L1-Optimal Filtering of Markov Jump Processes. I. Exact Solution and Numerical Implementation Schemes
Part I of this research work is devoted to the development of a class of numerical solution algorithms for the filtering problem of Markov jump processes by indirect continuous-time observations corrupted by Wiener noises. The expected L 1 norm of the estimation error is chosen as an optimality crit...
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| Published in | Automation and remote control Vol. 81; no. 11; pp. 1945 - 1962 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Moscow
Pleiades Publishing
01.11.2020
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0005-1179 1608-3032 |
| DOI | 10.1134/S0005117920110016 |
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| Abstract | Part I of this research work is devoted to the development of a class of numerical solution algorithms for the filtering problem of Markov jump processes by indirect continuous-time observations corrupted by Wiener noises. The expected
L
1
norm of the estimation error is chosen as an optimality criterion. The noise intensity depends on the state being estimated. The numerical solution algorithms involve not the original continuous-time observations, but the ones discretized by time. A feature of the proposed algorithms is that they take into account the probability of several jumps in the estimated state on the time interval of discretization. The main results are the statements on the accuracy of the approximate solution of the filtering problem, depending on the number of jumps taken into account for the estimated state, on the discretization step, and on the numerical integration scheme applied. These statements provide a theoretical basis for the subsequent analysis of particular numerical schemes to implement the solution of the filtering problem. |
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| AbstractList | Part I of this research work is devoted to the development of a class of numerical solution algorithms for the filtering problem of Markov jump processes by indirect continuous-time observations corrupted by Wiener noises. The expected
L
1
norm of the estimation error is chosen as an optimality criterion. The noise intensity depends on the state being estimated. The numerical solution algorithms involve not the original continuous-time observations, but the ones discretized by time. A feature of the proposed algorithms is that they take into account the probability of several jumps in the estimated state on the time interval of discretization. The main results are the statements on the accuracy of the approximate solution of the filtering problem, depending on the number of jumps taken into account for the estimated state, on the discretization step, and on the numerical integration scheme applied. These statements provide a theoretical basis for the subsequent analysis of particular numerical schemes to implement the solution of the filtering problem. |
| Author | Borisov, A. V. |
| Author_xml | – sequence: 1 givenname: A. V. surname: Borisov fullname: Borisov, A. V. organization: Institute of Informatics Problems, Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, Moscow Aviation Institute (National Research University), Moscow Center for Fundamental and Applied Mathematics, Moscow State University |
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| Keywords | local and global accuracy of approximation stable numerical solution algorithm Markov jump process |
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| References | ItoKRozovskiiBApproximation of the Kushner Equation for Nonlinear FilteringSIAM J.Control Optim.200038no.3893915175690010.1137/S0363012998344270 BäuerleNGilitschenskiIHanebeckUExact and Approximate Hidden Markov Chain Filters Based on Discrete ObservationsStatist. Risk Modeling201632no.3-415917635079781339.60040 ElliottRJAggounLMooreJBHidden Markov Models: Estimation and Control2008New YorkSpringer0819.60045 YinGZhangQLiuYDiscrete-time Approximation of Wonham FiltersJ.Control Theory Appl.2004no.211020938071260.60148 RabinerLRA Tutorial on Hidden Markov Models and Selected Applications in Speech RecognitionProc.IEEE19897725728610.1109/5.18626 CvitanićJLiptserRRozovskiiBA Filtering Approach to Tracking Volatility from Prices Observed at Random TimesAnnals Appl. Probab.200616no.316331652226007610.1214/105051606000000222 KloedenPPlatenENumerical Solution of Stochastic Differential Equations1992BerlinSpringer10.1007/978-3-662-12616-5 JamesMKrishnamurthyVLeGlandFTime Discretization of Continuous-Time Filters and Smoothers for HMM Parameter EstimationIEEE Trans. Autom. Control199642no.25936050852.62077 CappéOMoulinesERydenTInference in Hidden Markov Models2005BerlinSpringer10.1007/0-387-28982-8 BertsekasDPShreveSEStochastic Optimal Control: The Discrete-Time Case1978New YorkAcademic0471.93002Translated under the title Stokhasticheskoe optimal’noe upravlenie. Sluchai diskretnogo vremeni, Moscow: Fizmatlit, 1985. Wonham, W.Some Applications of Stochastic Differential Equations to Optimal Nonlinear Filtering, SIAM J. Control Optim., 1964, pp.347–369. AndersonBMooreJOptimal Filtering1979New JerseyPrentice Hill0688.93058 BorisovAVFiltering of Markov Jump Processes by Discretized ObservationsInform. Primenen.201812no.31151213869236 KushnerHJProbability Methods for Approximations in Stochastic Control and for Elliptic Equations1977New YorkAcademic0547.93076Translated under the title Veroyatnostnye metody approksimatsii v stokhasticheskikh zadachakh upravleniya i teorii ellipticheskikh uravnenii, Moscow: Fizmatlit, 1985. PlatenEBruti-LiberatiNNumerical Solution of Stochastic Differential Equations with Jumps in Finance2010BerlinSpringer10.1007/978-3-642-13694-8 KahanerDMolerCNashSNumerical Methods and Software1989Prentice HillNew Jersey0744.65002 DraganVMorozanTStoicaAMathematical Methods in Robust Control of Discrete-Time Linear StochasticSystems2010New YorkSpringer10.1007/978-1-4419-0630-4 DraganVAberkaneSH2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{H}}}_{2}$$\end{document}-optimal Filtering for Continuous-time Periodic Linear Stochastic Systems with State-dependent NoiseSyst. Control Lett.2014no.66354210.1016/j.sysconle.2013.12.020 IsaacsonEKellerHAnalysis of Numerical Methods1994New YorkDover0168.13101 BorisovAVWonham Filtering by Observations with Multiplicative NoisesAutom. Remote Control201879no.13950376904410.1134/S0005117918010046 HuberPRobust Statistics1981New YorkWiley10.1002/0471725250Translated under the title Robastnost’ v statistike, Moscow: Mir, 1984. BorisovAVApplication of Optimal Filtering Methods for On-Line Estimation of Queueing Network StatesAutom. Remote Control201677no.2277296365029610.1134/S0005117916020053 StoerJBulirschRIntroduction to Numerical Analysis1993New YorkSpringer10.1007/978-1-4757-2272-7 CrisanDKouritzinMXiongJNonlinear Filtering with Signal Dependent Observation NoiseElectron. J. Probab.2009no.1418631883254085110.1214/EJP.v14-687 KushnerHDupuisPNumerical Methods for Stochastic Control Problems in Continuous Time2001New YorkSpringer10.1007/978-1-4613-0007-6 TakeuchiYAkashiHLeast-squares State Estimation of Systems with State-dependent Observation NoiseAutomatica198521no.330331379306510.1016/0005-1098(85)90063-9 Malcolm, V., Elliott, R., and van der Hoek, J.On the Numerical Stability of Time-discretized State Estimation via Clark Transformations, Proc. 42nd IEEE Conf. Decis. Control, 2003, Maui, pp.1406–1412. EphraimYMerhavNHidden Markov ProcessesIEEE Trans. Inform. Theory200248no.615181569190947210.1109/TIT.2002.1003838 Clark, J.The Design of Robust Approximations to the Stochastic Differential Equations of Nonlinear Filtering, Communication Systems and Random Process Theory, Skwirzynski, J.K., Ed., Amsterdam: Sijthoff and Noordhoff, 1978. Borovkov, A.A.Asimptoticheskie metody v teorii massovogo obsluzhivaniya, Moscow: Fizmatlit, 1995, 2nd ed. Translated under the title Asymptotic Methods in Queuing Theory (Probability & Mathematical Statistics), New York: Wiley, 1984, 1st ed. RLiptserSh.ShiryaevANStatistika sluchainykh protsessov19741st edMoscowNaukaTranslated under the title Statistics of Random Processes I. General Theory, New York: Springer-Verlag, 1977. |
| References_xml | – reference: ElliottRJAggounLMooreJBHidden Markov Models: Estimation and Control2008New YorkSpringer0819.60045 – reference: KushnerHJProbability Methods for Approximations in Stochastic Control and for Elliptic Equations1977New YorkAcademic0547.93076Translated under the title Veroyatnostnye metody approksimatsii v stokhasticheskikh zadachakh upravleniya i teorii ellipticheskikh uravnenii, Moscow: Fizmatlit, 1985. – reference: StoerJBulirschRIntroduction to Numerical Analysis1993New YorkSpringer10.1007/978-1-4757-2272-7 – reference: BorisovAVApplication of Optimal Filtering Methods for On-Line Estimation of Queueing Network StatesAutom. Remote Control201677no.2277296365029610.1134/S0005117916020053 – reference: BorisovAVFiltering of Markov Jump Processes by Discretized ObservationsInform. Primenen.201812no.31151213869236 – reference: KahanerDMolerCNashSNumerical Methods and Software1989Prentice HillNew Jersey0744.65002 – reference: ItoKRozovskiiBApproximation of the Kushner Equation for Nonlinear FilteringSIAM J.Control Optim.200038no.3893915175690010.1137/S0363012998344270 – reference: RLiptserSh.ShiryaevANStatistika sluchainykh protsessov19741st edMoscowNaukaTranslated under the title Statistics of Random Processes I. General Theory, New York: Springer-Verlag, 1977. – reference: Malcolm, V., Elliott, R., and van der Hoek, J.On the Numerical Stability of Time-discretized State Estimation via Clark Transformations, Proc. 42nd IEEE Conf. Decis. Control, 2003, Maui, pp.1406–1412. – reference: Borovkov, A.A.Asimptoticheskie metody v teorii massovogo obsluzhivaniya, Moscow: Fizmatlit, 1995, 2nd ed. Translated under the title Asymptotic Methods in Queuing Theory (Probability & Mathematical Statistics), New York: Wiley, 1984, 1st ed. – reference: KloedenPPlatenENumerical Solution of Stochastic Differential Equations1992BerlinSpringer10.1007/978-3-662-12616-5 – reference: JamesMKrishnamurthyVLeGlandFTime Discretization of Continuous-Time Filters and Smoothers for HMM Parameter EstimationIEEE Trans. Autom. Control199642no.25936050852.62077 – reference: YinGZhangQLiuYDiscrete-time Approximation of Wonham FiltersJ.Control Theory Appl.2004no.211020938071260.60148 – reference: CvitanićJLiptserRRozovskiiBA Filtering Approach to Tracking Volatility from Prices Observed at Random TimesAnnals Appl. Probab.200616no.316331652226007610.1214/105051606000000222 – reference: AndersonBMooreJOptimal Filtering1979New JerseyPrentice Hill0688.93058 – reference: Wonham, W.Some Applications of Stochastic Differential Equations to Optimal Nonlinear Filtering, SIAM J. Control Optim., 1964, pp.347–369. – reference: IsaacsonEKellerHAnalysis of Numerical Methods1994New YorkDover0168.13101 – reference: EphraimYMerhavNHidden Markov ProcessesIEEE Trans. Inform. Theory200248no.615181569190947210.1109/TIT.2002.1003838 – reference: BorisovAVWonham Filtering by Observations with Multiplicative NoisesAutom. Remote Control201879no.13950376904410.1134/S0005117918010046 – reference: HuberPRobust Statistics1981New YorkWiley10.1002/0471725250Translated under the title Robastnost’ v statistike, Moscow: Mir, 1984. – reference: KushnerHDupuisPNumerical Methods for Stochastic Control Problems in Continuous Time2001New YorkSpringer10.1007/978-1-4613-0007-6 – reference: CrisanDKouritzinMXiongJNonlinear Filtering with Signal Dependent Observation NoiseElectron. J. Probab.2009no.1418631883254085110.1214/EJP.v14-687 – reference: DraganVAberkaneSH2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{H}}}_{2}$$\end{document}-optimal Filtering for Continuous-time Periodic Linear Stochastic Systems with State-dependent NoiseSyst. Control Lett.2014no.66354210.1016/j.sysconle.2013.12.020 – reference: PlatenEBruti-LiberatiNNumerical Solution of Stochastic Differential Equations with Jumps in Finance2010BerlinSpringer10.1007/978-3-642-13694-8 – reference: RabinerLRA Tutorial on Hidden Markov Models and Selected Applications in Speech RecognitionProc.IEEE19897725728610.1109/5.18626 – reference: BertsekasDPShreveSEStochastic Optimal Control: The Discrete-Time Case1978New YorkAcademic0471.93002Translated under the title Stokhasticheskoe optimal’noe upravlenie. Sluchai diskretnogo vremeni, Moscow: Fizmatlit, 1985. – reference: CappéOMoulinesERydenTInference in Hidden Markov Models2005BerlinSpringer10.1007/0-387-28982-8 – reference: TakeuchiYAkashiHLeast-squares State Estimation of Systems with State-dependent Observation NoiseAutomatica198521no.330331379306510.1016/0005-1098(85)90063-9 – reference: BäuerleNGilitschenskiIHanebeckUExact and Approximate Hidden Markov Chain Filters Based on Discrete ObservationsStatist. Risk Modeling201632no.3-415917635079781339.60040 – reference: Clark, J.The Design of Robust Approximations to the Stochastic Differential Equations of Nonlinear Filtering, Communication Systems and Random Process Theory, Skwirzynski, J.K., Ed., Amsterdam: Sijthoff and Noordhoff, 1978. – reference: DraganVMorozanTStoicaAMathematical Methods in Robust Control of Discrete-Time Linear StochasticSystems2010New YorkSpringer10.1007/978-1-4419-0630-4 |
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| Title | L1-Optimal Filtering of Markov Jump Processes. I. Exact Solution and Numerical Implementation Schemes |
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