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Prediction of stationary Gaussian random fields with incomplete quarterplane past

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  • Cheng, Raymond

Abstract

Let {Xm,n:(m,n)∈Z2} be a stationary Gaussian random field. Consider the problem of predicting X0,0 based on the quarterplane Q={(m,n):m≥0,n≥0}∖{(0,0)}, but with finitely many observations missing. Two solutions are presented. The first solution expresses the best predictor in terms of the moving average coefficients of {Xm,n}, under the assumption that the spectral density function has a strongly outer factorization. The second solution expresses the prediction error variance in terms of the autoregressive coefficients of {Xm,n}; it requires the reciprocal of the density function to have a strongly outer factorization, and relies on a modified duality argument. These solutions are extended by allowing the quarterplane past to be replaced with a much broader class of parameter sets. This enables the solution, for example, of the quarterplane interpolation problem.

Suggested Citation

  • Cheng, Raymond, 2015. "Prediction of stationary Gaussian random fields with incomplete quarterplane past," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 245-258.
  • Handle: RePEc:eee:jmvana:v:139:y:2015:i:c:p:245-258
    DOI: 10.1016/j.jmva.2015.03.007
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    References listed on IDEAS

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    1. Chiang, Tse-Pei, 1991. "The prediction theory of stationary random fields. III. Fourfold Wold decompositions," Journal of Multivariate Analysis, Elsevier, vol. 37(1), pages 46-65, April.
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    3. Cheng, R. & Pourahmadi, M., 1997. "Prediction with incomplete past and interpolation of missing values," Statistics & Probability Letters, Elsevier, vol. 33(4), pages 341-346, May.
    4. Bondon, Pascal, 2002. "Prediction with incomplete past of a stationary process," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 67-76, March.
    5. Kasahara, Yukio & Pourahmadi, Mohsen & Inoue, Akihiko, 2009. "Duals of random vectors and processes with applications to prediction problems with missing values," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1637-1646, July.
    6. Korezlioglu, Hayri & Loubaton, Philippe, 1986. "Spectral factorization of wide sense stationary processes on 2," Journal of Multivariate Analysis, Elsevier, vol. 19(1), pages 24-47, June.
    7. Pascal Bondon, 2005. "Influence of Missing Values on the Prediction of a Stationary Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(4), pages 519-525, July.
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