Black swans in space: modeling spatiotemporal processes with extremes

In ecological systems, extremes can happen in time, such as population crashes, or in space, such as rapid range contractions. However, current methods for joint inference about temporal and spatial dynamics (e.g., spatiotemporal modeling with Gaussian random fields) may perform poorly when underlying processes include extreme events. Here we introduce a model that allows for extremes to occur simultaneously in time and space. Our model is a Bayesian predictive‐process GLMM (generalized linear mixed‐effects model) that uses a multivariate‐t distribution to describe spatial random effects. The approach is easily implemented with our flexible R package glmmfields. First, using simulated data, we demonstrate the ability to recapture spatiotemporal extremes, and explore the consequences of fitting models that ignore such extremes. Second, we predict tree mortality from mountain pine beetle (Dendroctonus ponderosae) outbreaks in the U.S. Pacific Northwest over the last 16 yr. We show that our approach provides more accurate and precise predictions compared to traditional spatiotemporal models when extremes are present. Our R package makes these models accessible to a wide range of ecologists and scientists in other disciplines interested in fitting spatiotemporal GLMMs, with and without extremes. Applications of statistical models that allow for joint inference about spatial and temporal dynamics have advanced rapidly in ecology over the last several decades (e.g., Bascompte and Solé 1995, Latimer et al. 2009, Conn et al. 2015). Spatiotemporal models have also been widely used in other disciplines, including applications to weather, remote sensing, human disease dynamics, and crime (Cressie and Wikle 2011). When ecological data are spatially structured, explicitly accounting for spatial autocorrelation can improve model predictions and inference about parameters of interest (e.g., Shelton et al. 2014, Thorson et al. 2015, Ver Hoef et al. 2017). Including spatial components in statistical models involves extending models that most ecologists are familiar with, such as generalized linear models (GLMs) or generalized additive models (GAMs). Spatial relationships can be included as predictors in models of the mean (e.g., a two‐dimensional GAM) or can be included in models of the covariance (e.g., kriging). Recent extensions of these spatial covariance models include modeling spatial deviations in GLMs as random effects (Gaussian random fields, GRFs). GRFs represent a two‐dimensional version of Gaussian processes and define the expected value, variance, and