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 Expectation–Maximization 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 non-linear spectrum of scaling exponents ${\zeta }_q$. From our study, it thus appears that the spatial distribution of the wildfire ignition annual risk can be described by a non-trivial, 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.

我们引入简单且广泛的一类多重分形空间点模式，即强度为多重形的Cox过程，即，具有随机强度的Poisson过程类别对应于随机多重形分形。然后，我们通过标准的Expectation-Maximization程序提出最大似然方法，以便估算所有强度下这些强度的分布。正如在各种数值示例上所验证的那样，这提供了一个简单的框架来估计缩放定律，从而估计此类空间点过程的多重分形特性。在这种方法下，对法国Prométhée法国地中海野火数据库中收集的野火分布进行了调查，这尤其使我们能够计算与火灾事件每年可能性的空间分布相关的统计矩。我们显示每个订单$q$，这些时刻显示了定义良好的缩放行为，带有非线性的缩放指数谱 ${\zeta }_q$。从我们的研究中可以看出，野火点火年风险的空间分布可以用一个非平凡的，多重分形奇异谱来描述，并且这种风险不能减少到每公里提供多个事件${}^2$。我们的分析通过强度对数的直接空间相关估计得到证实，强度对数的奇特缓慢下降形状对应于多重分形级联的标志。多重分形特征似乎随时间恒定，并且在研究的三个区域中相似。