A Poisson process reparameterisation for Bayesian inference for extremes

A common approach to modelling extreme values is to consider the excesses above a high threshold as realisations of a non-homogeneous Poisson process. While this method offers the advantage of modelling using threshold-invariant extreme value parameters, the dependence between these parameters makes estimation more dicult. We present a novel approach for Bayesian estimation of the Poisson process model parameters by reparameterising in terms of a tuning parameter m. This paper presents a method for choosing the optimal value of m that near-orthogonalises the parameters, which is achieved by minimising the correlation between the asymptotic posterior distribution of the parameters. This choice of m ensures more rapid convergence and ecient sampling from the joint posterior distribution using Markov Chain Monte Carlo methods. Samples from the parameterisation of interest are then obtained by a simple transform. Results are presented in the cases of identically and non-identically distributed models for extreme rainfall in Cumbria, UK

Modelling heatwaves in central France:a case-study in extremal dependence

Heatwaves are phenomena that have large social and economic consequences. Understanding and estimating the frequency of such events are of great importance to climate scientists and decision makers. Heatwaves are a type of extreme event which are by definition rare and as such there are few data in the historical record to help planners. Extreme value theory is a general framework from which inference can be drawn from extreme events. When modelling heatwaves it is important to take into account the intensity and duration of events above a critical level as well as the interaction between both factors. Most previous methods assume that the duration distribution is independent of the critical level that is used to define a heatwave: a shortcoming that can lead to incorrect inferences. The paper characterizes a novel method for analysing the temporal dependence of heatwaves with reference to observed temperatures from Orleans in central France. This method enables estimation of the probabilities for heatwave events irrespectively of whether the duration distribution is independent of the critical level. The methods are demonstrated by estimating the probability of an event more severe than the 2003 European heatwave or an event that causes a specified increase in mortality

Assessing extremal dependence of North Sea storm severity

Extreme value theory provides asymptotically motivated methods for modelling the occurrences of extreme values of oceanographic variables, such as significant wave height at a single location. Problems are often multivariate or spatial in nature, where interest lies in the risk associated with joint occurrences of rare events, e.g. the risk to multiple offshore structures from a storm event. There are two different classes of joint tail behaviour that have very different implications: asymptotic independence suggesting that extreme events are unlikely to occur together, and asymptotic dependence implying that extreme events can occur simultaneously. It is vital to have good diagnostics to identify the appropriate dependence class. If variables are asymptotically independent, incorrectly assuming an asymptotically dependent model can lead to overestimation of the joint risk of extreme events, and hence to higher than necessary design costs of offshore structures. We develop improved diagnostics for differentiating between these two classes, which leads to increased confidence in model selection. Application to samples of North Sea sea-state and storm-peak significant wave height suggests that tail dependence changes with wave direction and distance between spatial locations, and that in most cases asymptotic independence seems to be the more appropriate assumption

Modelling record times in sport with extreme value methods

We exploit connections between extreme value theory and record processes to develop inference methods for record processes. Record processes have trends in them even when the underlying process is stationary. We study the problem of estimating the underlying trend in times achieved in sports from record data. We develop new methods for inference, simulating record series in non-stationary contexts, and assessing fit which account for the censored characteristic of record data and we apply these methods to athletics and swimming data

Modelling the effect of the El Nino-Southern Oscillation on extreme spatial temperature events over Australia

When assessing the risk posed by high temperatures, it is necessary to consider not only the temperature at separate sites but also how many sites are expected to be hot at the same time. Hot events that cover a large area have the potential to put a great strain on health services and cause devastation to agriculture, leading to high death tolls and much economic damage. South-eastern Australia experienced a severe heatwave in early 2009; 374 people died in the state of Victoria and Melbourne recorded its highest temperature since records began in 1859 (Nairn and Fawcett, 2013). One area of particular interest in climate science is the effect of large scale climatic phenomena, such as the El Nino-Southern Oscillation (ENSO), on extreme temperatures. Here, we develop a framework based upon extreme value theory to estimate the effect of ENSO on extreme temperatures across Australia. This approach permits us to estimate the change in temperatures with ENSO at important sites, such as Melbourne, and also whether we are more likely to observe hot temperatures over a larger spatial extent during a particular phase of ENSO. To this end, we design a set of measures that can be used to effectively summarise many important spatial aspects of an extreme temperature event. These measures are estimated using our extreme value framework and we validate whether we can accurately replicate the 2009 Australian heatwave, before using the model to estimate the probability of having a more severe event than has been observed

Characterising the changing behaviour of heatwaves with climate change

Understanding the impact of future heatwaves and the development of effective adaptation strategies requires knowledge of both the changes in heatwave temperatures and their durations. We develop a framework, utilising extreme value theory, which allows for the effect of a covariate on both the marginal quantiles and the temporal dependence structure of daily maximum temperatures enabling the changes in heatwave temperatures (marginal effects) to be identified separately from duration changes (dependence effects). To characterise future heatwave changes we use global mean temperature anomalies as a covariate to provide the metric for climate change. Future daily maximum temperatures and global mean temperature changes are provided by 13 general circulation models (GCMs) from the CMIP5 archive forced with predicted future emissions of radiative forcing agents from the RCP8.5 scenario. For Orleans, central France, we find that for all GCMs temporal dependence is unaffected by greenhouse gas induced climate change indicating that durations of heatwaves that exceed time varying high thresholds (i.e., the 1 year level) will not change in the future. However, all GCMs project significant changes in the temperature margins with events similar to the 2003 European heatwave increasing by 1.3C to 2.7C and (8.0C to 18.7C) for a 1C (5C) increase in global temperature. Collectively our results indicate there could be a significant increase in heatwave risk as the world warms with heatwaves increasing in temperature significantly faster than the global mean and local average temperatures

On spatial conditional extremes for ocean storm severity

We describe a model for the conditional dependence of a spatial process measured at one or more remote locations given extreme values of the process at a conditioning location, motivated by the conditional extremes methodology of Heffernan and Tawn. Compared to alternative descriptions in terms of max‐stable spatial processes, the model is advantageous because it is conceptually straightforward and admits different forms of extremal dependence (including asymptotic dependence and asymptotic independence). We use the model within a Bayesian framework to estimate the extremal dependence of ocean storm severity (quantified using significant wave height, HS) for locations on spatial transects with approximate east–west (E‐W) and north–south (N‐S) orientations in the northern North Sea (NNS) and central North Sea (CNS). For HS on the standard Laplace marginal scale, the conditional extremes “linear slope” parameter α decays approximately exponentially with distance for all transects. Furthermore, the decay of mean dependence with distance is found to be faster in CNS than NNS. The persistence of mean dependence is greatest for the E‐W transect in NNS, potentially because this transect is approximately aligned with the direction of propagation of the most severe storms in the region

kth-order Markov extremal models for assessing heatwave risks

Heatwaves are defined as a set of hot days and nights that cause a marked short-term increase in mortality. Obtaining accurate estimates of the probability of an event lasting many days is important. Previous studies of temporal dependence of extremes have assumed either a first-order Markov model or a particularly strong form of extremal dependence, known as asymptotic dependence. Neither of these assumptions is appropriate for the heatwaves that we observe for our data. A first-order Markov assumption does not capture whether the previous temperature values have been increasing or decreasing and asymptotic dependence does not allow for asymptotic independence, a broad class of extremal dependence exhibited by many processes including all non-trivial Gaussian processes. This paper provides a kth-order Markov model framework that can encompass both asymptotic dependence and asymptotic independence structures. It uses a conditional approach developed for multivariate extremes coupled with copula methods for time series. We provide novel methods for the selection of the order of the Markov process that are based upon only the structure of the extreme events. Under this new framework, the observed daily maximum temperatures at Orleans, in central France, are found to be well modelled by an asymptotically independent third-order extremal Markov model. We estimate extremal quantities, such as the probability of a heatwave event lasting as long as the devastating European 2003 heatwave event. Critically our method enables the first reliable assessment of the sensitivity of such estimates to the choice of the order of the Markov process

Modelling across extremal dependence classes

Different dependence scenarios can arise in multivariate extremes, entailing careful selection of an appropriate class of models. In bivariate extremes, the variables are either asymptotically dependent or are asymptotically independent. Most available statistical models suit one or other of these cases, but not both, resulting in a stage in the inference that is unaccounted for, but can substantially impact subsequent extrapolation. Existing modelling solutions to this problem are either applicable only on sub-domains, or appeal to multiple limit theories. We introduce a unified representation for bivariate extremes that encompasses a wide variety of dependence scenarios, and applies when at least one variable is large. Our representation motivates a parametric model that encompasses both dependence classes. We implement a simple version of this model, and show that it performs well in a range of settings

A step towards efficient inference for trends in UK extreme temperatures through distributional linkage between observations and climate model data

The aim of this paper is to set out a strategy for improving the inference for statistical models for the distribution of annual maxima observed temperature data, with a particular focus on past and future trend estimation. The observed data are on a 25 km grid over the UK. The method involves developing a distributional linkage with models for annual maxima temperatures from an ensemble of regional and global climate numerical models. This formulation enables additional information to be incorporated through the longer records, stronger climate change signals, replications over the ensemble and spatial pooling of information over sites. We find evidence for a common trend between the observed data and the average trend over the ensemble with very limited spatial variation in the trends over the UK. The proposed model, that accounts for all the sources of uncertainty, requires a very high dimensional parametric fit, so we develop an operational strategy based on simplifying assumptions and discuss what is required to remove these restrictions. With such simplifications we demonstrate more than an order of magnitude reduction in the local response of extreme temperatures to global mean temperature changes