Extrapolation problem for periodically correlated stochastic sequences with missing observations

Extrapolation problem for periodically correlated stochastic sequences

Authors

DOI:

https://doi.org/10.17721/1812-5409.2021/2.6

Keywords:

periodically correlated sequence, optimal linear estimate, mean square error, least favourable spectral density matrix, minimax spectral characteristic

Abstract

The problem of optimal estimation of the linear functionals $A{\zeta}=\sum_{j=1}^{\infty}{a}(j){\zeta}(j),$
which depend on the unknown values of a periodically correlated stochastic sequence ${\zeta}(j)$ from observations of the sequence ${\zeta}(j)+{\theta}(j)$ at points $j\in\{...,-n,...,-2,-1,0\}\setminus S$, $S=\bigcup _{l=1}^{s-1}\{-M_l\cdot T+1,\dots,-M_{l-1}\cdot T-N_{l}\cdot T\}$, is considered, where ${\theta}(j)$ is an uncorrelated with ${\zeta}(j)$ periodically correlated stochastic sequence. Formulas for calculation the mean square error and the spectral characteristic of the optimal estimate of the functional $A\zeta$ 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-robust spectral characteristics of the optimal estimates of functionals are proposed in the case of spectral uncertainty, where the spectral densities are not exactly known while some sets of admissible spectral densities are specified.

Pages of the article in the issue: 39 - 52

Language of the article: English

Author Biography

Mykhajlo Pavlovych Moklyachuk, Taras Shevchenko National University of Kyiv

Професор кафедри теорії ймовірностей, статистики і актуарної математики

References

W.R. Bennett, Statistics of regenerative digital transmission, Bell System Technical Journal, vol. 37, no. 6, pp. 1501--1542, 1958.

I.I. Dubovets'ka, O.Yu Masyutka and M.P. Moklyachuk, Interpolation of periodically correlated stochastic sequences, Theory of Probability and Mathematical Statistics, vol. 84, pp. 43--56, 2012.

I.I. Dubovets'ka and M.P. Moklyachuk, Filtration of linear functionals of periodically correlated sequences, Theory of Probability and Mathematical Statistics, vol. 86, pp. 51--64, 2013.

I.I. Dubovets'ka and M.P. Moklyachuk, Extrapolation of periodically correlated processes from observations with noise, Theory of Probability and Mathematical Statistics, vol. 88, pp. 67--83, 2014.

W.A. Gardner and L.E. Franks, Characterization of cyclostationary random signal processes, IEEE Transactions on information theory, vol. IT-21, no. 1, pp. 4--14, 1975.

W.A. Gardner, Cyclostationarity in communications and signal processing, New York: IEEE Press, 504 p., 1994.

W.A. Gardner, A.Napolitano and L.Paura, Cyclostationarity: Half a century of research, Signal Processing, vol. 86, pp. 639--697, 2006.

E. G. Gladyshev, Periodically correlated random sequences, Sov. Math. Dokl. vol, 2, pp. 385--388, 1961.

I. I. Golichenko and M.P. Moklyachuk, Interpolation problem for periodically correlated stochastic sequences with missing observations, Statistics, Optimization and Information Compututing, vol. 8, no. 2, pp. 631--654, 2020.

U. Grenander, A prediction problem in game theory, Arkiv f"or Matematik, vol. 3, pp. 371-379, 1957.

E.J. Hannan, Multiple time series. 2nd rev. ed., John Wiley & Sons, New York, 536 p., 2009.

H. Hurd and A. Miamee, Periodically correlated random sequences, John Wiley & Sons, New York, 353 p., 2007.

A.D, Ioffe and V.M. Tihomirov, Theory of extremal problems, Studies in Mathematics and its Applications, Vol. 6. Amsterdam, New York, Oxford: North-Holland Publishing Company. XII, 460 p., 1979.

S.A. Kassam and H.V. Poor, Robust techniques for signal processing: A survey, Proceedings of the IEEE, vol. 73, no. 3, pp. 433-481, 1985.

A.N. Kolmogorov, Selected works by A.N.~Kolmogorov. Vol. II: Probability theory and mathematical statistics. Ed. by A.N.~Shiryayev. Mathematics and Its Applications. Soviet Series. 26. Dordrecht etc. Kluwer Academic Publishers, 1992.

M.~Luz and M.~Moklyachuk, Estimation of stochastic processes with stationary increments and cointegrated sequences, London: ISTE; Hoboken, NJ: John Wiley & Sons, 282 p., 2019.

A. Makagon, Theoretical prediction of periodically correlated sequences, Probability and Mathematical Statistics, vol. 19, no. 2, pp. 287—322, 1999.

A. Makagon, A.G. Miamee, H. Salehi and A.R. Soltani, Stationary sequences associated with a periodically correlated sequence, Probability and Mathematical Statistics, vol. 31, no. 2, pp. 263--283, 2011.

O.Yu. Masyutka, M.P. Moklyachuk and M.I. Sidei Extrapolation problem for multidimensional stationary sequences with missing observations, Statistics, Optimization & Information Computing, vol. 7, no. 1, pp. 97--117, 2019.

O.Yu. Masyutka, M.P. Moklyachuk and M.I. Sidei Interpolation problem for multidimensional stationary processes with missing observations, Statistics, Optimization & Information Computing, vol. 7, no. 1, pp. 118--132, 2019.

O.Yu. Masyutka, M.P. Moklyachuk and M.I. Sidei Filtering of multidimensional stationary sequences with missing observations, Carpathian Mathematical Publications, vol. 11, no. 2, pp. 361—378, 2019.

M.P. Moklyachuk, Robust estimates for functionals of stochastic processes, Kyiv, 320~p., 2008.

M.P. Moklyachuk, Minimax-robust estimation problems for stationary stochastic sequences, Statistics, Optimization and Information Computing, vol. 3, no. 4, pp. 348--419, 2015.

M.P. Moklyachuk and I. I. Golichenko, Periodically correlated processes estimates, LAP Lambert Academic Publishing, 308 p., 2016.

M.P. Moklyachuk and A.Yu. Masyutka, Interpolation of multidimensional stationary sequences, Theory of Probability and Mathematical Statistics, vol. 73, pp. 125--133, 2006.

M.P. Moklyachuk and A.Yu. Masyutka, Minimax prediction problem for multidimensional stationary stochastic processes, Communications in Statistics-Theory and Methods, vol. 40, no. 19-20, pp. 3700--3710, 2011.

M.P. Moklyachuk and A.Yu. Masyutka, Minimax-robust estimation technique: For stationary stochastic processes, LAP Lambert Academic Publishing, 296 p., 2012.

M. Moklyachuk and M. Sidei, Extrapolation problem for stationary sequences with missing observations, Statistics, Optimization & Information Computing, vol. 5, no. 3, pp. 212--233, 2017.

M. Moklyachuk, M. Sidei and O. Masyutka, Estimation of stochastic processes with missing observations, Mathematics Research Developments. New York, NY: Nova Science Publishers, 336 p., 2019.

A. Napolitano, Cyclostationarity: Limits and generalizations, Signal Processing, vol. 120, pp. 323--347, 2016.

A. Napolitano, Cyclostationarity: New trends and applications, Signal Processing, vol. 120, pp. 385--408, 2016.

B.N. Pshenichnyi, Necessary conditions for an extremum, Pure and Applied mathematics. 4. New York: Marcel Dekker, 230 p., 1971.

R.T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press, 451 p., 1997.

Yu.A. Rozanov, Stationary stochastic processes, Holden-Day, San Francisco, 1967.

S.K. Vastola and H.V. Poor, An analysis of the effects of spectral uncertainty on Wiener filtering, Automatica, vol.~19, no.~3, pp. 289-293, 1983.

N. Wiener, Extrapolation, interpolation and smoothing of stationary tine series. With engineering applications, Cambridge, Mass.: The M. I. T. Press, 163 p., 1966.

A.M. Yaglom, Correlation theory of stationary and related random functions. Vol. 1: Basic results; Vol. 2: Suplementary notes and references, Springer Series in Statistics, Springer-Verlag, New York etc., 1987.

Downloads

Published

2021-11-04

How to Cite

Golichenko, I. I., Masyutka, O. Y., & Moklyachuk, M. P. (2021). Extrapolation problem for periodically correlated stochastic sequences with missing observations: Extrapolation problem for periodically correlated stochastic sequences. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (2), 39–52. https://doi.org/10.17721/1812-5409.2021/2.6

Issue

Section

Algebra, Geometry and Probability Theory