arXiv subjects

Mykhailo Moklyachuk

Publications and source records attributed to Mykhailo Moklyachuk.

11 recordsLinked to original sources

arXiv2026

Estimation of periodically correlated random fields that are isotropic on a sphere

The problem of optimal linear estimation of functionals depending on the unknown values of a spatial temporal isotropic random field $ζ(j,x)$, which is periodically correlated with respect to discrete time argument $j\in\mathrm Z$ and mean-square continuous isotropic on the unit sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Estimates are based on observations of the field $ζ(j,x)+θ(j,x)$ at points $(j,x):$ $j\in Z\backslash\{0, 1, .... , N\}$, $x\in S_{n}$, where $θ(j,x)$ is an uncorrelated with $ζ(t,x)$ spatial temporal isotropic random field, which is periodically correlated with respect to discrete time argument $j\in\mathrm Z$ and mean-square continuous isotropic on the sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of the functional are derived in the case where the spectral density matrices are exactly known. Formulas that determine the least favourable spectral density matrices and the minimax (robust) spectral characteristics are proposed in the case where the spectral density matrices are not exactly known but a class of admissible spectral density matrices is given.

math.ST↗

arXiv2025

On Minimax Estimation Problems for Periodically Correlated Stochastic Processes

The aim of this article is to overview the problem of mean square optimal estimation of linear functionals which depend on unknown values of periodically correlated stochastic process. Estimates are based on observations of this process and noise. These problems are investigated under conditions of spectral certainty and spectral uncertainty. Formulas for calculating the main characteristics (spectral characteristic, mean square error) of the optimal linear estimates of the functionals are proposed. The least favorable spectral densities and the minimax-robust spectral characteristics of optimal estimates of the functionals are presented for given sets of admissible spectral densities.

math.ST↗

arXiv2025

Filtering of periodically correlated processes

The problem of optimal linear estimation of a linear functional depending on the unknown values of periodically correlated stochastic process from observations of the process with additive noise is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed in the case where spectral densities are exactly known. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics are proposed for a given class of admissible spectral densities.

math.ST↗

arXiv2025

Robust interpolation of sequences with periodically stationary multiplicative seasonal increments

We consider stochastic sequences with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the interpolation problem for linear functionals constructed from unobserved values of a stochastic sequence of this type based on observations of the sequence with a 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 interpolation of the functionals. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear interpolation of the functionals are proposed in the case where spectral densities of the sequences are not exactly known while some sets of admissible spectral densities are given.

math.ST↗

arXiv2025

Robust Forecasting of Sequences with Periodically Stationary Long Memory Multiplicative Seasonal Increments Observed with Noise and Cointegrated Sequences

The problem of optimal estimation of linear functionals constructed from unobserved values of stochastic sequence with periodically stationary increments based on observations of the sequence with a periodically 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 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 given.

math.ST↗

arXiv2025

Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

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 $t\in\mathbb R$ and isotropic on the unit sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Estimates are based on observations of the field $ζ(t,x)+θ(t,x)$ at points $(t,x):t<0,x\in S_{n}$, 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 $t\in\mathbb R$ and isotropic on the sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. 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.

math.ST↗

arXiv2025

Minimax Estimation Problem for Periodically Correlated Stochastic Processes

The problem of optimal linear estimation of linear functionals depending on the unknown values of a periodically correlated stochastic process from observations of the process with additive noise is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functionals are proposed in the case where spectral densities are exactly known and in the case where the spectral densities are unknown while a class of admissible spectral densities is given. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics are proposed for a given class of admissible spectral densities.

math.ST↗

arXiv2025

Interpolation of functionals of stochastic sequences with stationary increments from observations with noise

The problem of optimal estimation of linear functional ${{A}_{N}}ξ=\sum\limits_{k=0}^{N}{a(k)ξ(k)}\,$ depending on the unknown values of a stochastic sequence $ξ(m)$ with stationary $n$-th increments from observations of the sequence $ξ(k)$ at points $k=-1,-2,\ldots $ and of the sequence $ξ(k)+η(k)$ at points of time $k=N+1,N+2,\ldots $ is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed under condition of spectral certainty, where spectral densities of the sequences $ξ(m)$ and $η(m)$ are exactly known. Minimax (robust) method of estimation is applied in the case where the spectral densities are not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some specific sets of admissible densities.

math.ST↗

arXiv2025

Robust extrapolation problem for stochastic sequences with stationary increments

The problem of optimal estimation of functionals $Aξ=\sum\nolimits_{k=0}^{\infty }{}a(k)ξ(k)$ and ${{A}_{N}}ξ=\sum\nolimits_{k=0}^{N}{}a(k)ξ(k)$ which depend on the unknown values of stochastic sequence $ξ(k)$ with stationary $n$th increments is considered. Estimates are based on observations of the sequence $ξ(m)$ at points of time $m=-1,-2,\ldots$. Formulas for calculating the value of the mean square error and the spectral characteristic of the optimal linear estimates of the functionals are derived in the case where spectral density of the sequence is exactly known. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case where the spectral density of the sequence is not known but a set of admissible spectral densities is given.

math.ST↗

arXiv2025

Filtering Problem for Functionals of Stationary Processes with Missing Observations

The problem of the mean-square optimal linear estimation of the functional $Aξ=\ \int\limits_{R^s}a(t)ξ(-t)dt,$ which depends on the unknown values of stochastic stationary process $ξ(t)$ from observations of the process $ξ(t)+η(t)$ at points $t\in\mathbb{R} ^{-} \backslash S $, $S=\bigcup\limits_{l=1}^{s}[-M_{l}-N_{l}, \, \ldots, \, -M_{l} ],$ $R^s=[0,\infty) \backslash S^{+},$ $S^{+}=\bigcup\limits_{l=1}^{s}[ M_{l}, \, \ldots, \, M_{l}+N_{l}]$ is considered. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functional are proposed under the condition of spectral certainty, where spectral densities of the processes $ξ(t)$ and $η(t)$ are exactly known. The minimax (robust) method of estimation is applied in the case where spectral densities are not known exactly, but sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some special sets of admissible spectral densities.

math.ST↗

arXiv2025

Filtering Problem for Random Processes with Stationary Increments

This paper deals with the problem of optimal mean-square filtering of the linear functionals $Aξ=\int_{0}^{\infty}a(t)ξ(-t)dt$ and $A_Tξ=\int_{0}^Ta(t)ξ(-t)dt$ which depend on the unknown values of random process $ξ(t)$ with stationary $n$th increments from observations of process $ξ(t)+η(t)$ at points $t\leq0$, where $η(t)$ is a stationary process uncorrelated with $ξ(t)$. We propose the values of mean-square errors and spectral characteristics of optimal linear estimates of the functionals when spectral densities of the processes are known. In the case where we can operate only with a set of admissible spectral densities relations that determine the least favorable spectral densities and the minimax spectral characteristics are proposed.

math.ST↗