Multifractal point processes and the spatial distribution of wildfires in French Mediterranean regions
Abstract
We introduce a simple and wide class of multifractal spatial point patterns as Cox processes which intensity is multifractal, i.e., the class of Poisson processes with a stochastic intensity corresponding to a random multifractal measure. We then propose a maximum likelihood approach by means of a standard ExpectationMaximization procedure in order to estimate the distribution of these intensities at all scales. This provides, as validated on various numerical examples, a simple framework to estimate the scaling laws and therefore the multifractal properties for this class of spatial point processes. The wildfire distribution gathered in the Prométhée French Mediterranean wildfire database is investigated within this approach that notably allows us to compute the statistical moments associated with the spatial distribution of annual likelihood of fire event occurrence. We show that for each order q, these moments display a well defined scaling behavior with a nonlinear spectrum of scaling exponents ζ_{q}. From our study, it thus appears that the spatial distribution of the wildfire ignition annual risk can be described by a nontrivial, multifractal singularity spectrum and that this risk cannot be reduced to providing a number of events per km^{2}. Our analysis is confirmed by a direct spatial correlation estimation of the intensity logarithms whose the peculiar slowly decreasing shape corresponds to the hallmark of multifractal cascades. The multifractal features appear to be constant over time and similar over the three regions that are studied.
 Publication:

Physica A Statistical Mechanics and its Applications
 Pub Date:
 April 2021
 DOI:
 10.1016/j.physa.2020.125697
 arXiv:
 arXiv:2006.02165
 Bibcode:
 2021PhyA..56825697B
 Keywords:

 Spatial point patterns;
 Cox processes;
 Multifractal measures;
 Natural fire;
 Prométhée database;
 Physics  Physics and Society;
 Physics  Data Analysis;
 Statistics and Probability
 EPrint:
 41 pages, 14 figures