Abstract
We introduce a statistically-inspired class of loss functions for scoring predictive models of wildfire risk, quantifying how well a model fits observed fire occurrence. These loss functions are derived as a weighted generalization of Poisson process deviance; this generalization unifies various existing approaches in the statistical wildfire literature and suggests new approaches, enabling improvements by relaxing requirements of probabilistic independence, using more of the historical information, and giving more importance to the largest fires. Nontrivially, we apply these tools to calibrating the parameters of wildland fire Monte Carlo simulations, and in particular the joint distribution of ignitions and fire durations. We argue that such an integrated approach is more reliable than optimizing the distribution of ignitions in isolation, because it optimizes the end results of simulations. We also describe a fast algorithm for approximating the loss function on candidate distributions of ignitions and durations without having to repeatedly run new simulations: using a sample-reweighting approach, a calibration simulation is run once, and the family of possible ignition distributions is defined and explored after the fact. In particular, distribution mixtures can be used for ensembling fire behavior models, and fire durations can be modeled and calibrated either explicitly via a conditional probability density function, or implicitly via a parametric hazard function that represents containment effectiveness. Furthermore, this method enables the use of gradient-based optimization algorithms for finding the best-fitting parameters. This enables a workflow similar to fitting parametric statistical models. We call this approach Plug-in Reweighted Poisson Likelihood Optimization (PiRPLO).