kth-order Markov extremal models for assessing heatwave risks

Abstract

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 firstorder 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 kthorder 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.

A model for the directional evolution of severe ocean storms

“…We estimate the model using a sample of m intervals { I k } of different lengths and data . The model is an extension of MEM of Winter and Tawn (, ), expressed for time series on a marginal standard Laplace scale. We therefore start the model description by outlining a nonstationary marginal model assumed applicable for all occurrences of H S used to transform the sample of H S values to the standard Laplace scale.…”

“…Winter () and Winter and Tawn () describe a general k th‐order Markov extremal model MEM(k) for the evolution of time series { X t } of extreme events on the standard Laplace scale exceeding threshold η . Here, we apply the model in turn to the “prepeak” and “postpeak” intervals of storm trajectories to describe respectively the evolution of a storm to and from its peak in time.…”

“…We further restrict the presentation of the model to the “postpeak” interval, noting that the corresponding analysis has also been performed for the “prepeak” interval. Finally, we note that MEM has been implemented by us as described in the works of Winter () and Winter and Tawn (). We therefore give an outline of the model and its estimation here, referring interested readers to the said articles for details.…”

“…This set of residuals is then used in kernel density estimation to estimate the joint distribution G 1:2 of [ Z 1 , Z 2 ] (and the temporary assumption that this distribution is Gaussian is relaxed). As shown in the work of Winter and Tawn (), to be able to simulate under the fitted model, we also require the conditional distribution G 2|1 , which is also estimated from the same kernel density estimation.…”

“…In the context of characterizing heat wave evolution, Winter and Tawn (, ) introduce a Markov extremal model (MEM) for an interval of extreme values of time series (transformed to a standard Laplace marginal scale) motivated by the conditional extremes model of Heffernan and Tawn (). In this work, we present a nonstationary extension of the work of Winter and Tawn incorporating evolution of sea state H S and direction.…”