Statistical modelling of extreme ocean environments for marine design: A review

We review aspects of extreme value modelling relevant to characterisation of ocean environments and the design of marine structures, summarising basic concepts, modelling with covariates and multivariate modelling (including conditional and spatial extremes). We outline Bayesian inference for extremes and reference software resources for extreme value modelling. Extreme value analysis is inherently dierent to other empirical modelling, in that estimating the tail (rather than the body) of a distribution from a sample of data, and extrapolation beyond the sample (rather than interpolation within) is demanded. Intuition accumulated from other areas of empirical modelling can be misleading. Careful consideration of the eects of sample size, measurement scale, threshold selection and serial dependence, associated uncertainties and implications of choices made is essential. Incorporation of covariate eects when necessary improves inference. Suitable tools (e.g. based on additive models, splines, random elds, spatial processes)

[1]  Philippe Naveau,et al.  Stochastic downscaling of precipitation: From dry events to heavy rainfalls , 2007 .

[2]  H. Rue,et al.  Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations , 2009 .

[3]  Kamil Feridun Turkman,et al.  Asymptotic models and inference for extremes of spatio-temporal data , 2010 .

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

[5]  Nigel G. Yoccoz,et al.  Hierarchical Modelling for the Environmental Sciences , 2007 .

[6]  Lee Fawcett,et al.  Improved estimation for temporally clustered extremes , 2007 .

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

[8]  L. de Haan,et al.  Stationary min-stable stochastic processes , 1986 .

[9]  Dani Gamerman,et al.  Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition , 2006 .

[10]  Fangjun Li,et al.  A comparison of extreme wave analysis methods with 1994–2010 offshore Perth dataset , 2012 .

[11]  P. J. Green,et al.  Probability and Statistical Inference , 1978 .

[12]  Inigo J. Losada,et al.  Estimation of the long‐term variability of extreme significant wave height using a time‐dependent Peak Over Threshold (POT) model , 2006 .

[13]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[14]  Malcolm R Leadbetter,et al.  On a basis for 'Peaks over Threshold' modeling , 1991 .

[15]  Philip Jonathan,et al.  Uncertainty due to choice of measurement scale in extreme value modelling of North Sea storm severity , 2012 .

[16]  Darmesah Gabda,et al.  Discussion of “Statistical Modeling of Spatial Extremes” by A. C. Davison, S. A. Padoan and M. Ribatet , 2012 .

[17]  Malik Chami,et al.  Impact of sea‐surface dust radiative forcing on the oceanic primary production: A 1D modeling approach applied to the West African coastal waters , 2009 .

[18]  J. Corcoran.  Modelling Extremal Events for Insurance and Finance , 2002 .

[19]  Philip Jonathan,et al.  Modeling the Seasonality of Extreme Waves in the Gulf of Mexico , 2011 .

[20]  Philip Jonathan,et al.  On the Estimation of Ocean Engineering Design Contours , 2014 .

[21]  Jonathan A. Tawn,et al.  A conditional approach for multivariate extreme values (with discussion) , 2004 .

[22]  L. de Haan,et al.  A Spectral Representation for Max-stable Processes , 1984 .

[23]  N. H. Bingham,et al.  Regular variation in more general settings , 1987 .

[24]  C. Guedes Soares,et al.  Prediction of extreme significant wave heights using maximum entropy , 2013 .

[25]  A. C. Davison,et al.  Statistical models: Name Index , 2003 .

[26]  D. Cox,et al.  A note on pseudolikelihood constructed from marginal densities , 2004 .

[27]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[28]  A. Davison,et al.  Statistical Modeling of Spatial Extremes , 2012, 1208.3378.

[29]  A. Davison,et al.  Bayesian Inference from Composite Likelihoods, with an Application to Spatial Extremes , 2009, 0911.5357.

[30]  Janet E. Heffernan,et al.  A conditional approach for multivariate extreme values , 2004 .

[31]  C. Wild,et al.  Vector Generalized Additive Models , 1996 .

[32]  S. Coles,et al.  Modelling Extreme Multivariate Events , 1991 .

[33]  Johan Segers,et al.  Inference for clusters of extreme values , 2003 .

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

[35]  C. Varin.  On composite marginal likelihoods , 2008 .

[36]  Richard L. Smith,et al.  Models for exceedances over high thresholds , 1990 .

[37]  Kevin Ewans,et al.  Statistical estimation of extreme ocean environments: The requirement for modelling directionality and other covariate effects , 2008 .

[38]  Martin Schlather,et al.  Examples for the coefficient of tail dependence and the domain of attraction of a bivariate extreme value distribution , 2001 .

[39]  Philip Jonathan,et al.  For Peer Review Threshold modelling of spatially-dependent non-stationary extremes with application to hurricane-induced wave heights , 2011 .

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

[41]  F. Mallor,et al.  An introduction to statistical modelling of extreme values. Application to calculate extreme wind speeds , 2009 .

[42]  Paul H. C. Eilers,et al.  Direct generalized additive modeling with penalized likelihood , 1998 .

[43]  Richard A. Davis,et al.  COMMENTS ON PAIRWISE LIKELIHOOD IN TIME SERIES MODELS , 2011 .

[44]  Jonathan A. Tawn,et al.  Bivariate extreme value theory: Models and estimation , 1988 .

[45]  H. Künsch.  Gaussian Markov random fields , 1979 .

[46]  N. Cook,et al.  Extreme wind speeds in mixed climates revisited , 2003 .

[47]  C. Guedes Soares,et al.  Modelling bivariate distributions of significant wave height and mean wave period. , 2002 .

[48]  S. E. Ahmed,et al.  Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference , 2008, Technometrics.

[49]  J. L. Wadsworth,et al.  Accounting for choice of measurement scale in extreme value modeling , 2010, 1011.3612.

[50]  Philip Jonathan,et al.  The Effect of Directionality on Northern North Sea Extreme Wave Design Criteria , 2008 .

[51]  James S. Clark,et al.  Hierarchical Modelling for the Environmental Sciences: Statistical Methods and Applications , 2006 .

[52]  J. Rueedi,et al.  Constraining the age distribution of highly mixed groundwater using 39Ar: A multiple environmental tracer (3H/3He, 85Kr, 39Ar, and 14C) study in the semiconfined Fontainebleau Sands Aquifer (France) , 2007 .

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

[54]  Jan Picek,et al.  Estimating extremes in climate change simulations using the peaks-over-threshold method with a non-stationary threshold , 2010 .

[55]  Philip Jonathan,et al.  Evaluating environmental joint extremes for the offshore industry , 2012, 1211.1365.

[56]  S. Nadarajah,et al.  Extreme Value Distributions: Theory and Applications , 2000 .

[57]  Emil Simiu,et al.  Estimating Uncertainty in the Extreme Value Analysis of Data Generated by a Hurricane Simulation Model , 2003 .

[58]  Jonathan A. Tawn,et al.  Statistics for Extreme Sea Currents , 1997 .

[59]  Philip Jonathan,et al.  A Spatiodirectional Model for Extreme Waves in the Gulf of Mexico , 2011 .

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

[61]  Philip Jonathan,et al.  Joint modelling of wave spectral parameters for extreme sea states , 2010 .

[62]  Richard E. Chandler,et al.  On the use of generalized linear models for interpreting climate variability , 2005 .

[63]  George Z. Forristall,et al.  On the Use of Directional Wave Criteria , 2004 .

[64]  Richard L. Smith,et al.  Estimating the Extremal Index , 1994 .

[65]  J. Tawn.  Modelling multivariate extreme value distributions , 1990 .

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

[67]  Eric Gilleland,et al.  Software for the analysis of extreme events: The current state and future directions , 2005 .

[68]  C. Guedes Soares,et al.  Bayesian inference for long-term prediction of significant wave height , 2007 .

[69]  J. Kyselý,et al.  Comparison of regional and at‐site approaches to modelling probabilities of heavy precipitation , 2011 .

[70]  G. Russell,et al.  A flexible extreme value mixture model , 2011, Comput. Stat. Data Anal..

[71]  Philip Jonathan,et al.  Uncertainties in Extreme Wave Height Estimates for Hurricane-Dominated Regions , 2007 .

[72]  Jonathan A. Tawn,et al.  Modelling non‐stationary extremes with application to surface level ozone , 2009 .

[73]  Philip Jonathan,et al.  Statistical measures of extremal dependence illustrated using measured sea surface elevations from a neighbourhood of coastal locations , 2013 .

[74]  Richard E. Chandler,et al.  Inference for clustered data using the independence loglikelihood , 2007 .

[75]  A. Ledford,et al.  Diagnostics for dependence within time series extremes , 2003 .

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

[77]  Douglas A. Mitchell,et al.  Grid Point Pooling in Extreme Value Analysis of Hurricane Hindcast Data , 2009 .

[78]  J. Tawn,et al.  Modelling Extreme-Value Dependence in International Stock Markets , 2002 .

[79]  Stuart Coles,et al.  Directional Modelling of Extreme Wind Speeds , 1994 .

[80]  S. Kotz,et al.  Multivariate Extreme‐Value Theory , 2014 .

[81]  Nicholas J. Cook,et al.  Towards better estimation of extreme winds , 1982 .

[82]  Philip Jonathan,et al.  Extreme Value Analysis of North Sea Storm Severity , 1998 .

[83]  Stuart G. Coles,et al.  Bayesian methods in extreme value modelling: a review and new developments. , 1996 .

[84]  Jonathan A. Tawn,et al.  Modelling Dependence within Joint Tail Regions , 1997 .

[85]  B. Silverman,et al.  Nonparametric regression and generalized linear models , 1994 .

[86]  A. O'Hagan,et al.  Accounting for threshold uncertainty in extreme value estimation , 2006 .

[87]  C. Guedes Soares,et al.  Modelling the long-term time series of significant wave height with non-linear threshold models , 2000 .

[88]  Philip Jonathan,et al.  The effect of directionality on extreme wave design criteria , 2007 .

[89]  J. Teugels,et al.  Statistics of Extremes , 2004 .

[90]  Kurt Hornik,et al.  The Comprehensive R Archive Network , 2012 .

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

[92]  Richard L. Smith,et al.  Approximate Bayesian computing for spatial extremes , 2011, Comput. Stat. Data Anal..

[93]  D. Cooley,et al.  Discussion of "Statistical Modeling of Spatial Extremes" by A. C. Davison, S. A. Padoan and M. Ribatet , 2012 .

[94]  Nicholas J. Cook,et al.  Exact and general FT1 penultimate distributions of extreme wind speeds drawn from tail-equivalent Weibull parents , 2004 .

[95]  Rosemary A. Roberts,et al.  Probability and Statistical Inference: Volume 2: Statistical Inference , 2011 .

[96]  Jonathan A. Tawn,et al.  Dependence modelling for spatial extremes , 2012 .

[97]  G. Wahba,et al.  Bootstrap confidence intervals for smoothing splines and their comparison to bayesian confidence intervals , 1995 .

[98]  Y. Pawitan.  In all likelihood : statistical modelling and inference using likelihood , 2002 .

[99]  J. Pickands.  Statistical Inference Using Extreme Order Statistics , 1975 .

[100]  G. J. G. Upton,et al.  An Introduction to Statistical Modelling , 1983 .

[101]  Harry Joe,et al.  Families of min-stable multivariate exponential and multivariate extreme value distributions , 1990 .

[102]  J. Vrijling,et al.  The estimation of extreme quantiles of wind velocity using L-moments in the peaks-over-threshold approach , 2001 .

[103]  Melisa Menéndez,et al.  Seasonality and duration in extreme value distributions of significant wave height , 2008 .

[104]  Sverre Haver,et al.  Wave climate off northern Norway , 1985 .

[105]  J. Hüsler,et al.  Laws of Small Numbers: Extremes and Rare Events , 1994 .

[106]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[107]  Julian Stander,et al.  Automated threshold selection methods for extreme wave analysis , 2009 .

[108]  T. Yee,et al.  Vector generalized linear and additive extreme value models , 2007 .

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

[110]  J. G. Kalbfleisch.  Probability and Statistical Inference , 1977 .

[111]  J. Hosking.  L‐Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics , 1990 .

[112]  Philip Jonathan,et al.  Modelling the Seasonality of Extreme Waves in the Gulf of Mexico , 2008 .

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

[114]  Tim Ramsay,et al.  Spline smoothing over difficult regions , 2002 .

[115]  Paul-Antoine Michelangeli,et al.  Probabilistic downscaling approaches: Application to wind cumulative distribution functions , 2009 .

[116]  M. Stone.  Cross‐Validatory Choice and Assessment of Statistical Predictions , 1976 .

[117]  Philip Jonathan,et al.  Joint modelling of vertical profiles of large ocean currents , 2012 .

[118]  H. Ritchie,et al.  Mapping the return periods of extreme sea levels: Allowing for short sea level records, seasonality, and climate change , 2007 .

[119]  Wei Liu,et al.  Bivariate maximum entropy distribution of significant wave height and peak period , 2013 .