Interpolation Problem for Multidimensional Stationary Processes with Missing Observations
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 processis considered.Estimates are based on observations of the process with an additive stationary stochastic noise process at points which do not belong to some finite intervals of a real line. The problem is investigated in the case of spectral certainty, where the spectral densities of the processes are exactly known. Formulas for calculating the mean-square errors and spectral characteristics of the optimal linear estimates of functionals are proposed under the condition of spectral certainty. 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 of the processes are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible spectral 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.
G. E. P. Box, G. M. Jenkins, G. C. Reinsel, and G. M. Ljung, Time series analysis. Forecasting and control. 5rd ed., Wiley, 2016.
P. J. Brockwell, and R. A. Davis, Time series: Theory and methods. 2nd ed.. New York: Springer, 1998.
R. Cheng, A. G. Miamee, and M. Pourahmadi, Some extremal problems in Lp(w), Proceedings of the American Mathematical Society, vol. 126, pp. 2333–2340, 1998.
R. Cheng, and M. Pourahmadi, Prediction with incomplete past and interpolation of missing values, Statistics & Probability Letters, vol. 33, pp. 341–346, 1996.
J. Franke, On the robust prediction and interpolation of time series in the presence of correlated noise. Journal of Time Series Analysis, vol. 5, no. 4, pp. 227–244, 1984.
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.
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 för Matematik, vol. 3, pp. 371–379, 1957.
E. J. Hannan, Multiple time series, Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & 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, Über lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae. Ser. A I, vol.37, 1947.
Y. Kasahara, 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, 1985.17.
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. M. Luz, and M. P. Moklyachuk, Minimax interpolation problem for stochastic processes with stationary increments, Statistics, Optimization & Information Computing, vol. 3, no. 1, pp. 30–41, 2015.
M. M. Luz, and M. P. Moklyachuk, Minimax interpolation of sequences with stationary increments and cointegrated sequences，Modern Stochastics: Theory and Applications, vol. 3, no. 1. pp. 59–87, 2016.
M. M. Luz, and M. P. Moklyachuk, Estimates of functionals from processes with stationary increments and cointegrated sequences, Kyiv: NVP ”Interservis”, 2016.
M. M. Luz, and M. P. Moklyachuk, Minimax interpolation of stochastic processes with stationary increments from observations with noise, Theory of Probability and Mathematical Statistics, vol. 94, pp. 121–135, 2017.
M. P. Moklyachuk, Nonsmooth analysis and optimization, Kyiv University, Kyiv, 2008.
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. P. Moklyachuk, and I. I. Golichenko, Periodically correlated processes estimates, LAP Lambert Academic Publishing, 2016.
M. P. Moklyachuk, and O. Yu. Masyutka, Interpolation of multidimensional stationary sequences, Theory of Probability and Mathematical Statistics, vol. 73, pp. 121–135, 2006.
M. P. Moklyachuk, and O. Yu. Masyutka, Robust estimation problems for stochastic processes, Theory of Stochastic Processes, vol.12, no. 3-4, pp. 88–113, 2006.
M. P. Moklyachuk, and O. Yu. Masyutka, Minimax-robust estimation technique for stationary stochastic processes, LAP Lambert Academic Publishing, 2012.
M. P. Moklyachuk, and M. I. Sidei, Interpolation of stationary sequences observed with the noise, Theory of Probability and Mathematical Statistics, vol. 93, pp. 143–156, 2015.30.M. P. Moklyachuk, and M. I. Sidei,Interpolation problem for stationary sequences with missing observations,Statistics,Optimization & Information Computing, vol. 3, no. 3, pp. 259–275, 2015.
M. P. Moklyachuk, and M. I. Sidei, Filtering problem for stationary sequences with missing observations. Statistics, Optimization & Information Computing, vol. 4, no. 4, pp. 308–325 , 2016.
M. P. Moklyachuk, and M. I. Sidei,Extrapolation problem for stationary sequences with missing observations,Statistics,
Optimization & Information Computing, vol. 5, no. 3, pp. 212–233, 2017.
M. M. Pelagatti, Time series modelling with unobserved components, New York: CRC Press, 2015.
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.
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, SpringerSeries in Statistics, Springer-Verlag, New York etc., 1987.
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