Kybernetika 47 no. 5, 696-714, 2011

Spatial prediction of the mark of a location-dependent marked point process: how the use of a parametric model may improve prediction.

Tomáš Mrkvička, François Goreaud and Joël Chadœuf


We discuss the prediction of a spatial variable of a multivariate mark composed of both dependent and explanatory variables. The marks are location-dependent and they are attached to a point process. We assume that the marks are assigned independently, conditionally on an unknown underlying parametric field. We compare (i) the classical non-parametric Nadaraya-Watson kernel estimator based on the dependent variable (ii) estimators obtained under an assumption of local parametric model where explanatory variables of the local model are estimated through kernel estimation and (iii) a kernel estimator of the result of the parametric model, supposed here to be a Uniformly Minimum Variance Unbiased Estimator derived under the local parametric model when complete and sufficient statistics are available. The comparison is done asymptotically and by simulations in special cases. The procedure for better estimator selection is then illustrated on a real-life data set.


kernel estimation, marked Poisson process, mean mark estimation, location-dependent mark distribution, segment process


62M30, 62G05


  1. J. Bouchon, Faille, G. Lemée, A. M. Robin and A. Schmitt: Cartes et notice des sols, du peuplement forestier et des groupements végétaux de la réserve biologique de la Tillaie en for\^et de Fontainebleau. University of Orsay 1973.   CrossRef
  2. C. Coudun and J. C. Gegout: Quantitative prediction of the distribution and abundance of Vaccinium myrtillus (L.) with climatic and edaphic factors. J. Vegetation Sci. 18 (2007), 4, 517-524.   CrossRef
  3. D. J. Finney: On the distribution of a variable whose logarithm is normally distributed. J. Roy. Statist. Soc. Ser. B 7 (1941), 155-161.   CrossRef
  4. F. Flénet, P. Villon and F. Ruget: Methodology of adaptation of the STICS model to a new crop: spring linseed (Linum usitatissimum, L.) Agronomie 24 (2004), 6-7, 367-381.   CrossRef
  5. W. H. Green: Econometric Analysis. Prentice Hall, New Jersey 2003.   CrossRef
  6. Ph. Guinier: Foresterie et protection de la nature. L'exemple de Fontainebleau. Rev. Forestière Française II (1950), 703-717.   CrossRef
  7. W. Härdle: Applied Non-parametric Regression. Cambridge University Press, Cambridge 1990.   CrossRef
  8. J. Illian, A. Penttinen, H. Stoyan and D. Stoyan: Statistical Analysis and Modelling of Spatial Point Patterns. Wiley, New York 2008.   CrossRef
  9. J. Kelsall and P. J. Diggle: Kernel estimation of relative risk. Bernoulli 1 (1995), 3-16.   CrossRef
  10. J. Kelsall and P. J. Diggle: Non-parametric estimation of spatial variation in relative risk. Statist. Medicine 14 (1995), 2335-2342.   CrossRef
  11. A. B. Lawson: Statistical Methods in Spatial Epidemiology. Wiley, Chichester 2001.   CrossRef
  12. E. L. Lehmann: Theory of Point Estimation. Wadsworth \et Brooks, California 1991.   CrossRef
  13. T. Mrkvička: Estimation variances for Poisson process of compact sets. Adv. Appl. Prob. (SGSA) 33 (2001), 765-772.   CrossRef
  14. T. Mrkvička: Estimation variances for parameterized marked point processes and for parameterized Poisson segment processes. Comment. Math. Univ. Carolin. 45,1 (2004), 109-117.   CrossRef
  15. T. Mrkvička: Estimation of intersection intensity in Poisson processes of segments. Comment. Math. Univ. Carolin. 48 (2007), 93-106.   CrossRef
  16. T. Mrkvička, S. Soubeyrand and J. Chad{øe}uf: Goodness-of-fit Test of the Mark Distribution in a Point Process with Non-stationary Marks. Research Report 36, Biostatistics and Spatial Processes Research Unit. INRA, Avignon 2009.   CrossRef
  17. N. de Noblet-Ducoudré, S. Gervois, P. Ciais, N. Viovy, N. Brisson, B. Seguin and A. Perrier: Coupling the soil-vegetation-atmosphere-transfer scheme ORCHIDEE to the agronomy model STICS to study the influence of croplands on the european carbon and water budgets. Agronomie 24 (2004), 6-7, 397-407.   CrossRef
  18. A. Penttinen, D. Stoyan and H. Hentonnen: Marked point processes in forests statistics. Forest Sci. 38 (1992), 4, 806-824.   CrossRef
  19. J. Y. Pontailler, A. Faille and G. Lemee: Storms drive successiinal dynamics in natural forests: a case study in Fontainebleau forest (France). Forest Ecology and Management 98 (1997), 1-15.   CrossRef
  20. B. W. Silverman: Density Estimation for Statistics and Data Analysis. Chapman and Hall, London 1986.   CrossRef
  21. D. Stoyan, W. S. Kendall and J. Mecke: Stochastic Geometry and Its Applications. Second edition. John Wiley and Sons, New York 1995.   CrossRef
  22. P. Van Bodegom, P. H. Verburg, A. Stein, S. Adiningsih and H. A. C. Denier Van Der Gon: Effects of interpolation and data resolution on methane emission estimates from rice paddies. Environ. Ecol. Statist. 9 (2002), 5-26.   CrossRef