Douglas G. Woolford

A model for predicting human-caused wildfire occurrence in the region of Madrid, Spain

This paper describes the development and validation of a spatio-temporal model for human-caused wildfire occurrence prediction at a regional scale. The study area is the 8028-km2 region of Madrid, located in central Spain, where more than 90% of wildfires are caused by humans. We construct a logistic generalised additive model to estimate daily fire ignition risk at a 1-km2 grid spatial resolution. Spatially referenced socioeconomic and weather variables appear as covariates in the model. Spatial and temporal effects are also included. The variables in the model were selected using an iterative approach, which we describe. We use the model to predict the expected number of fires in our study area during the 2002–05 period, by aggregating the estimated probabilities over space–time scales of interest. The estimated partial effects of the presence of railways, roads, and wildland–urban interface in forest areas were highly significant, as were the observed daily maximum temperature and precipitation.

The erlangization method for Markovian fluid flows

For applications of stochastic fluid models, such as those related to wildfire spread and containment, one wants a fast method to compute time dependent probabilities. Erlangization is an approximation method that replaces various distributions at a time t by the corresponding ones at a random time with Erlang distribution having mean t. Here, we develop an efficient version of that algorithm for various first passage time distributions of a fluid flow, exploiting recent results on fluid flows, probabilistic underpinnings, and some special structures. Some connections with a familiar Laplace transform inversion algorithm due to Jagerman are also noted up front.

Wildfire Prediction to Inform Fire Management: Statistical Science Challenges

Wildfire is an important system process of the earth that occurs across a wide range of spatial and temporal scales. A variety of methods have been used to predict wildfire phenomena during the past century to better our understanding of fire processes and to inform fire and land management decision-making. Statistical methods have an important role in wildfire prediction due to the inherent stochastic nature of fire phenomena at all scales. Predictive models have exploited several sources of data describing fire phenomena. Experimental data are scarce; observational data are dominated by statistics compiled by government fire management agencies, primarily for administrative purposes and increasingly from remote sensing observations. Fires are rare events at many scales. The data describing fire phenomena can be zero-heavy and nonstationary over both space and time. Users of fire modeling methodologies are mainly fire management agencies often working under great time constraints, thus, complex models have to be efficiently estimated. We focus on providing an understanding of some of the information needed for fire management decision-making and of the challenges involved in predicting fire occurrence, growth and frequency at regional, national and global scales.

A preemptive priority queue with balking

Abstract This paper analyzes a 2-class, single-server preemptive priority queueing model with low priority balking customers. Arrivals to each class are assumed to follow a Poisson process with exponentially distributed service times. The decision to balk or not is made on the basis of queue length, and two specific forms of balking behaviour are considered. The system under consideration is a quasi-birth and death process, and the steady-state joint distribution of the number of high and low priority customers in the system is derived explicitly via the method of generalized eigenvalues.

Forest Fire Risk Assessment: An Illustrative Example from Ontario, Canada

This paper presents an analysis of ignition and burn risk due to wildfire in a region of Ontario, Canada using a methodology which is applicable to the entire boreal forest region. A generalized additive model was employed to obtain ignition risk probabilities and a burn probability map using only historic ignition and fire area data. Constructing fire shapes according to an accurate physical model for fire spread, using a fuel map and realistic weather scenarios is possible with the Prometheus fire growth simulation model. Thus, we applied the Burn-P3 implementation of Prometheus to construct a more accurate burn probability map. The fuel map for the study region was verified and corrected. Burn-P3 simulations were run under the settings (related to weather) recommended in the software documentation and were found to be fairly robust to errors in the fuel map, but simulated fire sizes were substantially larger than those observed in the historic record. By adjusting the input parameters to reflect suppression effects, we obtained a model which gives more appropriate fire sizes. The resulting burn probability map suggests that risk of fire in the study area is much lower than what is predicted by Burn-P3 under its recommended settings.

Erlangized Fluid Queues with Application To Uncontrolled Fire Perimeter

Abstract The present paper develops an “Erlangization” method for fluid queues. It then applies the results we derive to a forestry problem: the evolution of an uncontrolled fire perimeter over time. Specifically, we focus on the probability of containing a fire prior to reaching a randomly distributed, finite time horizon. Transitions to lower non-zero levels are also investigated. A preliminary model is introduced to demonstrate the potential of the application, and numerical results are given for illustrative purposes.

Developing emergency department physician shift schedules optimized to meet patient demand

OBJECTIVES 1) To assess temporal patterns in historical patient arrival rates in an emergency department (ED) to determine the appropriate number of shift schedules in an acute care area and a fast-track clinic and 2) to determine whether physician scheduling can be improved by aligning physician productivity with patient arrivals using an optimization planning model. METHODS Historical data were statistically analyzed to determine whether the number of patients arriving at the ED varied by weekday, weekend, or holiday weekend. Poisson-based generalized additive models were used to develop models of patient arrival rate throughout the day. A mathematical programming model was used to produce an optimal ED shift schedule for the estimated patient arrival rates. We compared the current physician schedule to three other scheduling scenarios: 1) a revised schedule produced by the planning model, 2) the revised schedule with an additional acute care physician, and 3) the revised schedule with an additional fast-track clinic physician. RESULTS Statistical modelling found that patient arrival rates were different for acute care versus fast-track clinics; the patterns in arrivals followed essentially the same daily pattern in the acute care area; and arrival patterns differed on weekdays versus weekends in the fast-track clinic. The planning model reduced the unmet patient demand (i.e., the average number of patients arriving at the ED beyond the average physician productivity) by 19%, 39%, and 69% for the three scenarios examined. CONCLUSIONS The planning model improved the shift schedules by aligning physician productivity with patient arrivals at the ED.

Erlangian Approximations for the Transient Analysis of a Fluid Queue Model for Forest Fire Perimeter

Abstract This paper presents how the Erlangization method can be employed to estimate probabilities for the time-dependent behaviour of the level in a Markov modulated fluid flow. Formulas are given for estimating the probability of transitions to any fluid level, starting from an initial level in either an increasing or a decreasing stage in the phase process, prior to a finite timepoint of interest. This methodology is applied to a model developed to study the evolution of uncontrolled fire perimeter for forest fires in Ontario, Canada. The model estimates the relative effectiveness of fire suppression under various fire-weather and fuel type scenarios, focusing on containment and escape probabilities. Numerical results are provided for illustrative purposes.

A Simulation Study of Predicting Flush Date

For Canadas boreal forest region, the accurate modelling of the timing of the appearance of aspen leaves is important to forest fire management, as it signifies the end of the spring fire season that occurs after snowmelt. This article compares two methods, a midpoint rule and a conditional expectation method used to estimate the true flush date for interval-censored data from a large set of fire-weather stations in Alberta, Canada. The conditional expectation method uses the interval censored kernel density estimator of Braun et al. (2005). The methods are compared via simulation, where true flush dates were generated from a normal distribution and then converted into intervals by adding and subtracting exponential random variables. The simulation parameters were estimated from the data set and several scenarios were considered. The study reveals that the conditional expectation method is never worse than the midpoint method, and that there is a significant advantage to this method when the intervals are large. An illustration of the methodology applied to the Alberta data set is also provided.