectral certainty, where spectral densities of the sequences are exactly known. In the case of spectral uncertainty, where spectral densities of the sequences are not known exactly while sets of admissible spectral densities are given, the minimax-robust method of estimation is applied. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics of estimates are presented for some special classes of admissible spectral densities.

math.ST↗

arXiv2024

Filtering Problem for Functionals of Stationary Sequences

The problem of the mean-square optimal linear estimation of functionals which depend on the unknown values of a stationary stochastic sequence from observations of the sequence with noise is considered. In the case of spectral certainty, where the spectral densities of the sequences are exactly known, we propose formulas for calculating the spectral characteristic and value of the mean-square error of the estimate, which are determined using the Fourier coefficients of some functions from the spectral densities. The minimax-robust method of estimation is applied in the case of spectral uncertainty, where the spectral densities are not exactly known, but a class of admissible spectral densities is given. Formulas for determining the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of the functionals are proposed for some specific classes of admissible spectral densities.

math.ST↗

arXiv2024

Filtering of stochastic processes having periodically correlated increments

We deal with the problem of the mean square optimal estimation of linear transformations of the unobserved values of a continuous time stochastic process with periodically correlated increments. Estimates are based on observations of the process with a continuous time stochastic noise process which is periodically correlated increments as well. To solve the problem, we transform the processes to infinite dimensional vector valued stationary sequences. We obtain formulas for calculating the mean square errors and the spectral characteristics of the optimal estimates of the transformations. Formulas determining the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of transformations are derived.

math.ST↗

arXiv2023

Prediction problem for continuous time stochastic processes with periodically correlated increments observed with noise

We propose solution of the problem of the mean square optimal estimation of linear functionals which depend on the unobserved values of a continuous time stochastic process with periodically correlated increments based on observations of this process with periodically stationary noise. To solve the problem, we transform the processes to the sequences of stochastic functions which form an infinite dimensional vector stationary sequences. In the case of known spectral densities of these sequences, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.

math.ST↗

arXiv2023

Minimax interpolation of continuous time stochastic processes with periodically correlated increments observed with noise

We deal with the problem of optimal estimation of the linear functionals constructed from the missed values of a continuous time stochastic process $ξ(t)$ with periodically stationary increments at points $t\in[0;(N+1)T]$ based on observations of this process with periodically stationary noise. To solve the problem, a sequence of stochastic functions $ \{ξ^{(d)}_j(u)=ξ^{(d)}_j(u+jT,τ),\,\, u\in [0,T), j\in\mathbb Z\}. $ is constructed. It forms a $L_2([0,T);H)$-valued stationary increment sequence $\{ξ^{(d)}_j,j\in\mathbb Z\}$ or corresponding to it an (infinite dimensional) vector stationary increment sequence $\{\vecξ^{(d)}_j=(ξ^{(d)}_{kj}, k=1,2,\dots)^{\top}, j\in\mathbb Z\}$. In the case of a known spectral density, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.

math.ST↗

arXiv2023

Filtering problem for sequences with periodically stationary multiseasonal increments with spectral densities allowing canonical factorizations

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 $Aξ=\sum_{k=0}^{\infty}{a}(k) ξ(-k)$. 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.

math.ST↗

arXiv2023

Estimation problem for continuous time stochastic processes with periodically correlated increments

We deal with the problem of optimal estimation of the linear functionals constructed from unobserved values of a continuous time stochastic process with periodically correlated increments based on past observations of this process. To solve the problem, we construct a corresponding to the process sequence of stochastic functions which forms an infinite dimensional vector stationary increment sequence. In the case of known spectral density of the stationary increment sequence, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.

math.ST↗

arXiv2021

On minimax estimation problem for stationary stochastic sequences from observations in special sets of points

The problem of the mean-square optimal estimation of the linear functionals which depend on the unknown values of a stochastic stationary sequence from observations of the sequence in special sets of points is considered. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functionals are derived under the condition of spectral certainty, where the spectral density of the sequence is exactly known. The minimax (robust) method of estimation is applied in the case where the spectral density of the sequence is not known exactly while some 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.

math.ST↗

arXiv2021

Minimax-robust estimation problems for sequences with periodically stationary increments observed with noise

The problem of optimal estimation of linear functionals constructed from the unobserved values of a stochastic sequence with periodically stationary increments based on observations of the sequence with stationary noise is considered. For sequences with known 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 the minimax-robust spectral characteristics of the optimal linear estimates of functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are specified.

math.ST↗

arXiv2021

Robust filtering of sequences with periodically stationary multiplicative seasonal increments

We study stochastic sequences $ξ(k)$ 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 $ξ(k)$ 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.

math.ST↗

arXiv2021

Minimax extrapolation problem for periodically correlated stochastic sequences with missing observations

The problem of optimal estimation of the linear functionals which depend on the unknown values of a periodically correlated stochastic sequence $ζ(j)$ from observations of the sequence $ζ(j)+θ(j)$ at points $j\in\{\dots,-n,\dots,-2,-1,0\}\setminus S$, $S=\bigcup _{l=1}^{s-1}\{-M_l\cdot T+1,\dots,-M_{l-1}\cdot T-N_{l}\cdot T\}$, is considered, where $θ(j)$ is an uncorrelated with $ζ(j)$ periodically correlated stochastic sequence. Formulas for calculation the mean square error and the spectral characteristic of the optimal estimate of the functional $Aζ$ are proposed in the case where spectral densities of the sequences are exactly known. Formulas that determine the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of functionals are proposed in the case of spectral uncertainty, where the spectral densities are not exactly known while some sets of admissible spectral densities are specified.

math.ST↗

arXiv2020

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

We introduce stochastic sequences $ζ(k)$ 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 stochastic sequences $ζ(k)$ based on their observations at points $ k<0$. For sequences with known spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of functionals are proposed in the case where spectral densities of sequences are not exactly known while some sets of admissible spectral densities are given.

math.ST↗

arXiv2020

Interpolation problem for periodically correlated stochastic sequences with missing observations

The problem of mean square optimal estimation of linear functionals which depend on the unobserved values of a periodically correlated stochastic sequence is considered. The estimates are based on observations of the sequence with a noise. Formulas for calculation the mean square errors and the spectral characteristics of the optimal estimates of functionals are derived in the case of spectral certainty, where the spectral densities of the sequences are exactly known. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed in the case of spectral uncertainty, where the spectral densities of the sequences are not exactly known while some classes of admissible spectral densities are specified.

math.ST↗