Abstract
With vegetation data there are often physical reasons for believing that the response of neighbors has a direct influence on the response at a particular location. In terms of modeling such scenarios the family of auto-models or Markov random fields is a useful choice. If the observed responses are counts, the auto-Poisson model can be used. There are different ways to formulate the auto-Poisson model, depending on the biological context. A drawback of this model is that for positive autocorrelation the likelihood of the auto-Poisson model is not available in closed form. We investigate how this restriction can be avoided by right truncating the distribution. We review different parameter estimation techniques which apply to auto-models in general and compare them in a simulation study. Results suggest that the method which is most easily implemented via standard statistics software, maximum pseudo-likelihood, gives unbiased point estimates, but its variance estimates are biased. An alternative method, Monte Carlo maximum likelihood, works well but is computer-intensive and not available in standard software. We illustrate the methodology and techniques for model checking with clover leaf counts and seed count data from an agricultural experiment.
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References
Augustin, N. H., Mugglestone, M. A., and Buckland, S. T. (1998), “The Role of Simulation in Modelling Spatially Correlated Data,” Environmetrics, 9, 175–196.
Besag, J. (1972), “Nearest-Neighbour Systems and the Auto-Logistic Model for Binary Data,” Journal of the Royal Statistical Society, Series B, 34, 75–83.
— (1974), “Spatial Interaction and the Statistical Analysis of Lattice Systems” (with discussion). Journal of the Royal Statistical Society, Series B, 36, 192–236.
— (1975), “Statistical Analysis of Non-Lattice Data,” Statistician, 24, 179–195.
Besag, J., York, J., and Mollié, A. (1991), “Bayesian Image-Restoration, With Two Applications in Spatial Statistics,” Annals of the Institute of Statistical Mathematics, 43, 1–59.
Breslow, N. E., and Clayton, D. G. (1993), “Approximate Inference in Generalized Linear Mixed Models,” Journal of the American Statistical Association, 88, 9–25.
Brix, A. (1998), “Spatial and Spatio-temporal Models for Weed Abundance,” unpublished PhD thesis. Royal Veterinary and Agricultural University Copenhagen, Denmark.
Cressie, N. A. (1991), Statistics for Spatial Data, New York: Wiley.
Durbán, M. L., Hackett, C. A., and Currie, I. (1998), “Blocks, Trend and Interference in Field Trials,” in Statistical Modeling. Proceedings of the 14th International Workshop on Statistical Modeling, eds. B. Marx and H. Friedl, Graz, Austria.
Fahrmeir, L., and Lang, S. (2001), “Bayesian Inference for Generalized Additive Mixed Models Based on Markov Random Field Priors,” Applied Statistics, 50, 201–220.
Gamerman, D. (1997), Markov Chain Monte Carlo. Stochastic Simulation for Bayesian Inference. London: Chapman & Hall.
Gampe, J. (1998), “Trend or Correlation? Attributing Smoothness in Nonparametric Regression,” in Statistical Modeling. Proceedings of the 14th International Workshop on Statistical Modeling, eds. B. Marx, and H. Friedl, New Orleans, LA: pp. 216–221.
Gelman, A., Carlin, J., Stern, H., and Rubin, D. B. (2004), Bayesian Data Analysis, London: Chapman & Hall.
Geman, S., and Geman, D. (1984), “Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.
Geyer, C. J. (1999), “Likelihood Inference for Spatial Point Processess,” in Current Trends in Stochastic Geometry and Applications (to appear), eds. O. E. Barndorff-Nielsen and W. S. Kendall, London: Chapman & Hall.
Geyer, C. J., and Thompson, E. A. (1992), “Constrained Monte Carlo Maximum Likelihood for Dependent Data,” Journal of the Royal Statistical Society, Series B, 54, 657–699.
Gumpertz, M. L., Graham, J. M., and Ristaino, J. B. (1997), “Autologistic Model of Spatial Pattern of Phytophthora Epidemic in Bell Pepper,” Journal of Agricultural, Biological, and Environmental Statistics, 2, 131–156.
Hastie, T., and Tibshirani, R. J. (1990), Generalized Additive Models, London: Chapman and Hall.
Heikkinen, J., and Penttinen, A. (1999), “Bayesian Smoothing in the Estimation of the Pair Potential Function of Gibbs Point Processes,” Bernoulli, 5, 1119–1136.
Huffer, F. W., and Wu, H. L. (1998), “Markov Chain Monte Carlo for Autologistic Regression Models with Application to the Distribution of Plant Species,” Biometrics, 54, 509–524.
Ising, E. (1925), “Beitrag zur Theorie des Ferromagnetismus,” Zeitschrift für Physik, 31, 253–258.
Johnson, N. L., Kotz, S., and Kemp, A. W. (1992), Univariate Discrete Distributions, New York: Wiley.
Kaiser, M. S., and Cressie, N. (1997), “Modeling Poisson Variables with Positive Spatial Dependence,” Statistics and Probability Letters, 35, 423–432.
Kamman, E. E., and Wand, M. P. (2003), “Geoadditive Models,” Journal of the Royal Statistical Society, Series C, 52, 1–18.
Lin, X., and Zhang, D. (1999), “Inference in Generalized Additive Mixed Models by Using Smoothing Splines,” Journal of the Royal Statistical Society, Series B, 61, 381–400.
Marriott, C., Bolton, B., Barthram, G., Fisher, J., and Hood, K. (2002), “Early Changes in Species Composition of Upland Sown Grassland and Extensive Grazing Management,” Applied Vegetation Science, 5, 87–98.
McCullagh, P., and Nelder, J. A. (1989), Generalized Linear Models, London: Chapman and Hall.
Moran, P. A. P. (1950), “Notes on Continuous Stochastic Phenomena,” Biometrica, 37, 17–23.
Penttinen, A. (1984), “Modelling Interaction in Spatial Point Patters: Parameter Estimation by the Maximum Likelihood Method,” Jyvaskyla Studies in Computer Science, Economics, and Statistics, pp. 1–107.
Preisler, H. K. (1993), “Modelling Spatial Patterns of Trees Attacked by Bark-Beetles,” Applied Statistics, 42, 501–514.
Preisler, H. K., and Mitchell, R. G. (1993), “Colonization Patterns of the Mountain Pine-Beetle in Thinned and Unthinned Lodgepole Pine Stands,” Forest Science, 39, 528–545.
Whittle, P. (1954), “On Stationary Processes in the Plane,” Biometrika, 41, 434–439.
Wolpert, R. L., and Ickstadt, K. (1998), “Gamma/Poisson Random Field Models for Spatial Statistics,” Biometrika. 85, 251–267.
Wu, H. L., and Huffer, F. W. (1997), “Modeling the Distribution of Plant Species using the Autologistic Regression Model,” Environmental and Ecological Statistics, 4, 49–64.
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Augustin, N.H., McNicol, J. & Marriott, C.A. Using the truncated auto-Poisson model for spatially correlated counts of vegetation. JABES 11, 1–23 (2006). https://doi.org/10.1198/108571106X96871
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DOI: https://doi.org/10.1198/108571106X96871