Extremes Reading Group: Phil Jonathan

Phil Jonathan, Shell Global Solutions

Wednesday 16 October 2013, 1500-1600
Lecture Theatre 1, Fylde College

Non-stationary conditional extremes

Characterising the joint structure of extremes of environmental variables is important for improved understanding of those environments. Yet many applications of multivariate extreme value analysis adopt models that assume a particular form of extremal dependence between variables without justification, or restrict attention to regions in which all variables are extreme. The conditional extremes model of Heffernan and Tawn (2004) provides one approach to avoiding these particular restrictions.

Extremal marginal and dependence characteristics of environmental variables typically vary with covariates. Reliable descriptions of extreme environments should also therefore characterise any non-stationarity. Jonathan et al. (2013) extends the conditional extremes model of Heffernan and Tawn to include covariate effects, using Fourier representations of model parameters for single periodic covariates.

Here, we extend the work of Jonathan et al. (2013), introducing a general-purpose spline representation for model parameters as functions of multidimensional covariates, common to all inference step. Non-crossing quantile regression estimates appropriate non-stationary marginal quantiles simultaneously as functions of covariate; these are necessary as thresholds for extreme value modelling, and for standardisation of marginal distributions prior to application of the conditional extremes model. Marginal extreme value and conditional extremes modelling is performed within a roughness-penalised likelihood framework, with cross-validation to estimate suitable model parameter roughness. A bootstrap re-sampling procedure, encompassing all inferences, quantifies uncertainties in, and dependence structure of, parameter estimates and estimates of conditional extremes of one variate given large values of another.

We validate the approach using simulations from known joint distributions, the extremal dependence structures of which change with covariate. We apply the approach to joint modelling of storm peak significant wave height and associated storm peak period for extra-tropical storms at northern North Sea and South Atlantic Ocean locations, with storm direction as covariate. We evaluate the impact of incorporating directional effects on estimates for conditional return values.