  We develop a method for probabilistic prediction of extreme value hot-spots
in a spatio-temporal framework, tailored to big datasets containing important
gaps. In this setting, direct calculation of summaries from data, such as the
minimum over a space-time domain, is not possible. To obtain predictive
distributions for such cluster summaries, we propose a two-step approach. We
first model marginal distributions with a focus on accurate modeling of the
right tail and then, after transforming the data to a standard Gaussian scale,
we estimate a Gaussian space-time dependence model defined locally in the time
domain for the space-time subregions where we want to predict. In the first
step, we detrend the mean and standard deviation of the data and fit a
spatially resolved generalized Pareto distribution to apply a correction of the
upper tail. To ensure spatial smoothness of the estimated trends, we either
pool data using nearest-neighbor techniques, or apply generalized additive
regression modeling. To cope with high space-time resolution of data, the local
Gaussian models use a Markov representation of the Mat\'ern correlation
function based on the stochastic partial differential equations (SPDE)
approach. In the second step, they are fitted in a Bayesian framework through
the integrated nested Laplace approximation implemented in R-INLA. Finally,
posterior samples are generated to provide statistical inferences through
Monte-Carlo estimation. Motivated by the 2019 Extreme Value Analysis data
challenge, we illustrate our approach to predict the distribution of local
space-time minima in anomalies of Red Sea surface temperatures, using a gridded
dataset (11315 days, 16703 pixels) with artificially generated gaps. In
particular, we show the improved performance of our two-step approach over a
purely Gaussian model without tail transformations.
