Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 17 2 2010 10.5194/npg-17-237-2010 http://www.nonlin-processes-geophys.net/17/237/2010/ http://www.nonlin-processes-geophys.net/17/237/2010/npg-17-237-2010.html http://www.nonlin-processes-geophys.net/17/237/2010/npg-17-237-2010.pdf 237 244 2010-04-22 Kernel estimation and display of a five-dimensional conditional intensity function G. Adelfio adelfio@unipa.it Dipartimento di Scienze Statistiche e Matematiche "Silvio Vianelli", University of Palermo, Palermo, Italy The aim of this paper is to find a convenient and effective method of displaying some second order properties in a neighbourhood of a selected point of the process. The used techniques are based on very general high-dimensional nonparametric smoothing developed to define a more general version of the conditional intensity function introduced in earlier earthquake studies by Vere-Jones (1978). Adelfio, G.: An analysis of earthquakes clustering based on a second-order diagnostic approach, in: Data Analysis and Classification, Studies in Classification, Data Analysis, and Knowledge Organization, edited by: Palumbo, F., Greenacre, M., and Lauro, C. N., Springer-Verlag, 309–317, 2010. Adelfio, G. and Chiodi, M.: Second-order diagnostics for space-time point processes with application to seismic events, Environmetrics, 20, 895–911, 2009. Adelfio, G., Chiodi, M., De~Luca, L., Luzio, D., and Vitale, M.: Southern-tyrrhenian seismicity in space-time-magnitude domain, Ann. Geophys., 49(6), 1245–1257, 2006. Adelfio, G. and Ogata, Y.: Hybrid kernel estimates of space-time earthquake occurrance rates using the etas model, Ann. I. Stat. Math., 62(1), 127–143, 2010. Adelfio, G. and Schoenberg, F P.: Point process diagnostics based on weighted second-order statistics and their asymptotic properties, Ann. I. Stat. Math, 61(4), 929–948, 2009. Cressie, N.: Statistics for spatial data, Wiley series in probability and mathematical statistics, 1991. Daley, D J. and Vere-Jones, D.: An introduction to the theory of point processes, 2nd edn., Springer-Verlag, New York, 2003. Doguwa, S I.: On second order neighbourhood analysis of mapped point patterns. Biometrical J., 4, 451–457, 1989. Getis, A. and Franklin, J.: Second order neighbourhood analysis of mapped point patterns, Ecology, 68(3), 473–477, 1987. Grillenzoni, C.: Sequenzial kernel estimation of the conditional intensity of nonstationary point processes, Statistical inference for stochastic processes, 9, 135–160, 2006. Harte, D. and Vere-Jones, D.: Differences in coverage between the pde and new zealand local earthquake catalogues, New Zeal. J. Geol. Geop., 42(2), 237–253, 1999. Ogata, Y.: Statistical models for earthquake occurrences and residual analysis for point processes, J. Am. Stat. Assoc., 83(401), 9–27, 1988. Ripley, B D.: The second-order analysis of stationary point processes, J. Appl. Probab., 13(2), 255–266, 1976. Schoenberg, F P.: Testing Separability in Spatial-Temporal Marked Point Processes, Biometrics, 60, 471–481, 2004. Silverman, B W.: Density Estimation for Statistics and Data Analysis, Chapman and Hall, London, 1986. Utsu, T.: A statistical study on the occurrence of aftershocks, Geophys. Mag., 30, 521–605, 1961. Vere-Jones, D.: Space-time correlations for microearthquakes: a pilot study, Adv. Appl. Probab., 10, 73–87, 1978. Vincenty, T.: Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations, Survey review XXII, 176, 89–93, 1975.