Advertisement

Assessing Spatial Point Process Models Using Weighted K-functions: Analysis of California Earthquakes

  • Alejandro Veen
  • Frederic Paik Schoenberg
Part of the Lecture Notes in Statistics book series (LNS, volume 185)

Summary

We investigate the properties of a weighted analogue of Ripley’s Kfunction which was first introduced by Baddeley, Møller, and Waagepetersen. This statistic, called the weighted or inhomogeneous K-function, is useful for assessing the fit of point process models. The advantage of this measure of goodness-of-fit is that it can be used in situations where the null hypothesis is not a stationary Poisson model. We note a correspondence between the weighted K-function and thinned residuals, and derive the asymptotic distribution of the weighted K-function for a spatial inhomogeneous Poisson process. We then present an application of the use of the weighted K-function to assess the goodness-of-fit of a class of point process models for the spatial distribution of earthquakes in Southern California.

Key words

Goodness-of-fit of spatial point process models Inhomogeneity Spatial distribution of earthquakes Weighted Ripley’s K-function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Akaike. A new look at statistical model identification. IEEE Transactions on Automatic Control, AU-19:716–722, 1974.MathSciNetCrossRefGoogle Scholar
  2. [2]
    A.J. Baddeley, J. Møller and R.P. Waagepetersen. Non-and semiparametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica, 54(3):329–350, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A.J. Baddeley and B.W. Silverman. A cautionary example on the use of second-order methods for analyzing point patterns. Biometrics, 40:1089–1093, 1984.MathSciNetGoogle Scholar
  4. [4]
    M. Bartlett. The spectral analysis of two-dimensional point processes. Biometrika, 51:299–311, 1964.MathSciNetCrossRefGoogle Scholar
  5. [5]
    J.E. Besag. Comment on “modelling spatial patterns” by b.d. ripley. Journal of the Royal Statistical Society, Series B, 39:193–195, 1977.MathSciNetGoogle Scholar
  6. [6]
    N.A.C. Cressie. Statistics for spatial data, revised edition. Wiley, New York, 1993.Google Scholar
  7. [7]
    D.J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes, 2nd edition. Springer, New York, 2003.zbMATHGoogle Scholar
  8. [8]
    R. Davies. Testing the hypothesis that a point process is poisson. Adv. Appl. Probab., 9:724–746, 1977.zbMATHCrossRefGoogle Scholar
  9. [9]
    J. Dijkstra, T. Rietjens and F. Steutel. A simple test for uniformity. Statistica Neerlandica, 38:33–44, 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    R. Durret. Probability: Theory and Examples. Wadsworth, Belmont, CA, 1991.Google Scholar
  11. [11]
    L. Heinrich. Asymptotic gaussianity of some estimators for reduced factorial moment measures and product densities of stationary poisson cluster processes. Statistics, 19:87–106, 1988.zbMATHMathSciNetGoogle Scholar
  12. [12]
    L. Heinrich. Goodness-of-fit tests for the second moment function of a stationary multidimensional poisson process. Statistics, 22:245–278, 1991.zbMATHMathSciNetGoogle Scholar
  13. [13]
    Ohser J. and Stoyan D. On the second-order and orientation analysis of planar stationary point processes. Biometrical Journal, 23:523–533, 1981.MathSciNetzbMATHGoogle Scholar
  14. [14]
    A. Lawson. On tests for spatial trend in a non-homogeneous poisson process. Journal of Applied Statistics, 15:225–234, 1988.Google Scholar
  15. [15]
    B. Lisek and M. Lisek. A new method for testing whether a point process is poisson. Statistics, 16:445–450, 1985.MathSciNetzbMATHGoogle Scholar
  16. [16]
    E. Merzbach and Nualart. D. A characterization of the spatial poisson process and changing time. Annals of Probability, 14:1380–1390, 1986.MathSciNetzbMATHGoogle Scholar
  17. [17]
    Y. Ogata. Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83:9–27, 1988.CrossRefGoogle Scholar
  18. [18]
    Y. Ogata. Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50:379–402, 1998.zbMATHCrossRefGoogle Scholar
  19. [19]
    J. Ohser. On estimators for the reduced second-moment measure of point processes. Mathematische Operationsforschung und Statistik series Statistics, 14:63–71, 1983.zbMATHMathSciNetGoogle Scholar
  20. [20]
    C. Peek-Asa, J.F. Kraus, L.B. Bourque, D. Vimalachandra, J. Yu and J. Abrams. Fatal and hospitalized injuries resulting from the 1994 northridge earthquake. International Journal of Epidemiology, 27 (3):459–465, 1998.CrossRefGoogle Scholar
  21. [21]
    B.D Ripley. The second-order analysis of stationary point processes. Journal of Applied Probability, 13:255–266, 1976.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    B.D Ripley. Statistical Inference for Spatial Processes. Cambridge University Press, Cambridge, 1988.Google Scholar
  23. [23]
    J.G. Saw. Tests on intensity of a poisson process. Communications in Statistics, 4(8):777–782, 1975.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    F.P. Schoenberg. Transforming spatial point processes into poisson processes. Stochastic Processes and their Applications, 81:155–164, 1999.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    F.P. Schoenberg. Multi-dimensional residual analysis of point process models for earthquake occurrences. Journal of the American Statistical Association, 98:789–795, 2003.MathSciNetCrossRefGoogle Scholar
  26. [26]
    G. Schwartz. Estimating the dimension of a model. Annals of Statistics, 6:461–464, 1979.Google Scholar
  27. [27]
    B.W. Silverman. Distances on circles, toruses and spheres. Journal of Applied Probability, 15:136–143, 1978.zbMATHMathSciNetCrossRefGoogle Scholar
  28. [28]
    D. Stoyan and H. Stoyan. Improving ratio estimators of second order point process characteristics. Scandinavian Journal of Statistics, 27:641–656, 2000.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Inc. 2006

Authors and Affiliations

  • Alejandro Veen
    • 1
  • Frederic Paik Schoenberg
    • 1
  1. 1.UCLA Department of StatisticsLos AngelesUSA

Personalised recommendations