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Robust prediction and interpolation for vector stationary processes
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  • Published: July 1986

Robust prediction and interpolation for vector stationary processes

  • Haralampos Tsaknakis1,
  • Dimitri Kazakos2 &
  • P. Papantoni-Kazakos1 

Probability Theory and Related Fields volume 72, pages 589–602 (1986)Cite this article

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  • 4 Citations

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Summary

Robust multivariate prediction and interpolation problems for statistically contaminated vector valued second order stationary processes are considered. The statistical contamination is modeled by requiring that the spectral density matrices of the processes lie within certain nonparametric classes. Both prediction and interpolation are then formalized as games whose saddle point solutions are sought. Finally, such solutions are found and analyzed, for two specific multivariate spectral classes.

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References

  • Bellman, R.: Introduction to matrix analysis, 2nd edn. New York: McGraw Hill 1970

    Google Scholar 

  • Chen, C.T., Kassam, S.A.: Robust multiple-input matched filters. Proc. 19th Annual Allerton Conf. on Commun., Control and Computing, pp. 586–595 (1981)

  • Chen, C.T., Kassam, S.A.: Finite-length discrete-time matched filters for uncertain signal and noise. Proc. 1982 Conf. on Information Sciences and Systems, Princeton Univ., pp. 336–341 (1982)

  • Cimini, L.J., Kassam, S.A.: Robust and quantized Wiener filters for p-point spectral classes. Proceedings 14th Conf. on Info. Sciences and Systems, Princeton Univ., Princeton, N.J. (1980)

  • Franke, J.: Minimax robust prediction of discrete time series. Z. Wahrscheinlichkeitstheor. Verw. Geb. 68, 337–364 (1985)

    Google Scholar 

  • Franke, J., Poor, H.V.: Minimax robust filtering and finite-length robust predictors. Lecture Notes in Statistics 26. Robust and nonlinear time series analysis. Berlin Heidelberg New York Tokyo: Springer 1984

    Google Scholar 

  • Grenander, U., Szegö, G.: Toeplitz forms and their applications. University of California Press, Berkeley Los Angeles (1958)

    Google Scholar 

  • Hannan, E.J.: Multiple time series. Wiley: New York 1970

    Google Scholar 

  • Hardy, G.H., Rogosinski, W.W.: Fourier series, 3rd edn. Cambridge University Press, U.K. (1956)

    Google Scholar 

  • Helson, H., Lowdenslager, D.: Prediction theory and fourier series in Several Variables. Acta Math. 99, 165–202 (1958)

    Google Scholar 

  • Hosoya, Y.: Robust linear extrapolation of second order stationary processes. Ann. Probab. 6, 574–584 (1978)

    Google Scholar 

  • Kassam, S.A.: J. Time Ser. Anal. 3, 185–194 (1982)

    Google Scholar 

  • Kassam, S.A., Lim, T.L.: Robust Wiener filters. J. Franklin Inst. 304, 171–185 (1977)

    Google Scholar 

  • Kassam, S.A., Poor, H.V.: Robust techniques for signal processing-survey. IEEE Proceedings 73, 433–479

  • Katznelson, Y.: Harmonic analysis, 2nd edn. New York: Dover Publications, Inc. 1976

    Google Scholar 

  • Kolmogorov, A.: Interpolation and extrapolation. Bull. Acad. Sci. USSR, Ser. Math. 5, 3–14

  • Kolmogorov, A.: Stationary sequences in Hilbert space. Bull. Math. Univ., Moscow 2, 40 (1941)

    Google Scholar 

  • Martin, R.D., Debow, G.: Robust filtering with data dependent covariance. Proceedings of the 1976 Johns Hopkins Conf. on Information Sciences and Systems (1976)

  • Martin, R.D., Zeh, J.E.: Determining the character of time series outliers. Proceedings of the American Statistical Association (1977)

  • Masreliez, C.J., Martin, R.D.: Robust Bayesian estimation for the linear model and robustifying the Kalman filter. IEEE Trans. on Aut. Control, AC-22, 361–371 (1977)

    Google Scholar 

  • Papantoni-Kazakos, P.: A game theoretic approach to robust filtering. Information and Control 1735–1757 (1984)

  • Poor, H.V.: On robust Wiener filtering. IEEE Trans. on Aut. Control, AC-25, 531–536 (1980)

    Google Scholar 

  • Snyders, J.: On the error matrix in optimal linear filtering of stationary processes. IEEE Trans. on Infor. Theory IT-19, 593–599 (1973)

    Google Scholar 

  • Taniguchi, S.: J. Time Ser. Anal. 2, 53–62 (1981)

    Google Scholar 

  • Tsaknakis, H., Papantoni-Kazakos, P.: Robust linear filtering for multivariable stationary time series. University of Connecticut, EECS Dept., Technical Report TR-83-6, April (1983). Also, 1984 Conf. on Inf. Sciences and Systems proceedings

  • Vastola, K.S., Poor, H.V.: Robust Wiener-Kolmogorov theory. IEEE Trans. Inf. Th., IT-30, 316–327 (1984)

    Google Scholar 

  • Viterbi, A.J.: On the minimum mean square error resulting from linear filtering of stationary, signals in white noise. IEEE Trans. on Information Theory IT-11, 594–595 (1965)

    Google Scholar 

  • Whittle, P.: The analysis of multiple stationary time series. J. Roy, Statist. Soc., Ser. B. 15, 125–139 (1953)

    Google Scholar 

  • Wiener, N.: Extrapolation, interpolation and smoothing of stationary time series. Cambridge MIT Press (1949)

    Google Scholar 

  • Wiener, N., Masani, P.: The prediction theory of multivariate stochastic processes (1957, 1958) I. The regularity condition. Acta Math. 98, 111–150 (1957) II. The linear predictor. Acta Math. 99, 93–137 (1958)

    Google Scholar 

  • Zasuhin, V.: On the theory of multidimensional stationary random processes. Acad. Sci. USSR, 33, 435 (1941)

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of Connecticut, 06268, Storrs, CT, USA

    Haralampos Tsaknakis & P. Papantoni-Kazakos

  2. University of Virginia, 22901, Charlottesville, VA, USA

    Dimitri Kazakos

Authors
  1. Haralampos Tsaknakis
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  2. Dimitri Kazakos
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  3. P. Papantoni-Kazakos
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Additional information

Research supported by the Air Force Office of Scientific Research under Grants AFOSR-83-0229 and AFOSR-82-0030

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Tsaknakis, H., Kazakos, D. & Papantoni-Kazakos, P. Robust prediction and interpolation for vector stationary processes. Probab. Th. Rel. Fields 72, 589–602 (1986). https://doi.org/10.1007/BF00344722

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  • Received: 23 May 1985

  • Revised: 12 March 1986

  • Issue Date: July 1986

  • DOI: https://doi.org/10.1007/BF00344722

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Keywords

  • Stochastic Process
  • Stationary Process
  • Probability Theory
  • Spectral Density
  • Saddle Point
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