Filtering of Multidimensional Stationary Processes with Missing Observations

Year 2019, Volume: 2 Issue: 1, 24 - 32, 20.03.2019

Oleksandr Masyutka Mikhail Moklyachuk Maria Sidei

https://doi.org/10.32323/ujma.472929

Cited By: 1

Abstract

The problem of the mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional continuous time stationary stochastic process from observations of the process with a stationary noise is considered. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of the functionals are derived under the condition of spectral certainty, where spectral densities of the signal and the noise processes are exactly known. The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the processes are not known exactly, while some sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics of the optimal estimates are derived for some special sets of admissible spectral densities.

Keywords

Minimax-Robust estimate, Least favourable spectral density

References

• [1] A. N. Kolmogorov, Selected works by A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics, Ed. by A. N. Shiryayev. Math. Appl. Soviet Series., 26. Kluwer, 1992.
• [2] 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.
• [3] 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.
• [4] 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.
• [5] Yu. A. Rozanov, Stationary Stochastic Processes, San Francisco-Cambridge-London-Amsterdam: Holden-Day, 1967.
• [6] E. J. Hannan, Multiple time series, Wiley, 1970.
• [7] G. E. P. Box, G. M. Jenkins, G. C. Reinsel, G. M. Ljung, Time series analysis. Forecasting and control, 5th ed., Wiley, 2016.
• [8] P. J. Brockwell, R. A. Davis, Time series: Theory and methods, 2nd ed., Springer, 1998.
• [9] K. S. Vastola, H. V. Poor, An analysis of the effects of spectral uncertainty on Wiener filtering, Automatica, 28 (1983), 289–293.
• [10] U. Grenander, A prediction problem in game theory, Arkiv f¨or Matematik, 3 (1957), 371–379.
• [11] S. A. Kassam, H.V. Poor, Robust techniques for signal processing: A survey, Proceedings of the IEEE, 73(3) (1985), 433–481.
• [12] J. Franke, On the robust prediction and interpolation of time series in the presence of correlated noise, J. Time Ser. Analysis, 5(4) (1984), 227–244.
• [13] J. Franke, Minimax robust prediction of discrete time series, Z. Wahrscheinlichkeitstheor. Verw. Gebiete, 68 (1985), 337–364.
• [14] J. Franke, H.V. Poor, Minimax-robust filtering and finite-length robust predictors, Robust and Nonlinear Time Series Analysis. Lecture Notes in Statists., Springer-Verlag, 26 (1984), 87–126.
• [15] M. P. Moklyachuk, Robust Estimations of Functionals of Stochastic Processes, Kyiv University, Kyiv, 2008.
• [16] M. P. Moklyachuk, Minimax-robust estimation problems for stationary stochastic sequences, Statist. Optim. Inform. Comput., 3(4) (2015), 348–419.
• [17] M. P. Moklyachuk, O.Yu. Masyutka, Robust filtering of stochastic processes, Theory of Stochastic Processes, 13(1-2) (2007), 166–181.
• [18] M. P. Moklyachuk, O.Yu. Masyutka, Minimax-Robust Estimation Technique for Stationary Stochastic Processes, LAP Lambert Academic Publishing, 2012.
• [19] M. P. Moklyachuk, I. I. Golichenko, Periodically Correlated Processes Estimates, LAP Lambert Academic Publishing, 2016.
• [20] M. M. Luz, M. P. Moklyachuk, Filtering problem for functionals of stationary sequences, Statist. Optim. Inform. Comput., 4(1) (2016), 68–83.
• [21] P. Bondon, Prediction with incomplete past of a stationary process, Stochastic Process and their Applications. 98 (2002), 67–76.
• [22] P. Bondon, Influence of missing values on the prediction of a stationary time series, Journal of Time Series Analysis, 26(4) (2005), 519–525.
• [23] R. Cheng, A. G. Miamee, M. Pourahmadi, Some extremal problems in Lp(w), Proc. Amer. Math. Soc., 126 (1998), 2333–2340.
• [24] R. Cheng, M. Pourahmadi, Prediction with incomplete past and interpolation of missing values, Statist. Probab. Lett., 33 (1996), 341–346.
• [25] Y. Kasahara, M. Pourahmadi, A. Inoue, Duals of random vectors and processes with applications to prediction problems with missing values, Statist. Probab. Lett., 79(14) (2009), 1637–1646.
• [26] M. Pourahmadi, A. Inoue, Y. Kasahara, A prediction problem in L2(w), Proc. Amer. Math. Soc., 135(4) (2007), 1233–1239.
• [27] M. M. Pelagatti, Time Series Modelling with Unobserved Components, New York: CRC Press, 2015.
• [28] M. P. Moklyachuk, M. I. Sidei, Interpolation problem for stationary sequences with missing observations, Statist. Optim. Inform. Comput., 3(3) (2015), 259-275.
• [29] M. P. Moklyachuk, M. I. Sidei, Filtering problem for stationary sequences with missing observations. Statist. Optim. Inform. Comput., 4(4) (2016), 308–325.
• [30] M. P. Moklyachuk, M. I. Sidei, Filtering Problem for functionals of stationary processes with missing observations, Commun Optim. Theory, (2016), 1-18, Article ID 21 .
• [31] M. P. Moklyachuk, M. I. Sidei, Extrapolation problem for stationary sequences with missing observations, Statist. Optim. Inform. Comput., 5(3) (2017), 212–233.
• [32] I. I. Gikhman, A.V. Skorokhod, The Theory of Stochastic Processes. I., Springer, 2004.
• [33] H. Salehi, Algorithms for linear interpolator and interpolation error for minimal stationary stochastic processes, Ann. Probab., 7(5) (1979), 840–846.
• [34] B. N. Pshenichnyj, Necessary Conditions of an Extremum, Pure Appl. Math., 4. New York: Marcel Dekker, 1971.
• [35] A. D. Ioffe, V. M. Tikhomirov, Theory of extremal problems, Studies in Mathematics and its Applications, Vol. 6. North-Holland Publishing Company. Amsterdam, New York, Oxford, 1979.
• [36] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1997.
• [37] M. P. Moklyachuk, Nonsmooth Analysis and Optimization, Kyiv University, Kyiv, 2008.
• [38] M. M. Luz, M. P. Moklyachuk, Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences, Statist. Optim. Inform. Comput., 2(3) (2014), 176–199.