Understanding Spatial Point Patterns Through Intensity and Conditional Intensities

  • Jean-François CoeurjollyEmail author
  • Frédéric Lavancier
Part of the Lecture Notes in Mathematics book series (LNM, volume 2237)


This chapter deals with spatial statistics applied to point patterns. As well as in many specialized books, spatial point patterns are usually treated elaborately in books devoted to spatial statistics or stochastic geometry. Our aim is to propose a different point of view as we intend to present how we can understand and analyze a point pattern through intensity and conditional intensity functions. We present these key-ingredients theoretically and in an intuitive way. Then, we list some of the main spatial point processes models and provide their main characteristics, in particular, when available the form of their intensity and conditional intensity functions. Finally, we provide a non exhaustive list of statistical methodologies to estimate these functions and discuss the pros and cons of each method.



A part of the material presented here is the fruit of several collaborations. We take the opportunity to thank our main collaborators David Dereudre, Jesper Møller, Ege Rubak and Rasmus Waagepetersen. We are also grateful to Christophe Biscio, Achmad Choiruddin, Rémy Drouilhet, Yongtao Guan and Frédérique Letué. This contribution has been partly supported by the program ANR-11-LABX-0020-01.


  1. 1.
    A. Baddeley, Local composite likelihood for spatial point processes. Spat. Stat. 22, 261–295 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Baddeley, D. Dereudre, Variational estimators for the parameters of Gibbs point process models. Bernoulli 19(3), 905–930 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Baddeley, R. Turner, Practical maximum pseudolikelihood for spatial point patterns. Aust. N. Z. J. Stat. 42(3), 283–322 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Baddeley, J.-F. Coeurjolly, E. Rubak, R. Waagepetersen, Logistic regression for spatial Gibbs point processes. Biometrika 101(2), 377–392 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Baddeley, E. Rubak, R. Turner. Spatial Point Patterns: Methodology and Applications with R (CRC Press, Boca Raton, 2015)CrossRefGoogle Scholar
  6. 6.
    M. Berman, P. Diggle, Estimating weighted integrals of the second-order intensity of a spatial point process. J. R. Stat. Soc. Ser. B 51(1), 81–92 (1989)MathSciNetzbMATHGoogle Scholar
  7. 7.
    M. Berman, R. Turner, Approximating point process likelihoods with GLIM. Appl. Stat. 41, 31–38 (1992)CrossRefGoogle Scholar
  8. 8.
    J. Besag, Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B 36(2) 192–236 (1974).MathSciNetzbMATHGoogle Scholar
  9. 9.
    J.-M. Billiot, J.-F. Coeurjolly, R. Drouilhet, Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes. Electron. J. Stat. 2, 234–264 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    C.A.N. Biscio, J.-F. Coeurjolly, Standard and robust intensity parameter estimation for stationary determinantal point processes. Spat. Stat. 18, 24–39 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    C.A.N. Biscio, F. Lavancier, Quantifying repulsiveness of determinantal point processes. Bernoulli 22(4), 2001–2028 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    S.N. Chiu, D. Stoyan, W. S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, 3rd edn. (Wiley, Chichester, 2013)CrossRefGoogle Scholar
  13. 13.
    A. Choiruddin, J.-F. Coeurjolly, F. Letué, Convex and non-convex regularization methods for spatial point processes intensity estimation. Electron. J. Stat. 12(1), 1210–1255 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Clausel, J.-F. Coeurjolly, J. Lelong, Stein estimation of the intensity of a spatial homogeneous Poisson point process. Ann. Appl. Probab. 26(3), 1495–1534 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    J.-F. Coeurjolly, Median-based estimation of the intensity of a spatial point process. Ann. Inst. Stat. Math. 69, 303–313 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    J.-F. Coeurjolly, R. Drouilhet, Asymptotic properties of the maximum pseudo-likelihood estimator for stationary Gibbs point processes including the Lennard-Jones model. Electron. J. Stat. 4, 677–706 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    J.-F. Coeurjolly, Y. Guan, Covariance of empirical functionals for inhomogeneous spatial point processes when the intensity has a parametric form. Journal of Statistical Planning and Inference 155, 79–92 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    J.-F. Coeurjolly, F. Lavancier, Parametric estimation of pairwise Gibbs point processes with infinite range interaction. Bernoulli 23(2), 1299–1334 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    J.-F. Coeurjolly, J. Møller, Variational approach to estimate the intensity of spatial point processes. Bernoulli 20(3), 1097–1125 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    J.-F. Coeurjolly, N. Morsli, Poisson intensity parameter estimation for stationary Gibbs point processes of finite interaction range.Spat. Stat. 4, 45–56 (2013)CrossRefGoogle Scholar
  21. 21.
    J.-F. Coeurjolly, E. Rubak, Fast covariance estimation for innovations computed from a spatial Gibbs point process. Scand. J. Stat. 40(4), 669–684 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    J.-F. Coeurjolly, D. Dereudre, R. Drouilhet, F. Lavancier, Takacs–Fiksel method for stationary marked Gibbs point processes. Scand. J. Stat. 49(3), 416–443 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    J.-F. Coeurjolly, Y. Guan, M. Khanmohammadi, R. Waagepetersen, Towards optimal takacs–fiksel estimation. Spat. Stat. 18, 396–411 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    J.-F. Coeurjolly, J. Møller, R. Waagepetersen, Palm distributions for log Gaussian Cox processes. Scand. J. Stat. 44(1), 192–203 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    N.A.C. Cressie, Statistics for Spatial Data, 2nd edn. (Wiley, New York, 1993)zbMATHGoogle Scholar
  26. 26.
    N. Cressie, C.K. Wikle, Statistics for Spatio-Temporal Data (Wiley, Hoboken, 2015)zbMATHGoogle Scholar
  27. 27.
    O. Cronie, M.N.M. Van Lieshout, A non-model-based approach to bandwidth selection for kernel estimators of spatial intensity functions. Biometrika 105(2), 455–462 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes: Elementary Theory and Methods, vol. I, 2nd edn. (Springer, New York, 2003).Google Scholar
  29. 29.
    D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes: General Theory and Structure, vol. II, 2nd edn. (Springer, New York, 2008)Google Scholar
  30. 30.
    H.A. David, H.N. Nagaraja, Order Statistics, 3rd edn. (Wiley, Hoboken, 2003)CrossRefGoogle Scholar
  31. 31.
    D. Dereudre, F. Lavancier, Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes. Bernoulli 15(4), 1368–1396 (2009)MathSciNetCrossRefGoogle Scholar
  32. 32.
    D. Dereudre, F. Lavancier, Consistency of likelihood estimation for Gibbs point processes. Ann. Stat. 45(2), 744–770 (2017)MathSciNetCrossRefGoogle Scholar
  33. 33.
    D. Dereudre, F. Lavancier, K. S. Helisová, Estimation of the intensity parameter of the germ-grain Quermass-interaction model when the number of germs is not observed. Scand. J. Stat. 41(3), 809–929 (2014)MathSciNetCrossRefGoogle Scholar
  34. 34.
    P. Diggle, A kernel method for smoothing point process data. J. R. Stat. Soc. Ser. C 34(2), 138–147 (1985)zbMATHGoogle Scholar
  35. 35.
    P. Diggle, Statistical Analysis of Spatial and Spatio-Temporal Point Patterns (CRC Press, Boca Raton, 2013)CrossRefGoogle Scholar
  36. 36.
    P. Diggle, D. Gates, A. Stibbard, A nonparametric estimator for pairwise-interaction point processes. Biometrika 74(4), 763–770 (1987)MathSciNetCrossRefGoogle Scholar
  37. 37.
    T. Fiksel, Estimation of parameterized pair potentials of marked and non-marked Gibbsian point processes. Elektronische Informationsverarbeitung und Kybernetik 20, 270–278 (1984)MathSciNetzbMATHGoogle Scholar
  38. 38.
    H.-O. Georgii, Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys. 48, 31–51 (1976)MathSciNetCrossRefGoogle Scholar
  39. 39.
    C.J. Geyer, J. Møller, Simulation procedures and likelihood inference for spatial point processes. Scand. J. Stat. 21(4), 359–373 (1994)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Y. Guan, Fast block variance estimation procedures for inhomogeneous spatial point processes. Biometrika 96(1), 213–220 (2009)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Y. Guan, J.M. Loh, A thinned block bootstrap procedure for modeling inhomogeneous spatial point patterns. J. Am. Stat. Assoc. 102, 1377–1386 (2007)CrossRefGoogle Scholar
  42. 42.
    Y. Guan, Y. Shen, A weighted estimating equation approach for inhomogeneous spatial point processes. Biometrika 97, 867–880 (2010)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Y. Guan, A. Jalilian, R. Waagepetersen, Quasi-likelihood for spatial point processes. J. R. Stat. Soc. Ser. B 77(3), 677–697 (2015)MathSciNetCrossRefGoogle Scholar
  44. 44.
    L. Heinrich, M. Prokešová, On estimating the asymptotic variance of stationary point processes. Methodol. Comput. Appl. Probab. 12(3), 451–471 (2010)MathSciNetCrossRefGoogle Scholar
  45. 45.
    J. Illian, A. Penttinen, H. Stoyan, D. Stoyan, Statistical Analysis and Modelling of Spatial Point Patterns. Statistics in Practice (Wiley, Chichester, 2008)Google Scholar
  46. 46.
    J.L. Jensen, H.R. Künsch, On asymptotic normality of pseudolikelihood estimates of pairwise interaction processes. Ann. Inst. Stat. Math. 46, 475–486 (1994)zbMATHGoogle Scholar
  47. 47.
    J.L. Jensen, J. Møller, Pseudolikelihood for exponential family models of spatial point processes. Ann. Appl. Probab. 1, 445–461 (1991)MathSciNetCrossRefGoogle Scholar
  48. 48.
    J.F.C. Kingman, Poisson Processes, vol. 3 (Clarendon Press, Oxford, 1992)zbMATHGoogle Scholar
  49. 49.
    F. Lavancier, J. Møller, E. Rubak, Determinantal point process models and statistical inference. J. R. Stat. Soc. Ser. B 77(4), 853–877 (2015)MathSciNetCrossRefGoogle Scholar
  50. 50.
    J.E. Lennard-Jones, On the determination of molecular fields. Proc. R. Soc. Lond. A. 106(738), 463–477 (1924)CrossRefGoogle Scholar
  51. 51.
    O. Macchi, The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7, 83–122 (1975)MathSciNetCrossRefGoogle Scholar
  52. 52.
    J.A.F. Machado, J.M.C. Santos Silva, Quantiles for counts. J. Am. Stat. Assoc. 100(472), 1226–1237 (2005)MathSciNetCrossRefGoogle Scholar
  53. 53.
    S. Mase, Consistency of the maximum pseudo-likelihood estimator of continuous state space Gibbs processes. Ann. Appl. Probab. 5, 603–612 (1995)MathSciNetCrossRefGoogle Scholar
  54. 54.
    S. Mase, Marked Gibbs processes and asymptotic normality of maximum pseudo-likelihood estimators. Math. Nachr. 209, 151–169 (2000)MathSciNetCrossRefGoogle Scholar
  55. 55.
    J. Mateu, F. Montes, Likelihood inference for Gibbs processes in the analysis of spatial point patterns. Int. Stat. Rev. 69(1), 81–104 (2001)CrossRefGoogle Scholar
  56. 56.
    J. Møller, Shot noise Cox processes. Adv. Appl. Probab. 35, 614–640 (2003)MathSciNetCrossRefGoogle Scholar
  57. 57.
    J. Møller, K. Helisová, Likelihood inference for unions of interacting discs. Scand. J. Stat. 37(3), 365–381 (2010)MathSciNetCrossRefGoogle Scholar
  58. 58.
    J. Møller, R.P. Waagepetersen, Statistical Inference and Simulation for Spatial Point Processes (Chapman and Hall/CRC, Boca Raton, 2004).zbMATHGoogle Scholar
  59. 59.
    J. Møller, R.P. Waagepetersen, Modern spatial point process modelling and inference (with discussion). Scand. J. Stat. 34, 643–711 (2007)zbMATHGoogle Scholar
  60. 60.
    J. Møller, R. Waagepetersen, Some recent developments in statistics for spatial point patterns. Ann. Rev. Stat. Appl. 4(1), 317–342 (2017)CrossRefGoogle Scholar
  61. 61.
    J. Møller, A.R. Syversveen, R.P. Waagepetersen, Log Gaussian Cox processes. Scand. J. Stat. 25, 451–482 (1998)MathSciNetCrossRefGoogle Scholar
  62. 62.
    X. Nguyen, H. Zessin, Integral and differential characterizations of Gibbs processes. Math. Nachr. 88, 105–115 (1979)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Y. Ogata, K. Katsura, Maximum likelihood estimates of the fractal dimension for random spatial patterns. Biometrika 78(3), 463–474 (1991)MathSciNetCrossRefGoogle Scholar
  64. 64.
    M. Prokešová, E.B.V. Jensen, Asymptotic Palm likelihood theory for stationary point processes. Ann. Inst. Stat. Math. 65(2), 387–412 (2013)MathSciNetCrossRefGoogle Scholar
  65. 65.
    M. Prokešová, J. Dvořák, E. Jensen, Two-step estimation procedures for inhomogeneous shot-noise Cox processes. Ann. Inst. Stat. Math. 69(3), 513–542 (2017)MathSciNetCrossRefGoogle Scholar
  66. 66.
    S.L. Rathbun, N. Cressie, Asymptotic properties of estimators for the parameters of spatial inhomogeneous Poisson point processes. Adv. Appl. Probab. 26, 122–154 (1994)MathSciNetCrossRefGoogle Scholar
  67. 67.
    B. Ripley, Statistical Inference for Spatial Processes (Cambridge University Press, Cambridge, 1988)CrossRefGoogle Scholar
  68. 68.
    M. Rosenblatt, Remarks on some nonparametric estimates of a density function. Ann. Math. Stat. 27(3), 832–837 (1956)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Z. Sasvári, Multivariate Characteristic and Correlation Functions, vol. 50 (Walter de Gruyter, Berlin, 2013)CrossRefGoogle Scholar
  70. 70.
    F.P. Schoenberg, Consistent parametric estimation of the intensity of a spatial-temporal point process. J. Stat. Plann. Inference 128, 79–93 (2005)MathSciNetCrossRefGoogle Scholar
  71. 71.
    T. Shirai, Y. Takahashi, Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and Boson point processes. J. Funct. Anal. 2, 414–463 (2003)MathSciNetCrossRefGoogle Scholar
  72. 72.
    J.-L. Starck, F. Murtagh, Astronomical Image and Data Analysis (Springer, Berlin, 2007)zbMATHGoogle Scholar
  73. 73.
    R. Takacs, Estimator for the pair-potential of a Gibbsian point process. Math. Oper. Stat. Ser. Stat. 17, 429–433 (1986)MathSciNetzbMATHGoogle Scholar
  74. 74.
    U. Tanaka, Y. Ogata, D. Stoyan, Parameter estimation for Neyman-Scott processes. Biom. J. 50, 43–57 (2008)MathSciNetCrossRefGoogle Scholar
  75. 75.
    A.L. Thurman, R. Fu, Y. Guan, J. Zhu, Regularized estimating equations for model selection of clustered spatial point processes. Stat. Sin. 25(1), 173–188 (2015)MathSciNetzbMATHGoogle Scholar
  76. 76.
    M.N.M. Van Lieshout, Markov Point Processes and Their Applications (Imperial College Press, London, 2000)CrossRefGoogle Scholar
  77. 77.
    M.N.M. Van Lieshout, On estimation of the intensity function of a point process. Methodol. Comput. Appl. Probab. 14(3), 567–578 (2012)MathSciNetCrossRefGoogle Scholar
  78. 78.
    R. Waagepetersen, An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258 (2007)MathSciNetCrossRefGoogle Scholar
  79. 79.
    R. Waagepetersen, Estimating functions for inhomogeneous spatial point processes with incomplete covariate data. Biometrika 95(2), 351–363 (2008)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jean-François Coeurjolly
    • 1
    Email author
  • Frédéric Lavancier
    • 2
  1. 1.Université du Québec à MontréalDépartement de MathématiquesMontréalCanada
  2. 2.Université de NantesLaboratoire de Mathématiques Jean LerayNantesFrance

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