Advertisement

Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 517–543 | Cite as

A quadrat neighborhood estimator for intensity function of point processes

  • Azam Dehghani
  • Mohammad Q. Vahidi-AslEmail author
Original Research
  • 21 Downloads

Abstract

In nonparametric estimation of the intensity function of a point process, assigning the local event weight is particularly important. This paper describes a sequential quadrat partitioning of the study region to define a quadrat neighborhood of a point. Based on this idea, a quadrat neighborhood estimator of intensity function is introduced. We extend this method to estimate the product density. Meanwhile, we show that under infill asymptotics our proposed estimator is asymptotically unbiased for inhomogeneous Poisson point process. Simulations are also used to investigate the performance of our proposed estimator.

Keywords

Nonparametric estimation Adaptive estimation Point process Intensity function Product density 

Mathematics Subject Classification

62G05 62G99 60G55 

References

  1. 1.
    Baddeley, A., Chang, Y., Song, Y., Turner, R.: Nonparametric estimation of the dependence of a spatial point process on spatial covariates. Stat. Interface 5(2), 221–236 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berman, M., Diggle, P.: Estimating weighted integrals of the second-order intensity of a spatial point process. J. R. Stat. Soc. Ser. B. Stat. Methodol. 51, 81–92 (1989)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bernhardt, R., Meyer-Olbersleben, F., Kieback, B.: Fundamental investigation on the preparation of gradient structures by sedimentation of different powder fractions under gravity. In: Proceedings of the 4th International Conference on Composite Engineering, pp. 147–148 (1997)Google Scholar
  4. 4.
    Cressie, N.A.C.: Statistics for Spatial Data. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. A Wiley-Interscience Publication. Wiley, New York (1993)zbMATHGoogle Scholar
  5. 5.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2nd edn. Springer, New York (2003)zbMATHGoogle Scholar
  6. 6.
    Davies, T.M., Hazelton, M.L., Marshall, J.C.: Sparr: analyzing spatial relative risk using fixed and adaptive kernel density estimation in R. J. Stat. Softw. 39(i01), 1–14 (2011)Google Scholar
  7. 7.
    Diggle, P.: A Kernel method for smoothing point process data. Appl. Stat. 34, 138–147 (1985)CrossRefzbMATHGoogle Scholar
  8. 8.
    Diggle, P.: Statistical Analysis of Spatial Analysis of Spatial Point Patterns. Ox-ford University Press, NewYork (2003)zbMATHGoogle Scholar
  9. 9.
    Duong, T., Hazelton, M.L.: Plug-in bandwidth selectors for bivariate kernel density estimation. J. Nonparametric Stat. 15, 17–30 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guan, Y.: On consistent nonparametric intensity estimation for inhomogeneous spatial point process. J. Am. Stat. Assoc. 103, 1238–1247 (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hahn, U., Micheletti, A., Pohlink, R., Stoyan, D., Wendrock, H.: Stereological analysis and modelling of gradient structures. J. Microsc. 195(2), 113–124 (1999)CrossRefGoogle Scholar
  12. 12.
    Hahn, U., Jensen, E.B.V., Van Lieshout, M.N.M., Nielsen, L.S.: Inhomogeneous spatial point processes by location-dependent scaling. Adv. Appl. Probab. 35(2), 319–336 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Møller, J., Waagepetersen, R.P.: Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall-CRC, Boca Raton (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Møller, J., Waagepetersen, R.P.: Modern statistics for spatial point processes. Scand. Stat. Theory Appl. 34(4), 643–684 (2007)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Schaap, W.E.: DTFE: the Delaunay tessellation field estimator. Dissertation, University of Groningen (2007)Google Scholar
  16. 16.
    Schaap, W.E., Van de Waygeart, R.: Continuous fields and discrete samples: reconstruction through Delaunay tessellations. Astron. Astrophys. 363, L29–L32 (2000)Google Scholar
  17. 17.
    Stein, M.L.: Interpolation of Spatial Data. Some Theory for Kriging. Springer Series in Statistics. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  18. 18.
    Scott, D.W.: Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, New York (2015)zbMATHGoogle Scholar
  19. 19.
    van Lieshout, M.N.M.: On estimation of the intensity function of a point process. Methodol. Comput. Appl. Probab. 14(3), 567–578 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of StatisticsShahid Beheshti UniversityEvin, TehranIran

Personalised recommendations