Risks of Classification of the Gaussian Markov Random Field Observations

Abstract

Given the spatial lattice endowed with particular neighborhood structure, the problem of classifying a scalar Gaussian Markov random field (GMRF) observation into one of two populations specified by different regression coefficients and special parametric covariance (precision) matrix is considered. Classification rule based on the plug-in Bayes discriminant function with inserted ML estimators of regression coefficients, spatial dependence and scale parameters is studied. The novel closed-form expression for the actual risk and the approximation of the expected risk (AER) associated with the aforementioned classifier are derived. This is the extension of the previous study of GMRF classification to the case of complete parametric uncertainty. Derived AER is used as the main performance measure for the considered classifier. GMRF sampled on a regular 2-dimensional unit spacing lattice endowed with neighborhood structure based on the Euclidean distance between sites is used for a simulation experiment. The sampling properties of ML estimators and the accuracy of the derived AER for various values of spatial dependence parameters and Mahalanobis distance are studied. The influence of the neighborhood size on the accuracy of the proposed AER is examined as well.

This is a preview of subscription content, log in to check access.

References

  1. ABT, M. (1999), “Estimating the Prediction Mean Squared Error in Gaussian Stochastic Processes with Correlation Structure”, Scandinavian Journal of Statistics, 26, 563–578.

    MathSciNet  Article  Google Scholar 

  2. BATSIDIS, A., and ZOGRAFOS, K. (2011), “Errors of Misclassification in Discrimination of Dimensional Coherent Elliptic Random Field Observations”, Statistica Neerlandica, 65(4), 446–461.

    MathSciNet  Article  Google Scholar 

  3. BERRETT, C., and CALDER, C.A. (2016), “Bayesian Spatial Binary Classification”, Spatial Statistics, 16, 72–102.

    MathSciNet  Article  Google Scholar 

  4. BESAG, J. (1974), “Spatial Interaction and the Statistical Analysis of Lattice Systems (With Discussion)”, Journal of the Royal Statistical Society, 36, 192–236.

    MathSciNet  MATH  Google Scholar 

  5. CRESSIE, N.A.C., and CHAN, N. (1989), “Spatial Modelling of Regional Variables”, Journal of the American Statistical Association, 84 (406), 393–401.

    MathSciNet  Article  Google Scholar 

  6. DIGGLE, P.J., RIBEIRO, P.J., and CHRISTENSEN, O.F. (2003), “An Introduction to Model-Based Geostatistics”, in Spatial Statistics and Computational Methods: Lecture Notes in Statistics, Vol. 173, ed. J. Möller, pp. 43–86.

  7. DUČINSKAS, K. (2009), “Approximation of the Expected Error Rate in Classification of the Gaussian Random Field Observations”, Statistics & Probability Letters, 79, 138–144.

    MathSciNet  Article  Google Scholar 

  8. DUČINSKAS, K., and DREIŽIENĖ, L. (2011), “Supervised Classification of the Scalar Gaussian Random Field Observations Under a Deterministic Spatial Sampling Design”, Austrian Journal of Statistics, 40(1&2), 2536.

    Google Scholar 

  9. DUČINSKAS, K., and ZIKARIENE, E. (2015), “Actual Error Rates in Classification of the t-Distributed Random Field Observation Based on Plug-In Linear Discriminant Function”, Informatica, 26(4), 557–568.

    MathSciNet  Article  Google Scholar 

  10. DUČINSKAS, K., BORISENKO, I., and SIMKIENĖ I. (2013), “Statistical Classification of Gaussian Spatial Data Generated by Conditional Autoregressive Model”, Journal of Computer Science and Technology, 1(2), 69–79.

    Google Scholar 

  11. KHARIN, Y. (1996), Robustness in Statistical Pattern Recognition, Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  12. LAWOKO, C.R.O., and MCLACHLAN, G.L. (1985), “Discrimination with Autocorrelated Observations”, Pattern Recognition, 18(2), 145–149.

    MathSciNet  Article  Google Scholar 

  13. MARDIA, K.V., and MARSHALL, R.J. (1984), “Maximum Likelihood Estimation of Models for Residual Covariance in Spatial Regression”, Biometrika, 71, 135–146.

    MathSciNet  Article  Google Scholar 

  14. MCLACHLAN, G.J. (2004), Discriminant Analysis and Statistical Pattern Recognition, New York: Wiley.

    Google Scholar 

  15. RUE, H., and HELD, L. (2005), Gaussian Markov Random Fields: Theory and Applications, Boca Raton: Chapman and Hall.

    Google Scholar 

  16. RICHARDSON, S., GUIHENNEUC, C., and LASSERRE, V. (1992), “Spatial Linear Models with Autocorrelated Error Structure”, Statistician, 41, 539–557.

    Article  Google Scholar 

  17. DE OLIVEIRA, V., and FERREIRA, M.A.R. (2011), “Maximum Likelihood and Restricted Maximum Likelihood Estimation for a Class of Gaussian Markov Random Fields”, Metrika, 74(2), 167–183.

    MathSciNet  Article  Google Scholar 

  18. SWITZER, P. (1980), ”Extensions of Linear Discriminant Analysis for Statistical Classification of Remotely Sensed Satellite Imagery”, Mathematical Geology, 12(4), 367–376.

    MathSciNet  Article  Google Scholar 

  19. ZIMMERMAN, D.L., and CRESSIE, N. (1992), “Mean Squared Prediction Error in the Spatial Linear Model with Estimated Covariance Parameters”, Annals of the Institute of Statististics, 44, 27-43.

    MathSciNet  Article  Google Scholar 

  20. ZIMMERMAN, D.L. (2006), “Optimal Network Design for Spatial Prediction, Covariance Parameter Estimation, and Empirical Prediction”, Environmetrics, 17, 635–652.

    MathSciNet  Article  Google Scholar 

  21. ZHU, Z., and ZHANG, H. (2006), “Spatial Sampling Design Under Infill Asymptotic Framework”, Environmetrics, 17, 23–337.

    MathSciNet  Article  Google Scholar 

  22. ZHU, Z., and STEIN, M.L. (2006), “Spatial Sampling Design for Prediction with Estimated Parameters”, Journal of Agricultural, Biological and Environmental Statistics, 11(1), 24–44.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Lina Dreižienė.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dučinskas, K., Dreižienė, L. Risks of Classification of the Gaussian Markov Random Field Observations. J Classif 35, 422–436 (2018). https://doi.org/10.1007/s00357-018-9269-7

Download citation

Keywords

  • Bayes discriminant function
  • Training sample
  • Actual risk
  • Spatial weight