A composite likelihood based approach for max-stable processes using histogram-valued variables

Abstract

The analysis of Spatial Extremes has been given a growing interest over the last decade in a broad range of areas and especially in climatology with the goal to better understand the behaviour of events such as floods, heat waves or storms. Max-stable processes are a convenient and widely used tool to model such phenomena. In recent years, composite likelihood methods have appeared to bypass the intractability of the multivariate density function of such processes. However, the computational cost of these methods explodes as the number of temporal observations gets large. This is even more noticeable when working with a large number of spatial locations across a study region. To bypass this issue we introduce a symbolic data analysis (SDA) based approach which consists in aggregating data into histograms leading to a reduction of the complexity of the data. A symbolic version of the composite likelihood approach where observations are multivariate histogram-valued is provided and the classical results from Padoan, et al.(2010)[Likelihood-Based Inference for Max-Stable Processes. Journal of the American Statistical Association, 105, 263-277] are shown to be recovered as a limiting case.The performance of our procedure in terms of inferential and computational efficiency is studied in an extensive simulation study and the impact of coarsening the data and the design of the symbols (histograms) is discussed.Finally, the utility of the method is illustrated through the analysis of fortnightly maximum temperatures at $105$ locations across Australia using historical data and simulated data from two climate models.