×

zbMATH — the first resource for mathematics

A toolbox for fitting complex spatial point process models using integrated nested Laplace approximation (INLA). (English) Zbl 1257.62093
Summary: This paper develops a methodology that provides a toolbox for routinely fitting complex models to realistic spatial point pattern data. We consider models that are based on log-Gaussian Cox processes and include local interaction in these by considering constructed covariates. This enables us to use integrated nested Laplace approximation and to considerably speed up the inferential task. In addition, methods for model comparison and model assessment facilitate the modelling process. The performance of the approach is assessed in a simulation study. To demonstrate the versatility of the approach, models are fitted to two rather different examples, a large rainforest data set with covariates and a point pattern with multiple marks.

MSC:
62M30 Inference from spatial processes
65C60 Computational problems in statistics (MSC2010)
62P12 Applications of statistics to environmental and related topics
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Baddeley, A., Møller, J. and Waagepetersen, R. (2000). Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Stat. Neerl. 54 329-350. · Zbl 1018.62027 · doi:10.1111/1467-9574.00144
[2] Baddeley, A. and Turner, R. (2000). Practical maximum pseudolikelihood for spatial point patterns (with discussion). Aust. N. Z. J. Stat. 42 283-322. · Zbl 0981.62078 · doi:10.1111/1467-842X.00128
[3] Baddeley, A. J. and Turner, R. (2005). Spatstat: An R package for analyzing spatial point patterns. Journal of Statistical Software 12 1-42.
[4] Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005). Residual analysis for spatial point processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 617-666. · Zbl 1112.62302 · doi:10.1111/j.1467-9868.2005.00519.x
[5] Berman, M. and Turner, R. (1992). Approximating point process likelihoods with GLIM. Applied Statistics 41 31-38. · Zbl 0825.62614 · doi:10.2307/2347614
[6] Besag, J. E. (1977). Contribution to the discussion of Dr. Ripley’s paper. J. Roy. Statist. Soc. Ser. B 39 193-195.
[7] Burslem, D. F. R. P., Garwood, N. C. and Thomas, S. C. (2001). Tropical forest diversity-the plot thickens. Science 291 606-607.
[8] Condit, R. (1998). Tropical Forest Census Plots . Springer and R. G. Landes Company, Berlin and Georgetown, TX.
[9] Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns , 2nd ed. Hodder Arnold, London. · Zbl 1021.62076
[10] Diggle, P. J., Menezes, R. and Su, T.-l. (2010). Geostatistical inference under preferential sampling. J. R. Stat. Soc. Ser. C. Appl. Stat. 59 191-232. · doi:10.1111/j.1467-9876.2009.00701.x
[11] Forchhammer, M. C. and Boomsma, J. (1995). Foraging strategies and seasonal diet optimization of muskoxen in West Greenland. Oecologia 104 169-180.
[12] Forchhammer, M. C. and Boomsma, J. (1998). Optimal mating strategies in nonterritorial ungulates: A general model tested on muskoxen. Behavioural Ecology 9 136-143.
[13] Guan, Y. (2008). On consistent nonparametric intensity estimation for inhomogeneous spatial point processes. J. Amer. Statist. Assoc. 103 1238-1247. · Zbl 1205.62139 · doi:10.1198/016214508000000526
[14] Hanski, I. A. and Gilpin, M. E. (1997). Metapopulation Biology : Ecology , Genetics and Evolution . Academic Press, San Diego. · Zbl 0913.92025
[15] Hardy, O. J. and Vekemans, X. (2002). SPAGEDi: A versatile computer program to analyse spatial genetic structure at the individual or population levels. Molecular Ecology Notes 2 618-620.
[16] Ho, L. P. and Stoyan, D. (2008). Modelling marked point patterns by intensity-marked Cox processes. Statist. Probab. Lett. 78 1194-1199. · Zbl 1237.60040
[17] Huang, F. and Ogata, Y. (1999). Improvements of the maximum pseudo-likelihood estimators in various spatial statistical models. J. Comput. Graph. Statist. 8 519-530.
[18] Hubbell, S. P., Condit, R. and Foster, R. B. (2005). Barro Colorado forest census plot data. Available at . · www.ctfs.si.edu
[19] Hubbell, S. P., Foster, R. B., O’Brien, S. T., Harms, K. E., Condit, R., Wechsler, B., Wright, S. J. and Loo de Lao, S. (1999). Light gap disturbances, recruitment limitation, and tree diversity in a neotropical forest. Science 283 554-557.
[20] Illian, J. B. and Hendrichsen, D. K. (2010). Gibbs point process models with mixed effects. Environmetrics 21 341-353. · doi:10.1002/env.1008
[21] Illian, J. B. and Rue, H. (2010). A toolbox for fitting complex spatial point process models using integrated Laplace transformation (INLA). Technical report, Trondheim Univ. · Zbl 1257.62093
[22] Illian, J. B. and Simpson, D. (2011). Comment on Lindgren et al., an explicit link between Gaussian fields and Gaussian Markov random fields: The SPDE approach. J. Roy. Statist. Soc. Ser. B 73 423-498. · Zbl 1274.62360 · doi:10.1111/j.1467-9868.2011.00777.x
[23] Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns . Wiley, Chichester. · Zbl 1197.62135 · doi:10.1002/9780470725160
[24] Illian, J. B., Sørbye, S. H., Rue, H. and Hendrichsen, D. K. (2012). Fitting a log Gaussian Cox process with temporally varying effects-a case study. Journal of Environmental Statistics .
[25] John, R. C., Dalling, J. W., Harms, K. E., Yavitt, J. B., Stallard, R. F., Mirabello, M., Hubbell, S. P., Valencia, R., Navarrete, H., Vallejo, M. and Foster, R. B. (2007). Soil nutrients influence spatial distributions of tropical tree species. Proc. Natl. Acad. Sci. USA 104 864-869.
[26] Johnson, C. R. and Boerlijst, M. C. (2002). Selection at the level of the community: The importance of spatial structure. Trends in Ecology & Evolution 17 83-90.
[27] Killingback, T. and Doebeli, M. (1996). Spatial evolutionary game theory: Hawks and doves revisited. Proc. R. Soc. Lond. Ser. B 263 1135-1144.
[28] Latimer, A. M., Banerjee, S., Sang, S., Mosher, E. S. and SilanderJr., J. A. (2009). Hierarchical models facilitate spatial analysis of large data sets: A case study on invasive plant species in the northeastern United States. Ecology Letters 12 144-154.
[29] Law, R., Purves, D. W., Murrell, D. J. and Dieckmann, U. (2001). Causes and effects of small scale spatial structure in plant populations. In Integrating Ecology and Evolution in a Spatial Context (J. Silvertown and J. Antonovics, eds.) 21-44. Blackwell, Oxford.
[30] Law, R., Illian, J. B., Burslem, D. F. R. P., Gratzer, G., Gunatilleke, C. V. S. and Gunatilleke, I. A. U. N. (2009). Ecological information from spatial patterns of plants: Insights from point process theory. Journal of Ecology 97 616-628.
[31] Lawson, A. (1992). On fitting non-stationary Markov point process models on GLIM. In Computational Statistics , Volume 1 (Y. Dodge and J. Whittaker, eds.) 35-40. Physica Verlag, Heidelberg.
[32] Lindgren, F., Rue, H. and Lindström, J. (2011). An explicit link between Gaussian fields and Gaussian Markov random fields: The SPDE approach. J. Roy. Statist. Soc. Ser. B 73 423-498. · Zbl 1274.62360 · doi:10.1111/j.1467-9868.2011.00777.x
[33] Lunn, D. J., Thomas, A., Best, N. and Spiegelhalter, D. (2000). WinBUGS-a Bayesian modelling framework: Concepts, structure, and extensibility. Stat. Comput. 10 325-337.
[34] Menezes, R. (2005). Assessing spatial dependency under non-standard sampling. Ph.D. thesis, Universidad de Santiago de Compostela, Santiago de Compostela, Spain.
[35] Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines. Journal of Chemical Physics 6 1087-1092.
[36] Møller, J., Syversveen, A. R. and Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scand. J. Stat. 25 451-482. · Zbl 0931.60038 · doi:10.1111/1467-9469.00115
[37] Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability 100 . Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1044.62101
[38] Møller, J. and Waagepetersen, R. P. (2007). Modern statistics for spatial point processes. Scand. J. Stat. 34 643-711. · Zbl 1157.62067
[39] Moore, B. D., Lawler, I. R., Wallis, I. R., Beale, C. M. and Foley, W. J. (2010). Palatability mapping: A koala’s eye view of spatial variation in habitat quality. Ecology 91 3165-3176. · Zbl 0702.92026 · doi:10.1080/0022250X.1990.9990062
[40] Myllymäki, M. and Penttinen, A. (2009). Conditionally heteroscedastic intensity-dependent marking of log Gaussian Cox processes. Stat. Neerl. 63 450-473. · doi:10.1111/j.1467-9574.2009.00433.x
[41] Naylor, M., Greenhough, J., McCloskey, J., Bell, A. F. and Main, I. G. (2009). Statistical evaluation of characteristic earthquakes in the frequency-magnitude distributions of sumatra and other subduction zone regions. Geophysical Research Letters 36 . · dx.doi.org
[42] Neyman, J. and Scott, E. L. (1952). A theory of the spatial distribution of galaxies. Astrophys. J. 116 144-163. · doi:10.1086/145599
[43] Ogata, Y. (1999). Seismicity analysis through point-process modeling: A review. Pure and Applied Geophysics 155 471-507.
[44] R Development Core Team. (2009). R : A Language and Environment for Statistical Computing . R Foundation for Statistical Computing, Vienna, Austria. Available at . ISBN 3-900051-07-0. · www.R-project.org
[45] Rajala, T. A. and Illian, J. B. (2012). A family of spatial biodiversity measures based on graphs. Environ. Ecol. Stat.
[46] Ripley, B. D. (1976). The second-order analysis of stationary point processes. J. Appl. Probab. 13 255-266. · Zbl 0364.60087 · doi:10.2307/3212829
[47] Rue, H. and Held, L. (2005). Gaussian Markov Random Fields : Theory and Applications. Monographs on Statistics and Applied Probability 104 . Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1093.60003
[48] Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 319-392. · Zbl 1248.62156 · doi:10.1111/j.1467-9868.2008.00700.x
[49] Schoenberg, F. P. (2005). Consistent parametric estimation of the intensity of a spatial-temporal point process. J. Statist. Plann. Inference 128 79-93. · Zbl 1058.62069 · doi:10.1016/j.jspi.2003.09.027
[50] Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 583-639. · Zbl 1067.62010 · doi:10.1111/1467-9868.00353
[51] Steffan-Dewenter, I., Münzenberg, U., Thies, C. and Tscharntke, T. (2002). Scale dependent effects of landscape context on three pollinator guilds. Ecology 83 1421-1432.
[52] Stoyan, D. and Grabarnik, P. (1991). Second-order characteristics for stochastic structures connected with Gibbs point processes. Math. Nachr. 151 95-100. · Zbl 0726.60047 · doi:10.1002/mana.19911510108
[53] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications , 2nd ed. Wiley, London. · Zbl 0838.60002
[54] Strauss, D. J. (1975). A model for clustering. Biometrika 62 467-475. · Zbl 0313.62044 · doi:10.1093/biomet/62.2.467
[55] van Lieshout, M. N. M. (2000). Markov Point Processes and Their Applications . Imperial College Press, London. · Zbl 0968.60005 · www.worldscientific.com
[56] Waagepetersen, R. P. (2007). An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63 252-258, 315. · Zbl 1122.62073 · doi:10.1111/j.1541-0420.2006.00667.x
[57] Waagepetersen, R. and Guan, Y. (2009). Two-step estimation for inhomogeneous spatial point processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 685-702. · Zbl 1250.62047 · doi:10.1111/j.1467-9868.2008.00702.x
[58] Wiegand, T., Gunatilleke, S., Gunatilleke, N. and Okuda, T. (2007). Analysing the spatial structure of a Sri Lankan tree species with multiple scales of clustering. Ecology 88 3088-3012.
[59] Yue, Y. and Loh, J. M. (2011). Bayesian semiparametric intensity estimation for inhomogeneous spatial point processes. Biometrics 67 937-946. · Zbl 1226.62091 · doi:10.1111/j.1541-0420.2010.01531.x
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.