×

Filtering of multidimensional stationary sequences with missing observations. (English) Zbl 1458.60039

Summary: The problem of mean-square optimal linear estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence is considered. Estimates are based on observations of the sequence with an additive stationary stochastic noise sequence at points which do not belong to some finite intervals of a real line. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of the functionals are proposed 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 where spectral densities 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 are proposed for some special sets of admissible densities.

MSC:

60G10 Stationary stochastic processes
60G25 Prediction theory (aspects of stochastic processes)
60G35 Signal detection and filtering (aspects of stochastic processes)
62M20 Inference from stochastic processes and prediction
93E11 Filtering in stochastic control theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bondon P. Prediction with incomplete past of a stationary process. Stochastic Process. Appl. 2002, 98 , 67-76. doi: 10.1016/S0304-4149(01)00116-8 · Zbl 1058.62075 · doi:10.1016/S0304-4149(01)00116-8
[2] Bondon P. Influence of missing values on the prediction of a stationary time series. J. Time Ser. Anal. 2005, 26 (4), 519-525. doi: 10.1111/j.1467-9892.2005.00433.x · Zbl 1091.62094 · doi:10.1111/j.1467-9892.2005.00433.x
[3] Box G.E.P., Jenkins G.M., Reinsel G.C., Ljung G.M. Time series analysis. Forecasting and control. 5th ed. Wiley, 2016. · Zbl 1317.62001
[4] Brockwell P.J., Davis R.A. Time series: Theory and methods. 2nd ed. Springer, New York, 1998. · Zbl 0709.62080
[5] Cheng R., Miamee A.G., Pourahmadi M. Some extremal problems in \(L^p(w)\). Proc. Amer. Math. Soc. 1998, 126 , 2333-2340. doi: 10.1090/S0002-9939-98-04275-0 · Zbl 0903.60033 · doi:10.1090/S0002-9939-98-04275-0
[6] Cheng R., Pourahmadi M. Prediction with incomplete past and interpolation of missing values. Stat. Probab. Lett. 1996, 33 , 341-346. doi: 10.1016/S0167-7152(96)00146-0 · Zbl 0899.62121 · doi:10.1016/S0167-7152(96)00146-0
[7] Franke J. On the robust prediction and interpolation of time series in the presence of correlated noise. J. Time Ser. Anal. 1984, 5 (4), 227-244. doi: 10.1111/j.1467-9892.1984.tb00389.x · Zbl 0576.62090 · doi:10.1111/j.1467-9892.1984.tb00389.x
[8] Franke J. Minimax robust prediction of discrete time series. Z. Wahrscheinlichkeitstheor. Verw. Geb. 1985, 68 , 337-364. doi: 10.1007/BF00532645 · Zbl 0537.60034 · doi:10.1007/BF00532645
[9] Franke J., Poor H.V. Minimax-robust filtering and finite-length robust predictors. Robust and Nonlinear Time Series Analysis. Lecture Notes in Statistics, Springer-Verlag, 1984, 26 , 87-126. · Zbl 0569.62083
[10] Gikhman I.I., Skorokhod A.V. The theory of stochastic processes. I. Springer, Berlin, 2004. · Zbl 1068.60004
[11] Grenander U. A prediction problem in game theory. Ark. Mat. 1957, 26 , 371-379. · Zbl 0082.13302
[12] Hannan E.J. Multiple time series. Wiley, New York, 1970. doi: 10.1002/9780470316429 · Zbl 0211.49804 · doi:10.1002/9780470316429
[13] Ioffe A.D., Tihomirov V.M. Theory of extremal problems. North-Holland, Amsterdam, New York, Oxford, 1979. · Zbl 0407.90051
[14] Karhunen K. \"Uber lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fenn., Ser. A I, 1947, 37 , 1-79. · Zbl 0030.16502
[15] Kasahara Y., Pourahmadi M., Inoue A. Duals of random vectors and processes with applications to prediction problems with missing values. Stat. Probab. Lett. 2009, 79 (14), 1637-1646. doi: 10.1016/j.spl.2009.04.005 · Zbl 1456.62226 · doi:10.1016/j.spl.2009.04.005
[16] Kassam S.A., Poor H.V. Robust techniques for signal processing: A survey. Proc. IEEE 1985, 73 (3), 433-481. doi: 10.1109/PROC.1985.13167 · Zbl 0569.62084 · doi:10.1109/PROC.1985.13167
[17] Kolmogorov A.N. In: Shiryayev A. N. (Ed.) Selected works by A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics, Kluwer, Dordrecht etc., 1992. · Zbl 0743.60005
[18] Luz M. M., Moklyachuk M.P. Minimax-robust filtering problem for stochastic sequences with stationary increments. Theory Probab. Math. Statist. 2014, 89 , 127-142. doi: 10.1090/S0094-9000-2015-00940-6 (translation of Teor. Imovir. ta Matem. Statist. 2013, 89 , 115-129. (in English)) · Zbl 1332.60059 · doi:10.1090/S0094-9000-2015-00940-6
[19] Luz M. M., Moklyachuk M. P. Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences. Stat., Optim. Inf. Comput. 2014, 2 (3), 176-199. doi: 10.19139/soic.v2i3.56 · Zbl 1426.60046 · doi:10.19139/soic.v2i3.56
[20] Luz M. M., Moklyachuk M. P. Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences. Cogent Mathematics 2016, 3 , 1-21. doi: 10.1080/23311835.2016.1167811 · Zbl 1426.60046 · doi:10.1080/23311835.2016.1167811
[21] Luz M.M., Moklyachuk M.P. Filtering problem for functionals of stationary sequences. Stat., Optim. Inf. Comput. 2016, 4 (1), 68-83. doi: 10.19139/soic.v4i1.172 · Zbl 1474.60097 · doi:10.19139/soic.v4i1.172
[22] Luz M. M., Moklyachuk M. P. Estimates of functionals from processes with stationary increments and cointegrated sequences. NVP “Interservis”, Kyiv, 2016. (in Ukrainian)
[23] Moklyachuk M.P. On a filtering problem for vector-valued sequences. Theory Probab. Math. Statist. 1992, 47 , 107-118. (translation of Teor. Imovir. ta Matem. Statist. 1992, 47 , 104-117. (in Ukrainian)) · Zbl 0835.60038
[24] Moklyachuk M.P. Nonsmooth analysis and optimization. Kyiv University, Kyiv, 2008. (in Ukrainian) · Zbl 1224.49001
[25] Moklyachuk M.P. Robust estimations of functionals of stochastic processes. Kyiv University, Kyiv, 2008. (in Ukrainian) · Zbl 1249.62007
[26] Moklyachuk M.P. Minimax-robust estimation problems for stationary stochastic sequences. Stat., Optim. Inf. Comput. 2015, 3 (4), 348-419. doi: 10.19139/soic.v3i4.173 · doi:10.19139/soic.v3i4.173
[27] Moklyachuk M.P., Golichenko I.I. Periodically correlated processes estimates. LAP Lambert Academic Publishing, Saarbr\"ucken, 2016. · Zbl 1374.62001
[28] Moklyachuk M.P., Masyutka O.Yu. Robust filtering of stochastic processes. Theory Stoch. Process. 2007, 13 (1-2), 166-181. · Zbl 1142.60328
[29] Moklyachuk M.P., Masyutka O.Yu. Minimax-robust estimation technique for stationary stochastic processes. LAP Lambert Academic Publishing, Saarbr\"ucken, 2012. · Zbl 1289.62001
[30] Moklyachuk M.P., Sidei M.I. Interpolation problem for stationary sequences with missing observations. Stat., Optim. Inf. Comput. 2015, 3 (3), 259-275. doi: 10.19139/soic.v3i3.149 · doi:10.19139/soic.v3i3.149
[31] Moklyachuk M.P., Sidei M. Filtering problem for stationary sequences with missing observations. Stat., Optim. Inf. Comput. 2016, 4 (4), 308-325. doi: 10.19139/soic.v4i4.241 · doi:10.19139/soic.v4i4.241
[32] Moklyachuk M.P., Sidei M.I. Filtering Problem for functionals of stationary processes with missing observations. Commun. Optim. Theory 2016, Article ID 21, 1-18. · Zbl 1363.60051
[33] Moklyachuk M.P., Sidei M.I. Extrapolation problem for stationary sequences with missing observations. Stat., Optim. Inf. Comput. 2017, 5 (3), 212-233. doi: 10.19139/soic.v5i3.284 · Zbl 1363.60051 · doi:10.19139/soic.v5i3.284
[34] Pelagatti M.M. Time series modelling with unobserved components. CRC Press, New York, 2015.
[35] Pourahmadi M., Inoue A., Kasahara Y. A prediction problem in \(L^2(w)\). Proc. Amer. Math. Soc. 2007, 135 (4), 1233-1239. doi: 10.1090/S0002-9939-06-08575-3 · Zbl 1121.60041 · doi:10.1090/S0002-9939-06-08575-3
[36] Pshenichnyj B. N. Necessary conditions of an extremum. Marcel Dekker, New York, 1971. · Zbl 0212.23902
[37] Rockafellar R. T. Convex Analysis. Princeton University Press, Princeton, 1997. · Zbl 0932.90001
[38] Rozanov Yu.A. Stationary stochastic processes. Holden-Day, San Francisco-Cambridge-London-Amsterdam, 1967. · Zbl 0152.16302
[39] Salehi H. Algorithms for linear interpolator and interpolation error for minimal stationary stochastic processes. Ann. Probab. 1979, 7 (5), 840-846. · Zbl 0419.60032
[40] Vastola S. K., Poor H. V. An analysis of the effects of spectral uncertainty on Wiener filtering. Automatica 1983, 19 (3), 289-293. doi: 10.1016/0005-1098(83)90105-X · Zbl 0534.93062 · doi:10.1016/0005-1098(83)90105-X
[41] Wiener N. Extrapolation, interpolation and smoothing of stationary time series. With engineering applications. The M. I. T. Press, Cambridge, 1966. · Zbl 0138.12302
[42] Yaglom A.M. Correlation theory of stationary and related random functions. Vol. 1: Basic results. Springer-Verlag, New York etc., 1987. · Zbl 0685.62078
[43] Yaglom A.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.