Abstract
Intensity functions—which describe the spatial distribution of the occurrences of point processes—are useful for risk assessment. This paper deals with the robust nonparametric estimation of the intensity function of space–time data from events such as earthquakes. The basic approach consists of smoothing the frequency histograms with the local polynomial regression (LPR) estimator. This method allows for automatic boundary corrections, and its jump-preserving ability can be improved with robustness. We derive a robust local smoother from the weighted-average approach to M-estimation and we select its bandwidths with robust cross-validation (RCV). Further, we develop a robust recursive algorithm for sequential processing of the data binned in time. An extensive application to the Northern California earthquake catalog in the San Francisco, CA, area illustrates the method and proves its validity.
Similar content being viewed by others
References
Assaid, C.A., Birch, J.B.: Automatic bandwidth selection in robust nonparametric regression. J. Stat. Comput. Simul. 66, 259–272 (2000)
Bailey, T.C., Gatrell, A.C.: Interactive Spatial Data Analysis. Longman, Essex (1995)
Bouezmarni, T., Scaillet, O.: Consistency of asymmetric kernel density estimators and smoothed histograms with application to income data. Econom. Theory 21, 390–412 (2005)
Chaudhuri, P., Marron, J.S.: SiZer for exploration of structures in curves. J. Am. Stat. Assoc. 94, 807–823 (1999)
Chaudhuri, P., Marron, J.S.: Scale space view of curve estimation. Ann. Stat. 28, 408–428 (2000)
Cleveland, W.S.: Robust locally weighted regression and smoothing scatterplots. J. Am. Stat. Assoc. 74, 829–836 (1979)
Cheng, M.-Y., Fan, J., Marron, J.S.: On automatic boundary corrections. Ann. Stat. 25, 1691–1708 (1997)
Choi, E., Hall, P.: Nonparametric approach to the analysis of space–time data on earthquake occurrences. J. Comput. Graph. Stat. 8, 733–748 (1999)
Choi, E., Hall, P.: On the estimation of poles in intensity functions. Biometrika 87, 251–263 (2000)
Chu, C.K., Glad, I., Godtliebsen, F., Marron, J.S.: Edge-preserving smoothers for image processing. J. Am. Stat. Assoc. 93, 526–541 (1998)
Daley, D.A., Vere-Jones, D.: An Introduction to the Theory of Point Processes, vol. 1. Springer, New York (2003)
Fan, J., Gijbels, I.: Local Polynomial Modelling and its Applications. Chapman & Hall, London (1996)
Fan, J., Hu, C.T., Troung, Y.K.: Robust nonparametric function estimation. Scand. J. Stat. 21, 433–446 (1994)
Grillenzoni, C.: Nonparametric regression for nonstationary processes. J. Nonparametric Stat. 12, 265–282 (2000)
Grillenzoni, C.: Nonparametric smoothing of spatio-temporal point processes. J. Stat. Plan. Inference 128, 61–78 (2005)
Hall, P., Jones, M.C.: Adaptive M-estimation in nonparametric regression. Ann. Stat. 18, 1712–1728 (1990)
Hampel, F., Ronchetti, E., Rousseeuw, P., Stahel, W.: Robust Statistics: the Approach Based on Influence Functions. Wiley, New York (1986)
Härdle, W., Gasser, T.: Robust non-parametric function fitting. J. R. Stat. Soc. Ser. B 46, 42–51 (1984)
Härdle, W., Müller, M., Sperlich, S., Werwatz, A.: Nonparametric and Semiparametric Models. Springer, Berlin (2002)
Hillebrand, M., Müller, C.H.: On consistency of redescending M-kernel smoothers. Metrika 63, 71–90 (2006)
Huber, P.J.: Robust Statistics. Wiley, New York (1981)
Leung, D.H.-Y.: Cross-validation in nonparametric regression with outliers. Ann. Stat. 33, 2291–2310 (2005)
Leung, D.H.-Y., Marriott, F.H.C., Wu, E.K.H.: Bandwidth selection in robust smoothing. J. Nonparametric Stat. 2, 333–339 (1993)
Levine, N.: (2007). CrimeStat: a spatial statistics program for the analysis of crime incident locations. National Institute of Justice, Washington DC. Available at http://www.icpsr.umich.edu/CRIMESTAT
Rue, H., Chu, C.-K., Godtliebsen, F., Marron, J.S.: M-smoother with local linear fit. J. Nonparametric Stat. 14, 155–168 (2002)
Simonoff, J.S.: Smoothing Methods in Statistics. Springer, New York (1996)
Stock, C., Smith, E.: Adaptive kernel estimation and continuous probability representation of historical earthquake catalogs. Bull. Seismol. Soc. Am. 92, 901–912 (2002a)
Stock, C., Smith, E.: Comparison between seismicity models generated by different kernel estimations. Bull. Seismol. Soc. Am. 92, 913–922 (2002b)
Vere-Jones, D.: Statistical methods for the description and display of earthquake catalogs. In: Walden, A., Guttorp, P. (eds.) Statistics in the Environmental and Earth Sciences, pp. 220–246. Arnold, London (1992)
Wang, F., Scott, D.: The L1 method for robust nonparametric regression. J. Am. Stat. Assoc. 89, 65–76 (1994)
Zhuang, J., Ogata, Y., Vere-Jones, D.: Stochastic declustering of space-time earthquake occurrences. J. Am. Stat. Assoc. 97, 369–380 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Grillenzoni, C. Robust nonparametric estimation of the intensity function of point data. AStA 92, 117–134 (2008). https://doi.org/10.1007/s10182-008-0065-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10182-008-0065-2