Filtering Problem for Stationary Sequences with Missing Observations
Abstract
We deal with the problem of the mean-square optimal linear estimation of linear functionals which depend on the unknown values of a stationary stochastic sequence from observations of the sequence with a stationary noise sequence. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of functionals are derived 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 of spectral uncertainty, where spectral densities are not known exactly while sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some special sets of admissible densities.References
P. Bondon, Influence of missing values on the prediction of a stationary time series, Journal of Time Series Analysis, vol. 26, no. 4,pp. 519–525, 2005.
P. Bondon, Prediction with incomplete past of a stationary process, Stochastic Process and their Applications. vol.98, pp. 67–76, 2002.
R. Cheng, A.G. Miamee, M. Pourahmadi, Some extremal problems in Lp(w), Proc. Am. Math. Soc. vol.126, pp. 2333–2340, 1998.
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. 43–55, 2012.
J. Franke, Minimax robust prediction of discrete time series, Z. Wahrscheinlichkeitstheor. Verw. Gebiete, vol. 68, pp. 337–364,1985.
J. Franke and H. V. Poor, Minimax-robust filtering and finite-length robust predictors, Robust and Nonlinear Time Series Analysis. Lecture Notes in Statistics, Springer-Verlag, vol. 26, pp. 87–126, 1984.
F. R. Gantmacher, Applications of the theory of matrices., Interscience publishers, inc., New York., 1959.
I. I. Gikhman and A. V. Skorokhod, The theory of stochastic processes. I., Berlin: Springer, 2004.
U. Grenander, A prediction problem in game theory, Arkiv f¨or Matematik, vol. 3, pp. 371–379, 1957.
E. J. Hannan, Multiple time series, Wiley Series in Probability and Mathematical Statistics. New York etc.: JohnWiley & Sons, Inc.XI, 1970.
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, 1979.
K. Karhunen, U¨ ber lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae. Ser. A I, vol.37, 1947.
Y. Kasahara, and M. Pourahmadi, and A. Inoue, Duals of random vectors and processes with applications to prediction problems with missing values, Statistics & Probability Letters, vol. 79, no. 14, pp. 1637–1646, 2009.
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, Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences, Statistics, Optimization & Information Computing, vol. 2, no. 3, pp. 176–199, 2014.
M. M. Luz, and M. P. Moklyachuk, Filtering problem for functionals of stationary sequences, Statistics, Optimization & Information Computing, vol. 4, no. 1 pp. 68–83, 2016.
M. M. Luz, and M. P. Moklyachuk, Estimates of functionals from processes with stationary increments and cointegrated sequences, NVP ”Interservice”, Kyiv, 2016.
M. P. Moklyachuk, Robust procedures in time series analysis, Theory of Stochastic Processes, vol. 6, no. 3-4, pp. 127–147, 2000.
M. P. Moklyachuk, Game theory and convex optimization methods in robust estimation problems, Theory of Stochastic Processes, vol. 7, no. 1-2, pp. 253–264, 2001.
M. P. Moklyachuk, Robust estimations of functionals of stochastic processes, Kyiv University, Kyiv, 2008.
M. P. Moklyachuk, Minimax-robust estimation problems for stationary stochastic sequences, Statistics, Optimization & Information Computing, vol. 3, no. 4, pp. 348–419, 2015.
M. Moklyachuk and I. Golichenko, Periodically correlated processes estimates. LAP Lambert Academic Publishing, 2016.
M. Moklyachuk and O. Masyutka, Robust filtering of stochastic processes, Theory Stoch. Process., vol. 13, no. 1-2, pp. 166–181, 2007.
M. Moklyachuk and O. 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. Moklyachuk and O. Masyutka, Minimax-robust estimation technique for stationary stochastic processes, LAP Lambert Academic Publishing, 2012.
M. Moklyachuk and V. Ostapenko, Minimax interpolation problem for harmonizable stable sequences with noise observations, Journal of Allpied Mathematics and Statistics, vol. 2, no. 1, pp. 21–42, 2015.
M. Moklyachuk and V. Ostapenko, Minimax interpolation of harmonizable sequences, Theory of Probability and Mathematical Statistics, vol. 92, pp. 135146, 2016.
M. Moklyachuk and M. Sidei, Interpolation of stationary sequences observed with the noise, Theory of Probability and Mathematical Statistics, vol. 93, pp. 143-156, 2015.
M. Moklyachuk and M. Sidei, Interpolation problem for stationary sequences with missing observations, Statistics, Optimization & Information Computing, vol. 3, no. 3 pp. 259-275, 2015.
T. Nakazi Two problems in prediction theory. Studia Math., vol. 78, pp. 7–14, 1984.
M. Pourahmadi, A. Inoue and Y. Kasahara A prediction problem in L2(w). Proceedings of the American Mathematical Society, vol. 135, No. 4, pp. 1233–1239, 2007.
B. N. Pshenichnyj, Necessary conditions of an extremum, Pure and Applied mathematics. 4. New York: Marcel Dekker, 1971.
R. T. Rockafellar, Convex Analysis, Princeton University Press, 1997.
Yu. A. Rozanov, Stationary stochastic processes, San Francisco-Cambridge-London-Amsterdam: Holden-Day, 1967.
H. Salehi, Algorithms for linear interpolator and interpolation error for minimal stationary stochastic processes, The Annals of Probability, Vol. 7, No. 5, pp. 840–846, 1979.
K. S. Vastola and H. V. Poor, An analysis of the effects of spectral uncertainty on Wiener filtering, Automatica, vol. 28, pp. 289–293,1983.
N.Wiener, Extrapolation, interpolation and smoothing of stationary time series. With engineering applications, The M. I. T. Press, Massachusetts Institute of Technology, Cambridge, Mass., 1966.
H. Wold, A study in the analysis of stationary time series, Thesis University of Stockholm, 1938.
H. Wold, On prediction instationary time series, Ann. Math. Stat., vol. 19, no. 4, pp. 558–567, 1948.
A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. 1: Basic results, Springer Series in Statistics, Springer-Verlag, New York etc., 1987.
A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. 2: Supplementary notes and references, Springer Series in Statistics, Springer-Verlag, New York etc., 1987.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
