Maxima of moving maxima of continuous functions

Maxima of moving maxima of continuous functions (CM3) are max-stable processes aimed at modelling extremes of continuous phenomena over time. They are defined as Smith and Weissman’s M4 processes with continuous functions rather than vectors. After standardization of the margins of the observed process into unit-Fréchet, CM3 processes can model the remaining spatio-temporal dependence structure. CM3 processes have the property of joint regular variation. The spectral processes from this class admit particularly simple expressions given here. Furthermore, depending on the speed with which the parameter functions tend toward zero, CM3 processes fulfill the finite-cluster condition and the strong mixing condition. Processes enjoying these three properties also enjoy a simple expression for their extremal index. Next a method to fit CM3 processes to data is investigated. The first step is to estimate the length of the temporal dependence. Then, by selecting a suitable number of blocks of extremes of this length, clustering algorithms are used to estimate the total number of different profiles. The parameter functions themselves are estimated thanks to the output of the partitioning algorithms. The full procedure only requires one parameter which is the range of variation allowed among the different profiles. The dissimilarity between the original CM3 and the estimated version is evaluated by means of the Hausdorff distance between the graphs of the parameter functions.