Looking for Climate Change Signals in the Canadian Forest Fire Ignition Record Douglas G. Woolford1,2, Jiguo Cao1, Charmaine B. Dean1 and David L. Martell2 1. Department of Statistics & Actuarial Science, Simon Fraser University, Burnaby, Canada • 2. Faculty of Forestry, University of Toronto, Toronto, Canada Changes to fire ignition risk over time are modelled using the following logistic-GAM: logit ( p(twy) ) = s1(w) + s2(y) + s3(w, y) (Weber & Stocks, 1998) s1 represents the intra-annual seasonality pattern GENERALIZED ADDITIVE MODELS Generalized additive models (GAMs) extend generalized linear models, allowing for non-linear relationships by incorporating smooth functions. The smoothers are estimated as linear combinations of basis functions: s ( x) = where K ∑ k=1 ck φ k ( x ) the sum is over a finite number of knots k, partitioning the range of the covariate the ck are coefficients (to be estimated via maximum likelihood) for the set of basis functions, φk(x) s2 represents any inter-annual trends in ignitions s3 represents the interaction between intra and inter- In our analyses using the above model with standard normal errors we observed a heavy right tail in the distribution of the standardized residuals. To accommodate this feature in the model, we model the errors, ε, as a mixture of truncated normal and generalized Pareto(ξ, σ ) random variables. 2000 1990 0.00 10 1980 20 30 w ee k 1970 40 50 GOODNESS OF FIT & THE NEED FOR THE MIXTURE COMPONENT • Comparisons of observed fire day counts (points) versus those expected under the fitted model (lines) suggest a good fit: We identified the “extreme residuals” as the standardized residuals that exceeded 2 in magnitude and fit a generalized Pareto distribution via maximum likelihood. A visual examination of this fit is presented in the figure below. It suggests that the mixture framework for the error structure may be necessary. A Remark on Model Selection: A Mixture Distribution for the Error Terms: 0.05 In addition, there appears to be a drastic increase in ignition risk across years, combined with a lengthening of the fire season. However, this may be due to changes in the effectiveness of the detection system as described in the discussion section below. annual trends and allows for changes in the seasonal pattern in ignitions, including an extension of the fire season The above formulation easily permits one to test for the presence of inter-annual trends or changes to the pattern of fire season over years by using likelihood ratio tests to select between a series of nested models. 0.10 However, the need for a model that incorporates extreme events is apparent when we compare observed versus expected for individual fortnights across years. This is illustrated below for fortnight 12. Although the model does a good job at capturing the overall general trend, the more extreme events are being missed. 12 Goal of Our Work Explore for the latter two effects by looking for signals in historical records of forest fire ignitions using logistic generalized additive models. inherent to fire ignitions (i.e., the fire season) The seasonal nature of forest fire ignitions is evident: the ignition risk is zero during the late fall through early spring and then it increases, peaking in the early summer. 0.15 10 3.extending the fire season where p(twy) is the probability of a fire day at time twy. The plot on the right displays the fitted model, plotted in terms of the fitted probability of ignition. 0.20 8 2.increasing the number of forest fire ignitions Z(twy) ~ Binomial ( n = 7, p = p(twy) ) Model selection via likelihood ratio tests indicated that a model with all three smoother components are necessary. 6 1.increasing the amount of severe fire weather Let Z(twy) = # fire days during week w of year y. Then, 0.25 4 Increasing temperatures could alter a fire regime by: Define a fire day as a day where 1 or more fires occur. 2 The uncertainty of the potential impact of climate change on forest fire regimes needs to be addressed. THE FITTED MODEL 0 The management of Canada’s Boreal Forest is a challenging task, due in part, to the long planning horizons associated with renewable forest resources. MODEL FORMULATION observed INTRODUCTION 1970 1980 1990 2000 year MIXTURE MODELS Mixture models assume a random variable comes from a population that is composed of a set of distinct groups, each of which has a different distribution, e.g.: g Y ~ ∑ π i f ( y) i= 1 where the πi are mixing proportions (that sum to 1), representing the probability that Y comes from the component density fi(y) We use two-components for the residuals in our model: a mixture of normal and extreme value distributions this accounts for extreme events (e.g., weeks with an unusually large number of fire days, relative to usual ignition rates for that period) THE DATA AND STUDY AREA We analyze all lightning-caused fires in a 9,884,983 hectare ecoregion of northwestern Ontario for the period 1963 through 2004. The fire management strategy in this region has been to allow fires to burn unsuppressed, unless they pose an immediate threat to public safety or property. DISCUSSION Fire size at detection is a surrogate measure of detection system effectiveness. The median size at detection has decreased since the mid-1970s. Hence, the increase in fire activity that we are observing may be due to climate change, or improved detection system performance, or some combination of both of these factors. This is being investigated. Due to little human activity and fire intervention in this region, one could postulate that the data reflect the natural process of lightning fire ignitions in this section of the boreal forest ecosystem. Additional further work involves the development and implementation of a framework for the analysis of GAMMs with normal/extreme mixtures of error terms. However, organized detection does not occur in this region. Consequently, changes in unorganized detection patterns (e.g., increased amount of recreational fly-in fishing and/or public awareness) are potential confounding factors. Extensions to permit the joint analysis of several ecoregions would allow the coefficients of the spline smoothers and the probabilities of extreme events to very smoothly over space, and would provide a broad framework for the spatio-temporal analysis of extremes for a variety of discrete and continuous outcomes. ACKNOWLEDGEMENTS Funding from the following sources is gratefully acknowledged: The National Institute for Complex Data Structures GEOmatics for Informed Decisions The Natural Sciences and Engineering Research Council of Canada Thanks also to the Aviation and Forest Fire Management branch of the Ontario Ministry of Natural Resources for the use of their fire data. Thanks to Fletcher Quince of the Fire Management Systems Lab. at the University of Toronto for providing the photo used as the background.