# Minimax-robust Filtering Problem For Stochastic Sequences With Stationary Increments And Cointegrated Sequences

Maksym Luz, Mikhail P. Moklyachuk

Published 2014 · Mathematics

The problem of optimal estimation of the linear functionals $A {\xi}=\sum_{k=0}^{\infty}a (k)\xi(k)$ and $A_N{\xi}=\sum_{k=0}^{N}a (k)\xi(k)$ which depend on the unknown values of a stochastic sequence $\xi(m)$ with stationary $n$th increments is considered. Estimates are obtained which are based on observations of the sequence $\xi(m)+\eta(m)$ at points of time $m=-1,-2,\ldots$, where the sequence $\eta(m)$ is stationary and uncorrelated with the sequence $\xi(m)$. Formulas for calculating the mean-square errors and spectral characteristics of the optimal estimates of the functionals are derived in the case of spectral certainty, where spectral densities of the sequences $\xi(m)$ and $\eta(m)$ are exactly known. These results are applied for solving extrapolation problem for cointegrated sequences. In the case where spectral densities of the sequences are not known exactly, but sets of admissible spectral densities are given, the minimax-robust method of estimation is applied. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special classes of admissible densities.

This paper references

10.1007/BF02589429

A prediction problem in game theory

U. Grenander (1957)

10.1090/S0094-9000-2013-00888-6

Filtration of linear functionals of periodically correlated sequences

Iryna Dubovets’ka (2013)

PINSKER, The theory of curves with nth stationary incremens in Hilber spaces, Izv

M S. (1955)

10.7726/CMS.2014.1001

On Minimax Estimation Problems for Periodically Correlated Stochastic Processes

Iryna Dubovets’ka (2014)

10.1002/9781119205920.CH26

Time Series Analysis

D. Cox (2012)

ROBUST PROCEDURES IN TIME SERIES ANALYSIS

M. Moklyachuk (2000)

Correlation theory of stationary and related random processes with stationary n th increments

A. M. Y AGLOM (1987)

Über lineare Methoden in der Wahrscheinlichkeitsrechnung

K. Karhunen (1947)

10.1080/03610926.2011.581190

Minimax Prediction Problem for Multidimensional Stationary Stochastic Processes

Mikhail P. Moklyachuk (2011)

Time Series Analysis: Fore-casting and Control

G. Box (1994)

MASYUTKA, Minimax prediction problem for multidimensional stationary stochastic sequences, Theory Stoch

A. M. MOKLYACHUK (2008)

MOKLYACHUK, Interpolation of functionals of stochastic sequences with stationary increments for observations with noise, Prykl

M LUZM.P.M. (2012)

10.1090/S0094-9000-2014-00919-9

Extrapolation of periodically correlated stochastic processes observed with noise

Iryna Dubovets’ka (2014)

10.2307/1913236

Co-integration and Error Correction: Representation, Estimation, and Testing

R. Engle (1987)

Some clases of random fields in n-dimentional space related with random stationary processes, Teor

A. M. YAGLOM (2014)

10.2307/2333036

Stochastic processes

J. Doob (1953)

MOKLYACHUK, Nonsmooth analysis and optimization, Kyiv University, Kyiv

M P. (2008)

10.1007/978-1-4612-4620-6

Correlation Theory of Stationary and Related Random Functions I: Basic Results

A. Yaglom (1987)

ROBUST ESTIMATION PROBLEMS FOR STOCHASTIC PROCESSES 1

M. Moklyachuk (2006)

10.7551/mitpress/2946.001.0001

Extrapolation, Interpolation, and Smoothing of Stationary Time Series

N. Wiener (1964)

10.1515/156939706778239819

Extrapolation of multidimensional stationary processes

M. Moklyachuk (2006)

10.1515/9783111413426-016

Assessing L2 Writing in the Absence of Scoring Procedures: Construction of Rating Scales in a Cypriot-greek Efl in Class Context

Olivier Giraud (1824)

10.1090/S0094-9000-2014-00908-4

Interpolation of functionals of stochastic sequences with stationary increments

Maksym Luz (2014)

10.7726/CMS.2013.1009

Robust Extrapolation Problem for Stochastic Sequences with Stationary Increments

Mikhail P. Moklyachuk (2013)

MOKLYACHUK, Filtering of periodically correlated processes, Prykl

M.P.I.I. DUBOVETS’KA (2012)

Robust filtering of stochastic processes , Theory Stoch

A. M ASYUTKA (2006)

Co-Integration and Error Correction : Representation , Estimation , and Testing

D. Currie (2007)

10.1090/S0094-9000-2012-00862-4

Interpolation of periodically correlated stochastic sequences

Iryna Dubovets’ka (2012)

Extrapolation of periodically correlated processes from observations with noise

M. P. M OKLYACHUK UBOVETS’KA (2012)

10.1007/BF01066907

Minimax filtration of linear transformations of stationary sequences

Mikhail P. Moklyachuk (1991)

10.1109/PROC.1985.13167

Robust techniques for signal processing: A survey

S. Kassam (1985)

10.1214/aos/1176346512

Signal Extraction for Nonstationary Time Series

W. Bell (1984)

ROBUST FILTERING OF STOCHASTIC PROCESSES

M. Moklyachuk (2007)

10.19139/SOIC.V3I1.105

Minimax Interpolation Problem for Random Processes with Stationary Increments

Maksym Luz (2015)

MOKLYACHUK, Filtration of linear functionals of periodically correlated sequences, Theor

M.P.I.I. DUBOVETS’KA (2013)

10.1007/978-1-4615-7821-5_6

Minimax-Robust Filtering and Finite-Length Robust Predictors

J. Franke (1984)

Interpolation of functionals of stochactic sequanses with stationary increments

M. P. M OKLYACHUK (1992)

The theory of curves with n th stationary incremens in Hilber spaces , Izv

Y U. A. R OZANOV

Minimax prediction problem for multidimensional stationary stochastic sequences , Theory Stoch

A. M ASYUTKA (2007)

MOKLYACHUK, Interpolation of periodically correlated stochastic sequences, Theor

I. I. DUBOVETS’KA (2012)

Game theory and convex optimization methods in robust estimation problems , Theory Stoch

M. P. M OKLYACHUK (2000)

MASYUTKA, Extrapolation of multidimensional stationary processes, Random Operators and Stochastic Equations, 14(2006)

A. M. MOKLYACHUK (2006)

MOKLYACHUK, Game theory and convex optimization methods in robust estimation problems, Theory Stoch

M P. (2001)

YAGLOM, On linear extrapolaion of random processes with nth stationary incremens

A.M.M.S. PINSKER (1954)

10.1201/9781420039221.ch6

Stochastic Processes

S. Ross (1982)

10.2307/2343509

The Theory of Stochastic Processes

Z. Govindarajulu (1966)

MOKLYACHUK, Robust estimations of functionals of stochastic processes

M P. (2008)

10.2307/2981007

Extrapolation, Interpolation, and Smoothing of Stationary Time Series, with Engineering Applications

N. Wiener (1949)

10.1007/BF00532645

Minimax-robust prediction of discrete time series

J. Franke (1985)

10.2307/2005191

Necessary Conditions for an Extremum

B. N. Pshenichnyĭ (1971)

MOKLYACHUK, Minimax filtration of linear transformations of stationary sequences, Ukr

M P. (1991)

10.13189/MS.2014.020204

Robust Extrapolation Problem for Stochastic Processes with Stationary Increments

Maksym Luz (2014)

10.1007/978-3-642-61921-2

The Theory of Stochastic Processes II

I. I. Gikhman (1975)

10.1016/0005-1098(83)90105-X

An analysis of the effects of spectral uncertainty on wiener filtering

K. Vastola (1983)

MOKLYACHUK, Interpolation of functionals of stochactic sequanses with stationary increments

M LUZM.P.M. (2013)

10.1090/S0094-9000-2015-00940-6

Minimax-robust filtering problem for stochastic sequences with stationary increments

Maksym Luz (2015)

PSHENICHNYI, Necessary conditions of an extremum, “Nauka

B N. (1982)

10.7726/CMS.2015.1002

Filtering Problem for Random Processes with Stationary Increments

Maksym Luz (2015)

10.1007/978-3-662-46221-8_16

Probability Theory and Mathematical Statistics

I. Bronshteĭn (1987)

10.2307/1912100

Time Series Analysis, Forecasting And Control

G. P. Box (1970)

On linear extrapolaion of random processes with n th stationary incremens

A. M. Y AGLOM M. S. P INSKER (2012)

MOKLYACHUK, Robust procedures in time series analysis, Theory Stoch

M P. (2000)

Selected works by A

A. N. KOLMOGOROV (1992)

YAGLOM, Correlation theory of stationary and related random processes with stationary nth increments

A M. (1955)

MOKLYACHUK, Minimax-robust filtering problem for stochastic sequence with stationary increments

M LUZM.P.M. (2013)

10.4337/9780857931023.00013

Cointegration and error correction

J. Davidson (2013)

This paper is referenced by

10.1090/TPMS/1050

Estimates of functionals constructed from random sequences with periodically stationary increments

Petr Kozák (2019)

10.19139/SOIC.V3I3.149

Interpolation Problem for Stationary Sequences with Missing Observations

Mikhail P. Moklyachuk (2017)

MASYUTKA O.YU. , MOKLYACHUK M.P., SIDEI M.I. FILTERING OF MULTIDIMENSIONAL STATIONARY SEQUENCES WITH MISSING OBSERVATIONS 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.

(2020)

10.32323/UJMA.472929

Filtering of Multidimensional Stationary Sequences with Missing Observations

Oleksandr Masyutka (2018)

Minimax extrapolation of multidimensional stationary processes with missing observations

M. Moklyachuk (2018)

10.19139/soic-2310-5070-457

Minimax Estimation of Solutions of the First Order Linear Hyperbolic Systems with Uncertain Data

Olena A. Kapustian (2019)

10.19139/132

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

Maksym Luz (2015)

10.1080/23311835.2016.1167811

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

Maksym Luz (2016)

10.7726/JAMS.2015.1003

Minimax Extrapolation Problem For Harmonizable Stable Sequences With Noise Observations

Mikhail P. Moklyachuk (2015)

10.1007/s10559-019-00189-6

Approximate Guaranteed Mean Square Estimates of Functionals on Solutions of Parabolic Problems with Fast Oscillating Coefficients Under Nonlinear Observations

Oleksandr G. Nakonechnyi (2019)

10.15559/16-VMSTA51

Minimax interpolation of sequences with stationary increments and cointegrated sequences

Maksym Luz (2016)

10.19139/SOIC.V3I4.173

Minimax-robust estimation problems for stationary stochastic sequences

Mikhail P. Moklyachuk (2015)

10.7726/CMS.2015.1002

Filtering Problem for Random Processes with Stationary Increments

Maksym Luz (2015)

10.19139/SOIC.V4I1.172

Filtering Problem for Functionals of Stationary Sequences

Mikhail P. Moklyachuk (2016)

10.1080/23311835.2015.1133219

Minimax prediction of random processes with stationary increments from observations with stationary noise

Maksym Luz (2016)

10.1016/J.IFACOL.2018.09.568

Parity Space-Based Fault Detection by Minimum Error Minimax Probability Machine

Maiying Zhong (2018)

10.19139/soic-2310-5070-458

Interpolation problem for periodically correlated stochastic sequences with missing observations

Iryna Golichenko (2020)

10.19139/soic-2310-5070-551

Business Analytics using Dynamic Pricing based on Customer Entry-Exit Rates Tradeoff

Hamed Fazlollahtabar (2020)