Structured Space-Sphere Point Processes and K-Functions

  • Jesper MøllerEmail author
  • Heidi S. Christensen
  • Francisco Cuevas-Pacheco
  • Andreas D. Christoffersen


This paper concerns space-sphere point processes, that is, point processes on the product space of \(\mathbb {R}^{d}\) (the d-dimensional Euclidean space) and \(\mathbb {S}^{k}\) (the k-dimensional sphere). We consider specific classes of models for space-sphere point processes, which are adaptations of existing models for either spherical or spatial point processes. For model checking or fitting, we present the space-sphere K-function which is a natural extension of the inhomogeneous K-function for point processes on \(\mathbb {R}^{d}\) to the case of space-sphere point processes. Under the assumption that the intensity and pair correlation function both have a certain separable structure, the space-sphere K-function is shown to be proportional to the product of the inhomogeneous spatial and spherical K-functions. For the presented space-sphere point process models, we discuss cases where such a separable structure can be obtained. The usefulness of the space-sphere K-function is illustrated for real and simulated datasets with varying dimensions d and k.


First and second order separability Functional summary statistic Log Gaussian Cox process Pair correlation function Shot noise Cox process 

Mathematics Subject Classification (2010)

60G55 62M30 


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The authors are grateful to Jiří Dvořák for helpful comments and to Ali H. Rafati for collecting the pyramidal cell data.


  1. Baddeley A, Møller J, Waagepetersen R (2000) Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Stat Neerl 54:329–350MathSciNetCrossRefzbMATHGoogle Scholar
  2. Baddeley A, Rubak E, Turner R (2015) Spatial point patterns: methodology and applications with r. Chapman & Hall/CRC Press, Boca RatonCrossRefzbMATHGoogle Scholar
  3. Baddeley A, Nair G, Rakshit S, McSwiggan G (2017) “Stationary” point processes are uncommon on linear networks. Stat 6:68–78MathSciNetCrossRefGoogle Scholar
  4. Buxhoeveden DP, Casanova MF (2002) The minicolumn hypothesis in neuroscience. Brain 125:935–951CrossRefGoogle Scholar
  5. Cox DR (1955) Some statistical models related with series of events. J R Stat Soc Ser B 17:129–164zbMATHGoogle Scholar
  6. Daley DJ, Vere-Jones D (2003) An introduction to the theory of point processes volume I: elementary theory and methods, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  7. Diggle P, Chetwynd A, Häggkvist R (1995) Second-order analysis of space-time clustering. Stat Methods Med Res 4:124–136CrossRefGoogle Scholar
  8. Diggle P (2014) Statistical analysis of spatial and spatio-temporal point patterns. Chapman & Hall/CRC Press, Boca RatonzbMATHGoogle Scholar
  9. Dvořák J, Prokešová M (2016) Parameter estimation for inhomogeneous space-time shot-noise Cox point processes. Scand J Stat 43:939–961MathSciNetCrossRefzbMATHGoogle Scholar
  10. Fisher NI, Lewis T, Embleton BJJ (1987) Statistical analysis of spherical data. Cambridge University Press, New YorkCrossRefzbMATHGoogle Scholar
  11. Gabriel E, Diggle P (2009) Second-order analysis of inhomogeneous spatio-temporal point process data. Stat Neerl 63:43–51MathSciNetCrossRefGoogle Scholar
  12. Guan Y (2006) A composite likelihood approach in fitting spatial point process models. J Amer Stat Assoc 101:1502–1512MathSciNetCrossRefzbMATHGoogle Scholar
  13. Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns statistics in practice. Wiley, ChichesterzbMATHGoogle Scholar
  14. Koubek A, Pawlas Z, Brereton T, Kriesche B, Schmidt V (2016) Testing the random field model hypothesis for random marked closed sets. Spat Stat 16:118–136MathSciNetCrossRefGoogle Scholar
  15. Lavancier F, Møller J., Rubak E (2015) Determinantal point process models and statistical inference. J R Stat Soc Ser B 77:853–877MathSciNetCrossRefGoogle Scholar
  16. Lavancier F, Poinas A, Waagepetersen R (2018) Adaptive estimating function inference for non-stationary determinantal point processes Available on arXiv:1806.06231
  17. Lawrence T, Baddeley A, Milne R, Nair G (2016) Point pattern analysis on a region of a sphere. Stat 5:144–157MathSciNetCrossRefGoogle Scholar
  18. Li S (2011) Concise formulas for the area and volume of a hyperspherical cap. Asian J Math Stat 4:66–70MathSciNetCrossRefGoogle Scholar
  19. Lorente de Nó R (1938) The cerebral cortex: architecture, intracortical connections, motor projections. In: Fulton J F (ed) Physiology of the nervous system. Oxford University Press, Oxford, pp 274– 301Google Scholar
  20. Møller J, Syversveen AR, Waagepetersen R (1998) Log Gaussian Cox processes. Scand J Stat 25:451–482MathSciNetCrossRefzbMATHGoogle Scholar
  21. Møller J (2003) Shot noise Cox processes. Adv Appl Probab 35:614–640MathSciNetCrossRefzbMATHGoogle Scholar
  22. Møller J, Waagepetersen R (2007) Modern statistics for spatial point processes. Scand J Stat 34:643–684MathSciNetzbMATHGoogle Scholar
  23. Mountcastle VB (1978) The mindful brain: cortical organization and the group-selective theory of higher brain function. MIT Press, CambridgeGoogle Scholar
  24. Mrkvička T, Myllymäki M, Hahn U (2017) Multiple Monte Carlo testing, with applications in spatial point processes. Stat Comput 27:1239–1255MathSciNetCrossRefzbMATHGoogle Scholar
  25. Mrkvička T, Hahn U, Myllymäki M (2018) A one-way anova test for functional data with graphical interpretation Available on arXiv:1612.03608
  26. Myllymäki M, Mrkvička T, Grabarnik P, Seijo H, Hahn U (2017) Global envelope tests for spatial processes. J R Stat Soc Ser B 79:381–404MathSciNetCrossRefGoogle Scholar
  27. Møller J, Waagepetersen R (2004) Statistical inference and simulation for spatial point processes. Chapman & Hall/CRC Press, Boca RatonzbMATHGoogle Scholar
  28. Møller J, Ghorbani M (2012) Aspects of second-order analysis of structured inhomogeneous spatio-temporal point processes. Stat Neerl 66:472–491MathSciNetCrossRefGoogle Scholar
  29. Møller J, Rubak E (2016) Functional summary statistics for point processes on the sphere with an application to determinantal point processes. Spat Stat 18:4–23MathSciNetCrossRefGoogle Scholar
  30. Møller J, Nielsen M, Porcu E, Rubak E (2018) Determinantal point process models on the sphere. Bernoulli 24:1171–1201MathSciNetCrossRefzbMATHGoogle Scholar
  31. Ohser J (1983) On estimators for the reduced second moment measure of point processes. Math Oper Stat Ser Stat 14:63–71MathSciNetzbMATHGoogle Scholar
  32. Prokešová M, Dvořák J (2014) Statistics for inhomogeneous space-time shot-noise Cox processes. Spat Stat 10:76–86MathSciNetCrossRefzbMATHGoogle Scholar
  33. Waagepetersen R (2007) An estimating function approach to inference for inhomogeneous Neyman–Scott processes. Biometrics 63:252–258MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark

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