Hierarchical Transformed Scale Mixtures for Flexible Modeling of Spatial Extremes on Datasets With Many Locations

Abstract

Flexible spatial models that allow transitions between tail dependence classes have recently appeared in the literature. However, inference for these models is computationally prohibitive, even in moderate dimensions, due to the necessity of repeatedly evaluating the multivariate Gaussian distribution function. In this work, we attempt to achieve truly high-dimensional inference for extremes of spatial processes, while retaining the desirable flexibility in the tail dependence structure, by modifying an established class of models based on scale mixtures Gaussian processes. We show that the desired extremal dependence properties from the original models are preserved under the modification, and demonstrate that the corresponding Bayesian hierarchical model does not involve the expensive computation of the multivariate Gaussian distribution function. We fit our model to exceedances of a high threshold, and perform coverage analyses and cross-model checks to validate its ability to capture different types of tail characteristics. We use a standard adaptive Metropolis algorithm for model fitting, and further accelerate the computation via parallelization and Rcpp.Lastly, we apply the model to a dataset of a fire threat index on the Great Plains region of the US, which is vulnerable to massively destructive wildfires. We find that the joint tail of the fire threat index exhibits a decaying dependence structure that cannot be captured by limiting extreme value models.

Smooth copula‐based generalized extreme value model and spatial interpolation for extreme rainfall in Central Eastern Canada

“…It takes into account the spatiotemporal structure, using the distance between dependent clusters, by means of a dissimilarity measure designed to handle missing data. Note that other papers tackle the problem of missing data, using hierarchical max-stable process construction such as in Reich and Shaby (2012) or conditioning of a latent process in the context of extremes, such as in Zhang et al (2021).…”

Advances in Statistical Modeling of Spatial Extremes

“…Computational speed-up can be obtained for censored likelihood approaches by exploiting pseudo-Monte Carlo methods in the calculation of multivariate Gaussian or Student's t distributions (de Fondeville and Davison, 2018;Beranger et al, 2019); by adding a measurement error term to the model like in Zhang et al (2019); or even by using proper scoring rules instead of maximum likelihood as in de Fondeville and Davison (2018). However, to tackle problems in truly higher dimensions, sparse models with a fundamentally different probabilistic structure (Engelke and Hitz, 2020;Engelke and Ivanovs, 2021) need to be devised.…”

“…(2017) showed how to exploit (24) to perform censored likelihood inference based on high threshold exceedances for this class of models, but this remains fairly intensive in moderate dimensions (roughly D > 30) in cases where the random variable R cannot be integrated out in explicit form and (uni-dimensional) numerical integrals are thus required. Nevertheless, Zhang et al (2019) recently showed how to bypass the explicit integral in (24) and to fit these models more efficiently on many locations by adopting the Bayesian perspective and adding a measurement error term (i.e., a "nugget effect") to the model. Another appealing property of Gaussian scale mixture models is that they are easily amenable to unconditional or conditional simulation, which is typically required for the evaluation of spatial risk measures and for spatial prediction.…”

“…For practical convenience, the W process is typically chosen as a Gaussian process marginally transformed to the unit Pareto scale. Censored likelihood inference for peaks-over-threshold may be performed similarly to "classical" Gaussian scale mixtures; see Huser and Wadsworth (2019) and Zhang et al (2019). Gong and Huser (2019) extended this model to the time-dynamic framework to capture the upper and lower tail structures of bivariate cryptocurrency data jointly.…”

Contamination mapping in Bangladesh using a multivariate spatial Bayesian model for left-censored data

“…Here r ∈ [0, 1] is the ratio of spatial to total variation. The multivariate nugget term tackles the censoring in arsenic log-concentration, thereby circumnavigating computational burden occurring due to censored likelihoods (Hazra et al, 2018;Yadav et al, 2019;Zhang et al, 2021).…”