The Annals of Applied Statistics

Voronoi residual analysis of spatial point process models with applications to California earthquake forecasts

Andrew Bray, Ka Wong, Christopher D. Barr, and Frederic Paik Schoenberg

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Many point process models have been proposed for describing and forecasting earthquake occurrences in seismically active zones such as California, but the problem of how best to compare and evaluate the goodness of fit of such models remains open. Existing techniques typically suffer from low power, especially when used for models with very volatile conditional intensities such as those used to describe earthquake clusters. This paper proposes a new residual analysis method for spatial or spatial–temporal point processes involving inspecting the differences between the modeled conditional intensity and the observed number of points over the Voronoi cells generated by the observations. The resulting residuals can be used to construct diagnostic methods of greater statistical power than residuals based on rectangular grids.

Following an evaluation of performance using simulated data, the suggested method is used to compare the Epidemic-Type Aftershock Sequence (ETAS) model to the Hector Mine earthquake catalog. The proposed residuals indicate that the ETAS model with uniform background rate appears to slightly but systematically underpredict seismicity along the fault and to overpredict seismicity in along the periphery of the fault.

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Ann. Appl. Stat., Volume 8, Number 4 (2014), 2247-2267.

First available in Project Euclid: 19 December 2014

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Epidemic-Type Aftershock Sequence models Hector Mine residuals analysis point patterns seismology Voronoi tessellations


Bray, Andrew; Wong, Ka; Barr, Christopher D.; Schoenberg, Frederic Paik. Voronoi residual analysis of spatial point process models with applications to California earthquake forecasts. Ann. Appl. Stat. 8 (2014), no. 4, 2247--2267. doi:10.1214/14-AOAS767.

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  • Adelfio, G. and Schoenberg, F. P. (2009). Point process diagnostics based on weighted second-order statistics and their asymptotic properties. Ann. Inst. Statist. Math. 61 929–948.
  • Baddeley, A., Møller, J. and Pakes, A. G. (2008). Properties of residuals for spatial point processes. Ann. Inst. Statist. Math. 60 627–649.
  • Baddeley, A., Rubak, E. and Møller, J. (2011). Score, pseudo-score and residual diagnostics for spatial point process models. Statist. Sci. 26 613–646.
  • Baddeley, A. and Turner, R. (2000). Practical maximum pseudolikelihood for spatial point patterns (with discussion). Aust. N. Z. J. Stat. 42 283–322.
  • Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005). Residual analysis for spatial point processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 617–666.
  • Barr, C. D. and Diez, D. M. (2012). Sizes of Voronoi regions in a spatial network designed by an inhomogeneous Poisson process. Unpublished manuscript.
  • Barr, C. D. and Schoenberg, F. P. (2010). On the Voronoi estimator for the intensity of an inhomogeneous planar Poisson process. Biometrika 97 977–984.
  • Bray, A. and Schoenberg, F. P. (2013). Assessment of point process models for earthquake forecasting. Statist. Sci. 28 510–520.
  • Clements, R. A., Schoenberg, F. P. and Schorlemmer, D. (2011). Residual analysis methods for space-time point processes with applications to earthquake forecast models in California. Ann. Appl. Stat. 5 2549–2571.
  • Clements, R. A., Schoenberg, F. P. and Veen, A. (2012). Evaluation of space-time point process models using super-thinning. Environmetrics 23 606–616.
  • Czado, C., Gneiting, T. and Held, L. (2009). Predictive model assessment for count data. Biometrics 65 1254–1261.
  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • Dawid, A. P. (1984). Present position and potential developments: Some personal views: Statistical theory: The prequential approach. J. Roy. Statist. Soc. Ser. A 147 278–292.
  • Du, Q., Faber, V. and Gunzburger, M. (1999). Centroidal Voronoi tessellations: Applications and algorithms. SIAM Rev. 41 637–676.
  • Field, E. H. (2007). Overview of the working group for the development of regional earthquake models (RELM). Seismol. Res. Lett. 78 7–16.
  • Guan, Y. (2008). A goodness-of-fit test for inhomogeneous spatial Poisson processes. Biometrika 95 831–845.
  • Helmstetter, A., Kagan, Y. Y. and Jackson, D. D. (2007). High-resolution time-independent grid-based forecast $\mathrm{M}\geq 5$ earthquakes in California. Seismol. Res. Lett. 78 78–86.
  • Helmstetter, A. and Werner, M. (2012). Adaptive spatio-temporal smoothing of seismicity for long-term earthquake forecasts in California. Bull. Seismol. Soc. Amer. 102 2518–2529.
  • Hinde, A. L. and Miles, R. E. (1980). Monte Carlo estimates of the distributions of the random polygons of the Voronoi tessellation with respect to a Poisson process. J. Stat. Comput. Simul. 10 205–223.
  • Jackson, D. D. and Kagan, Y. Y. (1999). Testable earthquake forecasts for 1999. Seismol. Res. Lett. 70 393–403.
  • Johnson, E. A. and Miyanishi, K. (2001). Forest Fire: Behavior and Ecological Effects. Academic Press, San Diego.
  • Jordan, T. H. (2006). Earthquake predictability, brick by brick. Seismol. Res. Lett. 77 3–6.
  • Keeley, J. E., Safford, H., Fotheringham, C. J. et al. (2009). The 2007 Southern California wildfires: Lessons in complexity. J. For. September 287–296.
  • Lawson, A. (1993). A deviance residual for heterogeneous spatial Poisson processes. Biometrics 49 889–897.
  • Lawson, A. (2005). Comment on “Residual analysis for spatial point processes” by A. Baddeley, R. Turner, J. Møller and M. Hazelton. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 654.
  • Malamud, B. D., Millington, J. D. A. and Perry, G. L. W. (2005). Characterizing wildfire regimes in the United States. Proc. Natl. Acad. Sci. USA 102 4694–4699.
  • Massey, F. (1951). The Kolmogorov–Smirnov test for goodness of fit. J. Amer. Statist. Assoc. 42 68–78.
  • Meijering, J. L. (1953). Interface area, edge length, and number of vertices in crystal aggregation with random nucleation. Philips Res. Rep. 8 270–290.
  • Meyer, P. A. (1971). Démonstration simplifée d’un théorème de Knight. In Séminaire de Probabilités V Université de Strasbourg. Lecture Notes in Mathematics 191 191–195. Springer, Heidelberg.
  • Ogata, Y. (1998). Space-time point process models for earthquake occurrences. Ann. Inst. Statist. Math. 50 379–402.
  • Ogata, Y. (2011). Significant improvements of the space-time ETAS model for forecasting of accurate baseline seismicity. Earth, Planets and Space 63 217–229.
  • Ogata, Y., Jones, L. M. and Toda, S. (2003). When and where the aftershock activity was depressed: Contrasting decay patterns of the proximate large earthquakes in southern California. J. Geophys. Res. 108 2318–2329.
  • Okabe, A., Boots, B., Sugihara, K. and Chiu, S. (2000). Spatial Tessellations, 2nd ed. Wiley, Chichester.
  • Rhoades, D. A., Schorlemmer, D., Gerstenberger, M. C. et al. (2011). Efficient testing of earthquake forecasting models. Acta Geophys. 59 728–747.
  • Ripley, B. D. (1976). The second-order analysis of stationary point processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 39 172–212.
  • Schoenberg, F. (1999). Transforming spatial point processes into Poisson processes. Stochastic Process. Appl. 81 155–164.
  • Schoenberg, F. P. (2003). Multidimensional residual analysis of point process models for earthquake occurrences. J. Amer. Statist. Assoc. 98 789–795.
  • Schoenberg, F. P. (2013). Facilitated estimation of ETAS. Bull. Seismol. Soc. Amer. 103 1–7.
  • Schorlemmer, D. and Gerstenberger, M. C. (2007). RELM testing center. Seismol. Res. Lett. 78 30–35.
  • Schorlemmer, D., Gerstenberger, M. C., Weimer, S., Jackson, D. D. and Rhoades, D. A. (2007). Earthquake likelihood model testing. Seismol. Res. Lett. 78 17–27.
  • Schorlemmer, D., Zechar, J. D., Werner, M. J., Field, E. H., Jackson, D. D., Jordan, T. H. and the RELM Working Group (2010). First results of the regional earthquake likelihood models experiment. Pure Appl. Geophys. 167 859–876.
  • Tanemura, M. (2003). Statistical distributions of Poisson Voronoi cells in two and three dimensions. Forma 18 221–247.
  • Thorarinsdottir, T. L. (2013). Calibration diagnostics for point process models via the probability integral transform. Stat 2 150–158.
  • Veen, A. and Schoenberg, F. P. (2006). Assessing spatial point process models for California earthquakes using weighted K-functions: Analysis of California earthquakes. In Case Studies in Spatial Point Process Modeling (A. Baddeley, P. Gregori, J. Mateu, R. Stoica and D. Stoyan, eds.) 293–306. Springer, New York.
  • Xu, H. and Schoenberg, F. P. (2011). Point process modeling of wildfire hazard in Los Angeles County, California. Ann. Appl. Stat. 5 684–704.
  • Zechar, J. D., Gerstenberger, M. C. and Rhoades, D. A. (2010). Likelihood-based tests for evaluating space-rate-magnitude earthquake forecasts. Bull. Seismol. Soc. Amer. 100 1184–1195.
  • Zechar, J. D., Schorlemmer, D., Werner, M. J., Gerstenberger, M. C., Rhoades, D. A. and Jordan, T. H. (2013). Regional earthquake likelihood models I: First-order results. Bull. Seismol. Soc. Amer. 103 787–798.
  • Zhuang, J., Harte, D., Werner, M. J., Hainzl, S. and Zhou, S. (2012). Basic models of seismicity: Temporal models. Community Online Resource for Statistical Seismicity Analysis. DOI:10.5078/corssa-79905851.