Advertisement

A Bayesian Approach to Fitting Gibbs Processes with Temporal Random Effects

  • Ruth King
  • Janine B. Illian
  • Stuart E. King
  • Glenna F. Nightingale
  • Ditte K. Hendrichsen
Article

Abstract

We consider spatial point pattern data that have been observed repeatedly over a period of time in an inhomogeneous environment. Each spatial point pattern can be regarded as a “snapshot” of the underlying point process at a series of times. Thus, the number of points and corresponding locations of points differ for each snapshot. Each snapshot can be analyzed independently, but in many cases there may be little information in the data relating to model parameters, particularly parameters relating to the interaction between points. Thus, we develop an integrated approach, simultaneously analyzing all snapshots within a single robust and consistent analysis. We assume that sufficient time has passed between observation dates so that the spatial point patterns can be regarded as independent replicates, given spatial covariates. We develop a joint mixed effects Gibbs point process model for the replicates of spatial point patterns by considering environmental covariates in the analysis as fixed effects, to model the heterogeneous environment, with a random effects (or hierarchical) component to account for the different observation days for the intensity function. We demonstrate how the model can be fitted within a Bayesian framework using an auxiliary variable approach to deal with the issue of the random effects component. We apply the methods to a data set of musk oxen herds and demonstrate the increased precision of the parameter estimates when considering all available data within a single integrated analysis.

Key Words

Data augmentation Markov chain Monte Carlo Mixed effects model Musk oxen data Spatial and temporal point processes 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aarts, G., MacKenzie, M., McConnell, B., Fedak, M., and Matthiopoulos, J. (2008), “Estimating Space-Use and Habitat Preference from Wildlife Telemetry Data,” Ecography, 31, 140–160. CrossRefGoogle Scholar
  2. Babin, J.-S., Fortin, D., Wilmshurst, J. F., and Fortin, M.-E. (2011), “Energy Gains Predict the Distribution of Plains Bison Across Populations and Ecosystems,” Ecology, 92, 240–252. CrossRefGoogle Scholar
  3. Baddeley, A. J., and Turner, T. R. (2000), “Practical Maximum Pseudolikelihood for Spatial Point Patterns (with Discussion),” Australian and New Zealand Journal of Statistics, 32, 283–322. MathSciNetGoogle Scholar
  4. — (2006), “Modelling Spatial Point Patterns in R,” in Case Studies in Spatial Point Pattern Modelling, Lecture Notes in Statistics, Vol. 185, eds. A. Baddeley, P. Gregori, J. Mateu, R. Stoica, and D. Stoyan, New York: Springer, pp. 23–74. CrossRefGoogle Scholar
  5. — (2011), Spatstat Website. URL: www.spatstat.org.
  6. Baddeley, A. J., and van Lieshout, M. N. W. (1995), “Area-Interaction Point Processes,” Annals of the Institute of Statistical Mathematics, 47, 601–619. MathSciNetzbMATHCrossRefGoogle Scholar
  7. Berg, T. B. (2003), “Mammals,” in Zackenberg Ecological Research Operations. 8th Annual Report, eds. M. Rasch, and K. Canning, Copenhagen: Danish Polar Center, pp. 43–50. Google Scholar
  8. Berg, T. B., Schmidt, N.M., Høye, T.T., Aastrup, P., Hendrichsen, D.K., Forchhammer, M.C., and Klein, D.R. (2008), “High-Arctic Plant–Herbivore Interactions Under Climate Influence,” Advances in Ecological Research 40, 275–298. CrossRefGoogle Scholar
  9. Berman, M., and Turner, M. R. (1992), “Approximation Point Process Likelihoods with GLIM,” Journal of the Royal Statistical Society, Series C, 41, 31–38. zbMATHCrossRefGoogle Scholar
  10. Besbeas, P., Borysiewicz, R. S., and Morgan, B. J. T. (2008), “Completing the Ecological Jigsaw,” in Modeling Demographic Processes in Marked Populations, Springer—Series: Environmental and Ecological Statistics, Vol. 3, eds. D. L. Thomson, E. G. Cooch and M. J. Conroy, pp. 513–540. Google Scholar
  11. Besag, J. (1978), “Some Methods of Statistical Analysis for Spatial Data,” Bulletin of the International Statistical Institute, 47, 77–92. MathSciNetGoogle Scholar
  12. Besbeas, P., Lebreton, J. D., and Morgan, B. J. T. (2003), “The Efficient Integration of Abundance and Demographic Data,” Journal of the Royal Statistical Society, Series C, 52, 95–102. MathSciNetzbMATHCrossRefGoogle Scholar
  13. Blackwell, P. G. (2003), “Bayesian Inference for Markov Processes with Diffusion and Discrete Components,” Biometrika, 90, 613–627. MathSciNetCrossRefGoogle Scholar
  14. Bonner, S. J., Morgan, B. J. T., and King, R. (2010), “Continuous Covariates in Mark-Recapture-Recovery Analysis: A Comparison of Methods,” Biometrics, 66, 1256–1265. MathSciNetzbMATHCrossRefGoogle Scholar
  15. Breed, G. A., Jonsen, I. D., Myers, W. D., Bowen, W. D., and Leonard, M. L. (2009), “Sex-Specific, Seasonal Foraging Tactics of Adult Grey Seals (Halichoerus Grypus) Revealed by State-Space Analysis,” Ecology, 90, 3209–3221. CrossRefGoogle Scholar
  16. Brooks, S. P. (1998), “Markov Chain Monte Carlo Method and Its Application,” The Statistician, 47, 69–100. CrossRefGoogle Scholar
  17. Brooks, S. P., and Gelman, A. (1998), “Alternative Methods for Monitoring Convergence of Iterative Simulations,” Journal of Computational and Graphical Statistics, 7, 434–455. MathSciNetGoogle Scholar
  18. Brooks, S. P., King, R., and Morgan, B. J. T. (2004), “A Bayesian Approach to Combining Animal Abundance and Demographic Data,” Animal Biodiversity and Conservation, 27, 515–529. Google Scholar
  19. Brooker, R. W., Maestre, F. T., Callaway, R. M., Lortie, C. L., Cavieres, L. A., Kunstler, G., Liancourt, P., Tielborger, K., Travis, J. M. J., Anthelme, F., Armas, C., Coll, L., Corcket, E., Delzon, S., Forey, E., Kikvidze, Z., Olofsson, J., Pugnaire, F., Quiroz, Q. L., Saccone, P., Schiffers, K., Seifan, M., Touzard, B., and Michalet, R. (2008), “Facilitation in Plant Communities: The Past, the Present, and the Future,” Journal of Ecology, 96, 18–34. CrossRefGoogle Scholar
  20. Caraco, T., Martindale, S., and Pulliam, H. R. (1980), “Flocking: Advantages and Disadvantages,” Nature, 285, 400–401. CrossRefGoogle Scholar
  21. Cave, V. M., King, R., and Freeman, S. N. (2010), “An Integrated Population Model from Constant Effort Bird-Ringing Data,” Journal of Agricultural, Biological, and Environmental Statistics, 15, 119–137. MathSciNetCrossRefGoogle Scholar
  22. Clutton-Brock, T. H., and Pemberton, J. (2004), Soay Sheep, Cambridge: Cambridge University Press. Google Scholar
  23. Cornulier, T., and Bretagnolle, V. (2006), “Assessing the Influence of Environmental Heterogeneity on Bird Spacing Patterns: A Case Study with Two Raptors,” Ecography, 29, 240–250. CrossRefGoogle Scholar
  24. Elberling, B., Tamstorf, M. P., Michelsen, A., Arndal, M. F., Sigsgaard, C., Illeris, L., Bay, C., Hansen, B. U., Christensen, T. R., Hansen, E. S., Jakobsen, B. H., and Beyens, L. (2008), “Soil and Plant Community-Characteristics and Dynamics at Zackenberg,” Advances in Ecological Research, 40, 223–248. CrossRefGoogle Scholar
  25. Forchhammer, M. C., Post, E., Berg, T. B., Høye, T. T., and Schmidt, N. M. (2005), “Local-Scale and Short-Term Herbivore–Plant Spatial Dynamics Reflect Influences of Large-Scale Climate,” Ecology, 86, 2644–2651. CrossRefGoogle Scholar
  26. Forchhammer, M. C., Schmidt, N. M., Høye, T. T., Berg, T. B., Hendrichsen, D. K., and Post, E. (2008), “Population Dynamical Responses to Climate Change,” Advances in Ecological Research, 40, 391–419. CrossRefGoogle Scholar
  27. Gauthier, G., Besbeas, P., Lebreton, J. D., and Morgan, B. J. T. (2007), “Population Growth in Greater Snow Geese: A Modeling Approach to Integrating Demographic and Population Survey Information,” Ecology, 88, 1420–1429. CrossRefGoogle Scholar
  28. Gelman, A. (2006), “Prior Distributions for Variance Parameters in Hierarchical Models,” Bayesian Analysis, 1, 515–534. MathSciNetCrossRefGoogle Scholar
  29. Green, P. J. (1995), “Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination,” Biometrika, 82, 711–732. MathSciNetzbMATHCrossRefGoogle Scholar
  30. Hansen, J., Hansen, L. H., Christoffersen, K. S., Albert, K. R., Skovgaard, M. S., Bay, C., Kristensen, D. K., Berg, T. B., Lund, M., Boulanger-Lapointe, N., Sørensen, P. L., Christensen, M. U., and Schmidt, N. M. (2011), “Zackenberg Basic: The BioBasis Programme,” in Zackenberg Ecological Research Operations, 16th Annual Report, 2010, eds. L. M. Jensen, and M. Rasch, pp. 43–66. Aarhus University, DCE—Danish Centre for Environment and Energy, 2011. Google Scholar
  31. Hassell, M. P. (1975), “Density-Dependence in Single-Species Populations,” Journal of Animal Ecology, 44, 283–295. CrossRefGoogle Scholar
  32. Illera, J. C., von Wehrden, H., and Wehner, J. (2010), “Nest Site Selection and the Effects of Land Use in a Multi-scale Approach on the Distribution of a Passerine in an Island Arid Environment,” Journal of Arid Environments, 74, 1408–1412. CrossRefGoogle Scholar
  33. Illian, J. B., and Hendrichsen, D. K. (2010), “Gibbs Point Process Models with Mixed Effects,” Environmetrics, 21, 341–353. MathSciNetCrossRefGoogle Scholar
  34. Illian, J. B., Møller, J., and Waagepetersen, R. P. (2009), “Hierarchical Spatial Point Process Analysis for a Plant Community with High Biodiversity,” Environmental and Ecological Statistics, 16, 389–405. MathSciNetCrossRefGoogle Scholar
  35. Illian, J. B., Penttinen, A., Stoyan, H., and Stoyan, D. (2008), Statistical Analysis and Modelling of Spatial Point Patterns, Chichester: Wiley. zbMATHGoogle Scholar
  36. Illian, J. B., Sørbye, S. H., and Rue, H. (2012a). “A Toolbox for Fitting Complex Spatial Point Process Models Using Integrated Laplace Transformation (INLA),” Annals of Applied Statistics, in press. Google Scholar
  37. Illian, J. B., Sørbye, S. H., Rue, H., and Hendrichsen, D. (2012b), “Using INLA to Fit a Complex Point Process Model with Temporally Varying Effects—A Case Study,” Journal of Environmental Statistics, 3, 7. Google Scholar
  38. Janson, C. H. (1988), “Food Competition in Brown Capuchin Monkeys (Cebus Apella): Quantitative Effects of Group Size and Tree Productivity,” Behaviour, 105, 53–76. CrossRefGoogle Scholar
  39. Jetz, W., Carbone, C., Fulford, J., and Brown, J. H. (2004), “The Scaling of Animal Space Use,” Science, 306, 266–268. CrossRefGoogle Scholar
  40. Jonsen, I. D., Flemming, J. M., and Myers, R. A. (2005), “Robust State-Space Modeling of Animal Movement Data,” Ecology, 86, 2874–2880. CrossRefGoogle Scholar
  41. Johnson, D. S., London, J. M., Lea, M.-A., and Durban, J. W. (2008), “Continuous-Time Correlated Random Walk Model for Animal Telemetry Data,” Ecology, 89, 1208–1215. CrossRefGoogle Scholar
  42. Kass, R. E., and Raftery, A. E. (1995), “Bayes Factors,” Journal of the American Statistical Association, 90, 773–793. zbMATHCrossRefGoogle Scholar
  43. King, R., Brooks, S. P., Mazzetta, C., Freeman, S. N., and Morgan, B. J. T. (2008), “Identifying and Diagnosing Population Declines: A Bayesian Assessment of Lapwings in the UK,” Journal of the Royal Statistical Society, Series C, 57, 609–632. MathSciNetCrossRefGoogle Scholar
  44. King, R., Morgan, B. J. T., Gimenez, O., and Brooks, S. P. (2009), Bayesian Analysis for Population Ecology, Boca Raton: CRC Press. Google Scholar
  45. Langrock, R., King, R., Matthiopulos, J., Thomas, L., Fortin, D., and Morales, J. M. (2012), “Flexible and Practical Modeling of Animal Telemetry Data: Hidden Markov Models and Extensions,” Ecology (Statistical Reports), in press. Google Scholar
  46. Law, R., Purves, D., Murrell, D., and Dieckmann, U. (2001), “Causes and Effects of Small Scale Spatial Structure in Plant Populations,” in Integrating Ecology and Evolution in a Spatial context, eds. J. Silvertown and J. Antonovics, Oxford: Blackwell Science, pp. 21–44. Google Scholar
  47. Li, C., Jiang, Z. G., Li, L. L., Li, Z. Q., Fang, H. X., Li, C. W., and Bauchamp, G. (2012), “Effects of Reproductive Status, Social Rank, Sex and Group Size on Vigilance Patterns in Przewalski’s Gazelle,” PLoS ONE, 7, e32607. doi: 10.1371/journal.pone.0032607. CrossRefGoogle Scholar
  48. Lindgren, F., Lindström, J., and Rue, H. (2011), “An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach (with Discussion),” Journal of the Royal Statistical Society, Series B, 73, 423–498. zbMATHCrossRefGoogle Scholar
  49. McClintock, B. T., King, R., Thomas, L., Matthiopulos, J., McConnell, B. J., and Morales, J. M. (2012), “A General Discrete-Time Modeling Framework for Animal Movement Using Multi-State Random Walks,” Ecological Monographs, 82, 335–349. CrossRefGoogle Scholar
  50. McCrea, R. S., Morgan, B. J. T., Gimenez, O., Besbeas, P., Lebreton, J. D., and Bregnballe, T. (2010), “Multi-site Integrated Population Modelling,” Journal of Agricultural, Biological, and Environmental Statistics, 15, 539–561. MathSciNetCrossRefGoogle Scholar
  51. Møller, J., and Waagepetersen, R. P. (2007), “Modern Statistics for Spatial Point Processes (with Discussion),” Scandinavian Journal of Statistics, 34, 643–711. Google Scholar
  52. Morales, J. M., Haydon, D. T., Frair, J., Holsinger, K. E., and Fryxell, J. M. (2004), “Extracting More out of Relocation Data: Building Movement Models as Mixtures of Random Walks,” Ecology, 89, 2436–2445. CrossRefGoogle Scholar
  53. Nicholson, A. J. (1954), “An Outline of the Dynamics of Animal Populations,” Australian Journal of Zoology, 2, 9–65. CrossRefGoogle Scholar
  54. Reynolds, T. J., King, R., Harwood, J., Frederikesen, M., Harris, M. P., and Wanless, S. (2009), “Integrated Data Analyses in the Presence of Emigration and Tag-Loss,” Journal of Agricultural, Biological, and Environmental Statistics, 14, 411–431. MathSciNetCrossRefGoogle Scholar
  55. Rue, H., Martino, S., and Chopin, N. (2009), “Approximate Bayesian Inference for Latent Gaussian Models Using Integrated Nested Laplace Approximations (with Discussion),” Journal of the Royal Statistical Society, Series B, 71, 319–392. MathSciNetzbMATHCrossRefGoogle Scholar
  56. Schaub, M., Gimenez, O., Sierro, S., and Arlettaz, R. (2007), “Assessing Population Dynamics from Limited Data with Integrated Modeling: Life History of the Endangered Greater Horseshoe Bat,” Conservation Biology, 21, 945–955. CrossRefGoogle Scholar
  57. Stoyan, D., Kendall, W., and Mecke, J. (1995), Stochastic Geometry and Its Applications (2nd ed.), London: Wiley. zbMATHGoogle Scholar
  58. Tanner, M. A., and Wong, W. H. (1987), “The Calculation of Posterior Distribution by Data Augmentation,” Journal of the American Statistical Association, 82, 528–540. MathSciNetzbMATHCrossRefGoogle Scholar
  59. van Lieshout, M. N. M. (2000), Markov Point Processes and Their Applications, London: Imperial College Press. zbMATHCrossRefGoogle Scholar
  60. Wikelski, M., Moxley, J., Eaton-Mordas, A., López-Uribe, M. M., Holland, R., Moskowitz, D., Roubik, D. W., and Kays, R. (2010), “Large-Range Movements of Neotropical Orchid Bees Observed via Radio Telemetry,” PLoS ONE, 5, e10738. CrossRefGoogle Scholar

Copyright information

© International Biometric Society 2012

Authors and Affiliations

  • Ruth King
    • 1
  • Janine B. Illian
    • 1
  • Stuart E. King
    • 1
  • Glenna F. Nightingale
    • 1
  • Ditte K. Hendrichsen
    • 2
  1. 1.School of Mathematics and StatisticsUniversity of St AndrewsSt Andrews, FifeUK
  2. 2.Norwegian Institute For Nature ResearchTrondheimNorway

Personalised recommendations