A geostatistical extreme-value framework for fast simulation of natural hazard events

We develop a statistical framework for simulating natural hazard events that combines extreme value theory and geostatistics. Robust generalized additive model forms represent generalized Pareto marginal distribution parameters while a Student’s t-process captures spatial dependence and gives a continuous-space framework for natural hazard event simulations. Efficiency of the simulation method allows many years of data (typically over 10 000) to be obtained at relatively little computational cost. This makes the model viable for forming the hazard module of a catastrophe model. We illustrate the framework by simulating maximum wind gusts for European windstorms, which are found to have realistic marginal and spatial properties, and validate well against wind gust measurements.

[1]  P. Green Penalized Likelihood for General Semi-Parametric Regression Models. , 1987 .

[2]  Francesco Pauli,et al.  Penalized likelihood inference in extreme value analyses , 2001 .

[3]  Raphael Huser,et al.  Space–time modelling of extreme events , 2012, 1201.3245.

[4]  Laura C. Dawkins,et al.  The XWS open access catalogue of extreme European windstorms from 1979 to 2012 , 2014 .

[5]  Caroline Keef,et al.  Spatial risk assessment for extreme river flows , 2009 .

[6]  H. Rue,et al.  An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach , 2011 .

[7]  Philip Jonathan,et al.  Threshold modelling of spatially dependent non‐stationary extremes with application to hurricane‐induced wave heights , 2011 .

[8]  Jonathan A. Tawn,et al.  The multivariate Gaussian tail model: an application to oceanographic data , 2000 .

[9]  M. Hulme,et al.  A high-resolution data set of surface climate over global land areas , 2002 .

[10]  Holger Rootzén,et al.  Univariate and bivariate GPD methods for predicting extreme wind storm losses , 2009 .

[11]  C. J. Neumann,et al.  The International Best Track Archive for Climate Stewardship (IBTrACS): unifying tropical cyclone data. , 2010 .

[12]  H. Kunreuther,et al.  Catastrophe modeling : a new approach to managing risk , 2013 .

[13]  Stephen Jewson,et al.  The spatial structure of European wind storms as characterized by bivariate extreme-value Copulas , 2012 .

[14]  Stuart G. Coles,et al.  Spatial Regression Models for Extremes , 1999 .

[15]  S. Padoan,et al.  Likelihood-Based Inference for Max-Stable Processes , 2009, 0902.3060.

[16]  Anders Moberg,et al.  Daily dataset of 20th‐century surface air temperature and precipitation series for the European Climate Assessment , 2002 .

[17]  S. Wood Thin plate regression splines , 2003 .

[18]  R. Koenker Quantile Regression: Name Index , 2005 .

[19]  C. R. Deboor,et al.  A practical guide to splines , 1978 .

[20]  Sidney I. Resnick,et al.  Tail estimates motivated by extreme-value theory , 1984, Advances in Applied Probability.

[21]  Janet E. Heffernan,et al.  Dependence Measures for Extreme Value Analyses , 1999 .

[22]  Kevin I. Hodges,et al.  Feature Tracking on the Unit Sphere , 1995 .

[23]  J. Thepaut,et al.  The ERA‐Interim reanalysis: configuration and performance of the data assimilation system , 2011 .

[24]  K. Born,et al.  Can dynamically downscaled windstorm footprints be improved by observations through a probabilistic approach? , 2014 .

[25]  D. Cox,et al.  An Analysis of Transformations , 1964 .

[26]  Alan Y. Chiang,et al.  Generalized Additive Models: An Introduction With R , 2007, Technometrics.

[27]  A. Ledford,et al.  Statistics for near independence in multivariate extreme values , 1996 .

[28]  G. Wahba Spline models for observational data , 1990 .

[29]  Martin Schlather,et al.  Models for Stationary Max-Stable Random Fields , 2002 .

[30]  Jonathan A. Tawn,et al.  A dependence measure for multivariate and spatial extreme values: Properties and inference , 2003 .

[31]  Richard L. Smith,et al.  MAX-STABLE PROCESSES AND SPATIAL EXTREMES , 2005 .

[32]  Harry Joe,et al.  Composite Likelihood Methods , 2012 .

[33]  P. Diggle,et al.  Model‐based geostatistics , 2007 .

[34]  U. Ulbrich,et al.  A model for the estimation of storm losses and the identification of severe winter storms in Germany , 2003 .

[35]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[36]  Masaaki Sibuya,et al.  Bivariate extreme statistics, I , 1960 .

[37]  A. Frigessi,et al.  A Dynamic Mixture Model for Unsupervised Tail Estimation without Threshold Selection , 2002 .

[38]  S. Coles,et al.  An Introduction to Statistical Modeling of Extreme Values , 2001 .

[39]  T. Opitz,et al.  Extremal tt processes: Elliptical domain of attraction and a spectral representation , 2012, J. Multivar. Anal..

[40]  P. Guttorp,et al.  Geostatistical Space-Time Models, Stationarity, Separability, and Full Symmetry , 2007 .

[41]  R. Tibshirani,et al.  Generalized additive models for medical research , 1986, Statistical methods in medical research.

[42]  Kevin I. Hodges,et al.  Adaptive Constraints for Feature Tracking , 1999 .

[43]  Philippe Naveau,et al.  A statistical rainfall‐runoff mixture model with heavy‐tailed components , 2009 .

[44]  Martin Schlather,et al.  Some covariance models based on normal scale mixtures , 2011 .

[45]  M. Dolores Ugarte,et al.  Statistical Methods for Spatio-temporal Systems , 2006 .

[46]  Thomas Mikosch,et al.  Copulas: Tales and facts , 2006 .

[47]  A. Davison,et al.  Generalized additive modelling of sample extremes , 2005 .

[48]  D. Nychka,et al.  Bayesian Spatial Modeling of Extreme Precipitation Return Levels , 2007 .

[49]  David R. Cox,et al.  A simple spatial-temporal model of rainfall , 1988, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[50]  A. Wood,et al.  Simulation of Stationary Gaussian Processes in [0, 1] d , 1994 .

[51]  Anthony C. Davison,et al.  Spatial modeling of extreme snow depth , 2011, 1111.7091.

[52]  S. Wood Generalized Additive Models: An Introduction with R , 2006 .

[53]  Alan E. Gelfand,et al.  Hierarchical modeling for extreme values observed over space and time , 2009, Environmental and Ecological Statistics.

[54]  Anthony C. Davison,et al.  Geostatistics of extremes , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.