Filtering problem for periodically correlated stochastic sequences with missing observations

Authors

  • I. I. Golichenko National Technical University of Ukraine ”Igor Sikorsky Kyiv Politechnic Institute”, Kyiv https://orcid.org/0000-0001-5639-8271
  • M. P. Moklyachuk Taras Shevchenko National University of Kyiv

DOI:

https://doi.org/10.17721/1812-5409.2023/2.4

Keywords:

periodically correlated stochastic sequence, minimax (robust) estimate, least favorable spectral density, minimax spectral characteristics

Abstract

The problem of the mean-square optimal estimation of the linear functionals which depend on the unknown values of a periodically correlated stochastic sequence from observations of the sequence with missings is considered. Formulas for calculation the mean-square error and the spectral characteristic of the optimal estimate of the functionals 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 spectral characteristics are proposed in the case of spectral uncertainty, when spectral densities of sequences are not exactly known but the class of admissible spectral densities is given.

Pages of the article in the issue: 30 - 43

Language of the article: English

References

BENNETT, W.R. (1958) Statistics of regenerative digital transmission. Bell System Technical Journal 37(6). p. 1501–1542.

DUBOVETS’KA, I.I. (2013) Filtration of linear functionals of periodically correlated sequences Theory Probab. Math. Stat. 86. p. 51–64.

GARDNER, W.A., FRANKS, L.E. (1975) Characterization of cyclostationary random signal processes IEEE Transactions on information theory IT-21(1) .p. 4–14.

GARDNER, W.A. (1994) Cyclostationarity in communications and signal processing New York: IEEE Press.

GARDNER, W.A., NAPOLITANO. A. and PAURA. L. (2006) Cyclostationarity: Half a century of research Signal Processing 86. p. 639–697.

GLADYSHEV, E.G. (1961) Periodically correlated random sequences Sov. Math. Dokl. 2. p. 385–388.

GOLICHENKO, I.I., MOKLYACHUK, M.P. (2020) Interpolation Problem for Periodically Correlated Stochastic Sequences with Missing Observations/ I.I. Golichenko, M.P. Moklyachuk// Statistics, Optimization & Information Computing. 8(2). p. 631–654.

GOLICHENKO, I. I., MASYUTKA, A. YU. and MOKLYACHUK, M. P. (2021) Extrapolation problem for periodically correlated stochastic sequences with missing observations. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics. 2. p. 39–52.

GOLICHENKO, I., MOKLYACHUK, M. (2023) Estimation problems for periodically correlated stochastic sequences with missed observations. In: M. Moklyachuk (ed.) Stochastic Processes: Fundamentals and Emerging Applications. Nova Science Publishers, New York. p. 111–162.

GRENANDER, U. (1957) A prediction problem in game theory. Ark. Mat. 3. p. 371–379.

HANNAN, E. J. (1970) Multiple time series. Wiley, New York.

HURD, H. L., MIAMEE, A. (2007) Periodically correlated random sequences. John Wiley & Sons, Inc., Publication.

IOFFE, A.D. and TIHOMIROV, V.M. (1979) Theory of extremal problems. North-Holland Publishing Company.

KASSAM, S. A. and POOR, H. V. (1985) Robust techniques for signal processing: A survey. Proc. IEEE. 73(3). p. 433–481.

KOLMOGOROV, A. N. (1992) In: Shiryayev A. N. (Ed.) Selected works by A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics Kluwer Academic Publishers.

LUZ, M., MOKLYACHUK, M. (2019) Estimation of Stochastic Processes with Stationary Increments and Cointegrated Sequences. London: ISTE; Hoboken, NJ: John Wiley & Sons.

MAKAGON, A. (1999) Theoretical prediction of periodically correlated sequences. Probab. Math. Statist. 19(2). p. 287–322.

MAKAGON, A., SALEHI, H. and SOLTANI, A.R. (2011) Stationary sequences associated with a periodically correlated sequence Probab. Math. Statist. 31(2). p. 263–283.

MASYUTKA, O. YU., MOKLYACHUK, M. P. and SIDEI, M. I. (2011) Interpolation problem for multidimensional stationary sequences with missing observations Stochastic Modeling and Applications. 22(2). p. 85–103.

MASYUTKA, O. YU., MOKLYACHUK, M. P. and SIDEI, M. I. (2019) Extrapolation problem for multidimensional stationary sequences with missing observations. Statistics, Optimization & Information Computing. 7(1). p. 97–117.

MASYUTKA, O. YU., MOKLYACHUK, M. P. and SIDEI, M. I. (2019) Filtering of multidimensional stationary sequences with missing observations. Carpathian Mathematical Publications. 11(2). p. 361–378.

MASYUTKA, O. YU., GOLICHENKO, I.I. and MOKLYACHUK, M. P. (2022) On estimation problem for continuous time stationary processes from observations in special sets of points. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics. 1. p. 20–33.

MOKLYACHUK, M. P. (2015) Minimax-robust estimation problems for stationary stochastic sequences. Statistics, Optimization & Information Computing. 3(4). p. 348–419.

MOKLYACHUK, M. P. and MASYUTKA, A. YU. (2012) Minimax-robust estimation technique for stationary stochastic processes. LAP LAMBERT Academic Publishing.

MOKLYACHUK, M. P., MASYUTKA, A. YU. and GOLICHENKO, I.I. (2018) Estimates of periodically correlated isotropic random fields. Nova Science Publishers Inc. New York.

MOKLYACHUK, M. P., SIDEI, M. I. and MASYUTKA, O. YU. (2019) Estimation of stochastic processes with missing observations. New York, NY: Nova Science Publishers.

MOKLYACHUK, M. P., GOLICHENKO, I.I. (2016) Periodically correlated processes estimates. LAP Lambert Academic Publishing.

NAPOLITANO A. (2016) Cyclostationarity: Limits and generalizations Signal processing. 120. p. 323–347.

NAPOLITANO A. (2016) Cyclostationarity: New trends and applications Signal processing. 120. p. 385–408.

PSHENICHNYI, B.N. (1971) Necessary conditions of an extremum. New York: Marcel Dekker.

ROCKAFELLAR, R. T. (1997) Convex Analysis. Princeton University Press.

ROZANOV, YU.A. (1967) Stationary stochastic processes. San Francisco-Cambridge-London-Amsterdam: Holden-Day.

VASTOLA, S.K., POOR H.V. (1983) An analysis of the effects of spectral uncertainty on Wiener filtering. Automatica. 19(3). p. 289–293.

WIENER, N. (1966) Extrapolation, interpolation and smoothing of stationary time series. With engineering applications. The M. I. T. Press, Massachusetts Institute of Technology, Cambridge.

YAGLOM, A. M. (1987) Correlation theory of stationary and related random functions. Vol. 1: Basic results; Vol. 2: Supplementary notes and references. Springer-Verlag, New York etc.

Downloads

Published

2023-12-23

How to Cite

Golichenko, I. I., & Moklyachuk, M. P. (2023). Filtering problem for periodically correlated stochastic sequences with missing observations. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (2), 30–43. https://doi.org/10.17721/1812-5409.2023/2.4

Issue

Section

Algebra, Geometry and Probability Theory

Most read articles by the same author(s)