Bayesian inference for the Brown–Resnick process, with an application to extreme low temperatures
Models for spatial extremes must account appropriately for asymptotic dependence, and this motivates the use of max-stable processes, which are the only non-trivial limits of properly rescaled pointwise maxima of random functions. The Brown–Resnick max-stable process has proven to be well-suited for modeling extremes of complex environmental processes, but in many applications its likelihood function is unobtainable and inference must be based on a composite likelihood, thereby preventing the use of classical Bayesian techniques. In this paper we exploit a case in which the full likelihood of a Brown–Resnick process can be calculated, using componentwise maxima and their partitions in terms of individual events, and we propose two new approaches to inference. The first estimates the partition using declustering, while the second uses the partition in a Markov chain Monte Carlo algorithm. We use these approaches to construct a Bayesian hierarchical max-stable model that can model complex spatial patterns in the marginal distributions of extremes and can describe their joint extremal dependence appropriately. We use this to model extreme low temperatures in northern Fennoscandia, in order to consider the likely effects of climate change on the risk of forest damage caused by outbreaks of pest insect populations.
Reference: Thibaud, E., Aalto, J., Cooley, D. S., Davison, A. C. & Heikkinen, J. 2015. Bayesian inference for the Brown–Resnick process, with an application to extreme low temperatures. arXiv preprint arXiv:1506.07836.