Second-order analysis of marked inhomogeneous spatiotemporal point processes: applications to earthquake data. (English) Zbl 1434.62232
Point processes describe a lot of real events as diseases, fires, earthquake, etc. In this paper point process tools are developed which allow to perform second-order nonparametric analyses of marked spatiotemporal point patterns. Unbiased and consistent estimators for specified statistics are defined and applied to the earthquake data wanting to analyze space-time interactions between large and small earthquakes. The obtained methods are applied to analyze seismic activity in the Andaman Sea region. The paper is organized in six sections where the necessary notions are introduced before defining the marked inhomogeneous spatiotemporal second-order reduced moment measure and obtaining some properties of these statistics. Nonparametric estimators for intensity functions are proposed as well as for new second-order summary statistics. The obtained results are applied to give the second-order analysis of the earthquake data set. The proofs of the results are contained in the Appendix. Also some special cases are analyzed, namely when a marked spatiotemporal point process is stationary, or multivariate and stationary.
Reviewer: Ilie Valuşescu (Bucureşti)
MSC:
| 62P12 | Applications of statistics to environmental and related topics |
| 62H12 | Estimation in multivariate analysis |
| 62H11 | Directional data; spatial statistics |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 62G05 | Nonparametric estimation |
| 86A15 | Seismology (including tsunami modeling), earthquakes |
Keywords:
adaptive intensity estimation; earthquakes; marked inhomogeneous spatiotemporal point process; marked (second-order) intensity-reweighted stationarity; marked spatiotemporal \(K\)-function; marked spatiotemporal second-order reduced moment measure; nonparametric estimatorsReferences:
| [1] | Baddeley, A., Møller, J., & Waagepetersen, R. (2000). Non‐ and semi‐parametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica, 54, 329-350. · Zbl 1018.62027 |
| [2] | Baddeley, A., Rubak, E., & Turner, R. (2015). Spatial point patterns: Methodology and applications with R. London: Chapman and Hall/CRC Press. |
| [3] | Barr, C. D., & Schoenberg, F. P. (2010). On the Voronoi estimator for the intensity of an inhomogeneous planar Poisson process. Biometrika, 97, 977-984. · Zbl 1204.62147 |
| [4] | Chiu, S. N., Stoyan, D., Kendall, W. S., & Mecke, J. (2013). Stochastic geometry and its applications, (3rd ed.). Chichester, England: John Wiley & Sons. · Zbl 1291.60005 |
| [5] | Choi, E., & Hall, P. (1999). Nonparametric approach to analysis of space‐time data on earthquake occurrences. Journal of Computational and Graphical Statistics, 8, 733-748. |
| [6] | Cronie, O., & Särkkä, A. (2011). Some edge correction methods for marked spatio‐temporal point process models. Computational Statistics & Data Analysis, 55, 2209-2220. · Zbl 1328.62548 |
| [7] | Cronie, O., & Van Lieshout, M. N. M. (2015). A J‐function for inhomogeneous spatio‐temporal point processes. Scandinavian Journal of Statistics, 42, 562-579. · Zbl 1364.60063 |
| [8] | Cronie, O., & Van Lieshout, M. N. M. (2016). Summary statistics for inhomogeneous marked point processes. Annals of the Institute of Statistical Mathematics, 68, 905-928. · Zbl 1400.62209 |
| [9] | Cronie, O., & Van Lieshout, M. N. M. (2018). A non‐model‐based approach to bandwidth selection for kernel estimators of spatial intensity functions. Biometrika, 105, 455-462. · Zbl 07072424 |
| [10] | Daley, D. J., & Vere‐Jones, D. (2003). An introduction to the theory of point processes: Volume I: Elementary theory and methods, (2nd ed.). New York, NY: Springer, New York. · Zbl 1026.60061 |
| [11] | Daley, D. J., & Vere‐Jones, D. (2008). An introduction to the theory of point processes: Volume II: General theory and structure, (2nd ed.). New York, NY: Springer, New York. · Zbl 0657.60069 |
| [12] | Diggle, P. (2014). Statistical analysis of spatial and spatio‐temporal point patterns. Boca Raton, FL: CRC Press, Taylor & Francis Group. · Zbl 1435.62004 |
| [13] | Diggle, P. J., Chetwynd, A. G., Häggkvist, R., & Morris, S. E. (1995). Second‐order analysis of space‐time clustering. Statistical Methods in Medical Research, 4, 124-136. |
| [14] | Ferguson, T. S. (1996). A course in large sample theory. Boca Raton, FL: Chapman & Hall/CRC. · Zbl 0871.62002 |
| [15] | Gabriel, E. (2014). Estimating second‐order characteristics of inhomogeneous spatio‐temporal point processes: Influence of edge correction methods and intensity estimates. Methodology and Computing in Applied Probability, 16, 411-431. · Zbl 1308.60061 |
| [16] | Gabriel, E., & Diggle, P. J. (2009). Second‐order analysis of inhomogeneous spatio‐temporal point process data. Statistica Neerlandica, 63, 43-51. · Zbl 07882057 |
| [17] | Haenggi, M. (2012). Stochastic geometry for wireless networks. New York, NY: Cambridge University Press. |
| [18] | Halmos, P. R. (1974). Measure theory. New York, NY: Springer, New York. · Zbl 0283.28001 |
| [19] | Harte, D. (2010). PtProcess: An R package for modelling marked point processes indexed by time. Journal of Statistical Software, 35, 1-32. |
| [20] | Heinrich, L. (2013). Asymptotic methods in statistics of random point processes. Berlin, Germany: Springer. · Zbl 1296.62163 |
| [21] | Illian, J., Penttinen, A., Stoyan, H., & Stoyan, D. (2008). Statistical analysis and modelling of spatial point patterns, Chichester, England: John Wiley & Sons. · Zbl 1197.62135 |
| [22] | Kanamori, H. (1977). The energy release of great earthquakes. Journal of Geophysical Research, 82, 2981-2987. |
| [23] | Lotwick, H. W., & Silverman, B. W. (1982). Methods for analysing spatial processes of several types of points. Journal of the Royal Statistical Society: Series B (Methodological), 44, 406-413. |
| [24] | Marsan, D., & Lengliné, O. (2008). Extending earthquakes’ reach through cascading. Science, 319, 1076-1079. |
| [25] | Møller, J., & Ghorbani, M. (2012). Aspects of second‐order analysis of structured inhomogeneous spatio‐temporal point processes. Statistica Neerlandica, 66, 472-491. |
| [26] | Møller, J., Safavimanesh, F., & Rasmussen, J. G. (2016). The cylindrical K‐function and Poisson line cluster point processes. Biometrika, 103, 937-954. · Zbl 1506.62380 |
| [27] | Møller, J., & Waagepetersen, R. P. (2004). Statistical inference and simulation for spatial point processes. New York, NY: Chapman and Hall/CRC. · Zbl 1044.62101 |
| [28] | Myllymäki, M., Mrkvička, T., Grabarnik, P., Seijo, H., & Hahn, U. (2017). Global envelope tests for spatial processes. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79, 381-404. · Zbl 1414.62404 |
| [29] | Ogata, Y. (1998). Space‐time point‐process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50, 379-402. · Zbl 0947.62061 |
| [30] | Ripley, B. D. (1976). The second‐order analysis of stationary point processes. Journal of Applied Probability, 13, 255-266. · Zbl 0364.60087 |
| [31] | Schneider, R., & Weil, W. (2008). Stochastic and integral geometry. Berlin, Germany: Springer‐Verlag Berlin Heidelberg. · Zbl 1175.60003 |
| [32] | Schoenberg, F. P. (2005). Consistent parametric estimation of the intensity of a spatial-temporal point process. Journal of Statistical Planning and Inference, 128, 79-93. · Zbl 1058.62069 |
| [33] | Schoenberg, F. P., Brillinger, D. R., & Guttorp, P. (2002). Point processes, spatialtemporal. Chichester, UK: John Wiley & Sons, Ltd. |
| [34] | Silverman, B. W. (1986). Density estimation for statistics and data analysis. London, England: Chapman & Hall. · Zbl 0617.62042 |
| [35] | Snyder, J. P. (1987). Map projections — A working manual. Washington, D.C.: United States Government Printing Office. |
| [36] | Stoyan, D., & Stoyan, H. (2000). Improving ratio estimators of second order point process characteristics. Scandinavian Journal of Statistics, 27, 641-656. · Zbl 0963.62089 |
| [37] | USGS. (2012). Earthquake hazards program. Earthquake Glossary. Retrieved from http://pubs.usgs.gov/circ/1324/ |
| [38] | Van Lieshout, M. N. M. (2000). Markov point processes and their applications. London, England: Imperial College Press. · Zbl 0968.60005 |
| [39] | Van Lieshout, M. N. M. (2006). A J‐function for marked point patterns. Annals of the Institute of Statistical Mathematics, 58, 235-259. · Zbl 1097.62094 |
| [40] | Van Lieshout, M. N. M. (2012). On estimation of the intensity function of a point process. Methodology and Computing in Applied Probability, 14, 567-578. · Zbl 1274.60157 |
| [41] | Van Lieshout, M. N. M. (2018). Nonparametric indices of dependence between components for inhomogeneous multivariate random measures and marked sets. Scandinavian Journal of Statistics, 45, 985-1015. · Zbl 1408.62157 |
| [42] | Van Lieshout, M. N. M., & Baddeley, A. J. (1999). Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics, 26, 511-532. · Zbl 0942.62061 |
| [43] | Vere‐Jones, D. (2009). Some models and procedures for space‐time point processes. Environmental and Ecological Statistics, 16, 173-195. |
| [44] | Zhao, J., & Wang, J. (2010). Asymptotic properties of an empirical K‐function for inhomogeneous spatial point processes. Statistics: A Journal of Theoretical and Applied Statistics, 44, 261-267. · Zbl 1282.62052 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
