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Some prediction problems for stationary random fields with quarter-plane past

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  • Kohli, P.
  • Pourahmadi, M.

Abstract

We study several nonstandard prediction problems where a number of observations are added to the quarter-plane past of a stationary random field. The goal is to provide informative and explicit prediction error variance formulas in terms of either the autoregressive or moving average parameters of the random fields. However, unlike the time series situation, the prediction error variances for random fields seem to be expressible only in terms of the moving average parameters, and attempts to express them formally in terms of the autoregressive parameters lead to a new and mysterious projection operator which captures the nature of the “edge-effects” encountered in the estimation of the spectral density function of stationary random fields. The approach leads to a number of technical issues and open problems.

Suggested Citation

  • Kohli, P. & Pourahmadi, M., 2014. "Some prediction problems for stationary random fields with quarter-plane past," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 112-125.
    • Handle: RePEc:eee:jmvana:v:127:y:2014:i:c:p:112-125
    DOI: 10.1016/j.jmva.2014.02.009
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    References listed on IDEAS

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    1. Kallianpur, G. & Miamee, A. G. & Niemi, H., 1990. "On the prediction theory of two-parameter stationary random fields," Journal of Multivariate Analysis, Elsevier, vol. 32(1), pages 120-149, January.
    2. 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.
    3. Bondon, Pascal, 2002. "Prediction with incomplete past of a stationary process," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 67-76, March.
    4. 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.
    5. Fuchun Huang & Yosihiko Ogata, 2002. "Generalized Pseudo-Likelihood Estimates for Markov Random Fields on Lattice," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(1), pages 1-18, March.
    6. Mohsen Pourahmadi, 1989. "Estimation And Interpolation Of Missing Values Of A Stationary Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 10(2), pages 149-169, March.
    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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    Cited by:

    1. Cheng, Raymond, 2015. "Prediction of stationary Gaussian random fields with incomplete quarterplane past," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 245-258.

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