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.
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References
H. Akaike. A new look at statistical model identification. IEEE Transactions on Automatic Control, AU-19:716–722, 1974.
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.
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.
M. Bartlett. The spectral analysis of two-dimensional point processes. Biometrika, 51:299–311, 1964.
J.E. Besag. Comment on “modelling spatial patterns” by b.d. ripley. Journal of the Royal Statistical Society, Series B, 39:193–195, 1977.
N.A.C. Cressie. Statistics for spatial data, revised edition. Wiley, New York, 1993.
D.J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes, 2nd edition. Springer, New York, 2003.
R. Davies. Testing the hypothesis that a point process is poisson. Adv. Appl. Probab., 9:724–746, 1977.
J. Dijkstra, T. Rietjens and F. Steutel. A simple test for uniformity. Statistica Neerlandica, 38:33–44, 1984.
R. Durret. Probability: Theory and Examples. Wadsworth, Belmont, CA, 1991.
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.
L. Heinrich. Goodness-of-fit tests for the second moment function of a stationary multidimensional poisson process. Statistics, 22:245–278, 1991.
Ohser J. and Stoyan D. On the second-order and orientation analysis of planar stationary point processes. Biometrical Journal, 23:523–533, 1981.
A. Lawson. On tests for spatial trend in a non-homogeneous poisson process. Journal of Applied Statistics, 15:225–234, 1988.
B. Lisek and M. Lisek. A new method for testing whether a point process is poisson. Statistics, 16:445–450, 1985.
E. Merzbach and Nualart. D. A characterization of the spatial poisson process and changing time. Annals of Probability, 14:1380–1390, 1986.
Y. Ogata. Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83:9–27, 1988.
Y. Ogata. Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50:379–402, 1998.
J. Ohser. On estimators for the reduced second-moment measure of point processes. Mathematische Operationsforschung und Statistik series Statistics, 14:63–71, 1983.
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.
B.D Ripley. The second-order analysis of stationary point processes. Journal of Applied Probability, 13:255–266, 1976.
B.D Ripley. Statistical Inference for Spatial Processes. Cambridge University Press, Cambridge, 1988.
J.G. Saw. Tests on intensity of a poisson process. Communications in Statistics, 4(8):777–782, 1975.
F.P. Schoenberg. Transforming spatial point processes into poisson processes. Stochastic Processes and their Applications, 81:155–164, 1999.
F.P. Schoenberg. Multi-dimensional residual analysis of point process models for earthquake occurrences. Journal of the American Statistical Association, 98:789–795, 2003.
G. Schwartz. Estimating the dimension of a model. Annals of Statistics, 6:461–464, 1979.
B.W. Silverman. Distances on circles, toruses and spheres. Journal of Applied Probability, 15:136–143, 1978.
D. Stoyan and H. Stoyan. Improving ratio estimators of second order point process characteristics. Scandinavian Journal of Statistics, 27:641–656, 2000.
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Veen, A., Schoenberg, F.P. (2006). Assessing Spatial Point Process Models Using Weighted K-functions: Analysis of California Earthquakes. In: Baddeley, A., Gregori, P., Mateu, J., Stoica, R., Stoyan, D. (eds) Case Studies in Spatial Point Process Modeling. Lecture Notes in Statistics, vol 185. Springer, New York, NY. https://doi.org/10.1007/0-387-31144-0_16
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DOI: https://doi.org/10.1007/0-387-31144-0_16
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