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.