  Recent extreme value theory literature has seen significant emphasis on the
modelling of spatial extremes, with comparatively little consideration of
spatio-temporal extensions. This neglects an important feature of extreme
events: their evolution over time. Many existing models for the spatial case
are limited by the number of locations they can handle; this impedes extension
to space-time settings, where models for higher dimensions are required.
Moreover, the spatio-temporal models that do exist are restrictive in terms of
the range of extremal dependence types they can capture. Recently, conditional
approaches for studying multivariate and spatial extremes have been proposed,
which enjoy benefits in terms of computational efficiency and an ability to
capture both asymptotic dependence and asymptotic independence. We extend this
class of models to a spatio-temporal setting, conditioning on the occurrence of
an extreme value at a single space-time location. We adopt a composite
likelihood approach for inference, which combines information from full
likelihoods across multiple space-time conditioning locations. We apply our
model to Red Sea surface temperatures, show that it fits well using a range of
diagnostic plots, and demonstrate how it can be used to assess the risk of
coral bleaching attributed to high water temperatures over consecutive days.
