scholarly journals Robust Filtering of Sequences with Periodically Stationary Multiplicative Seasonal Increments

2021 ◽  
Vol 9 (4) ◽  
pp. 1010-1030
Author(s):  
Maksym Luz ◽  
Mikhail Moklyachuk
Keyword(s):  
Spectral Characteristics ◽  
Linear Filtering ◽  
Linear Functionals ◽  
Spectral Densities ◽  
Optimal Linear ◽  
The Mean ◽  
Noise Sequence ◽  
Mean Square Errors ◽  
Filtering Problem ◽  
Optimal Linear Filtering

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 filtering 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 filtering of the functionals. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal linear filtering of the 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.

scholarly journals Filtering of multidimensional stationary sequences with missing observations

2019 ◽  
Vol 11 (2) ◽  
pp. 361-378
Author(s):  
O.Yu. Masyutka ◽  
M.P. Moklyachuk ◽  
M.I. Sidei
Keyword(s):  
Spectral Characteristics ◽  
Stochastic Noise ◽  
Mean Square ◽  
Missing Observations ◽  
Linear Functionals ◽  
Spectral Densities ◽  
Optimal Linear ◽  
The Mean ◽  
Noise Sequence ◽  
Mean Square Errors

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

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

2020 ◽  
Vol 8 (3) ◽  
pp. 684-721
Author(s):  
Maksym Luz ◽  
Mikhail Moklyachuk
Keyword(s):  
Long Memory ◽  
Spectral Characteristics ◽  
Optimal Estimation ◽  
Mean Square ◽  
Linear Functionals ◽  
Spectral Densities ◽  
Optimal Estimates ◽  
Optimal Linear ◽  
The Mean ◽  
Mean Square Errors

We introduce a stochastic sequence $\zeta(k)$ 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 the stochastic sequence $\zeta(k)$  based on its  observations at points $ k<0$. 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 the sequence are not exactly known while some sets of admissible spectral densities are given.

scholarly journals Minimax prediction of sequences with periodically stationary increments

2021 ◽  
Vol 13 (2) ◽  
pp. 352-376
Author(s):  
P.S. Kozak ◽  
M.M. Luz ◽  
M.P. Moklyachuk
Keyword(s):  
Spectral Characteristics ◽  
Optimal Estimation ◽  
Mean Square ◽  
Linear Functionals ◽  
Spectral Densities ◽  
Stationary Increments ◽  
Optimal Estimates ◽  
Optimal Linear ◽  
The Mean ◽  
Mean Square Errors

The problem of optimal estimation of linear functionals constructed from unobserved values of a stochastic sequence with periodically stationary increments based on its observations at points $ k<0$ 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 favourable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.

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

2020 ◽  
pp. 68-83
Author(s):  
M. P. Moklyachuk ◽  
M. M. Luz
Keyword(s):  
Spectral Characteristics ◽  
Optimal Estimation ◽  
Mean Square ◽  
Linear Functionals ◽  
Spectral Densities ◽  
Stationary Increments ◽  
Estimation Problems ◽  
Optimal Linear ◽  
The Mean ◽  
Mean Square Errors

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.

Extrapolation Problem for Continuous Time Periodically Correlated Isotropic Random Fields

2017 ◽  
Vol 19 ◽  
pp. 1-23
Author(s):  
Iryna Golichenko ◽  
Oleksand Masyutka ◽  
Mikhail Moklyachuk
Keyword(s):  
Random Field ◽  
Spectral Characteristics ◽  
Mean Square ◽  
Spectral Densities ◽  
Optimal Linear ◽  
Extrapolation Problem ◽  
Time Argument ◽  
The Mean ◽  
Mean Square Errors ◽  
Isotropic Random

The problem of optimal linear estimation of functionals depending on the unknown values of a random fieldζ(t,x), which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the unit sphere Sn with respect to spatial argumentxєSn. Estimates are based on observations of the fieldζ(t,x) +Θ(t,x) at points (t,x) :t< 0;xєSn, whereΘ(t,x) is an uncorrelated withζ(t,x) random field, which is mean-square continuous periodically correlated with respect to time argumenttє R and isotropic on the sphereSnwith respect to spatial argumentxєSn. 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.

scholarly journals Interpolation Problem for Periodically Correlated Stochastic Sequences with Missing Observations

2020 ◽  
Vol 8 (2) ◽  
pp. 631-654
Author(s):  
Iryna Golichenko ◽  
Mikhail Moklyachuk
Keyword(s):  
Interpolation Problem ◽  
Spectral Characteristics ◽  
Optimal Estimation ◽  
Mean Square ◽  
Missing Observations ◽  
Linear Functionals ◽  
Spectral Densities ◽  
Optimal Estimates ◽  
The Mean ◽  
Mean Square Errors

The problem of mean square optimal estimation of linear functionals which depend on the unknown 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 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 spectral densities of the sequences are not exactly known while some classes of admissible spectral densities are specified.

scholarly journals Optimal linear filtering of general multidimensional Gaussian processes and its application to Laplace transforms of quadratic functionals

2001 ◽  
Vol 14 (3) ◽  
pp. 215-226 ◽  
Author(s):  
M. L. Kleptsyna ◽  
A. Le Breton
Keyword(s):  
Gaussian Processes ◽  
Laplace Transforms ◽  
Quadratic Functional ◽  
Linear Filtering ◽  
Volterra Type ◽  
Conditional Mean ◽  
Optimal Linear ◽  
Filtering Problem ◽  
Optimal Linear Filtering ◽  
Optimal Filtering Problem

The optimal filtering problem for multidimensional continuous possibly non-Markovian, Gaussian processes, observed through a linear channel driven by a Brownian motion, is revisited. Explicit Volterra type filtering equations involving the covariance function of the filtered process are derived both for the conditional mean and for the covariance of the filtering error. The solution of the filtering problem is applied to obtain a Cameron-Martin type formula for Laplace transforms of a quadratic functional of the process. Particular cases for which the results can be further elaborated are investigated.

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