Climate Change Impacts on Forest Fire Ignitions
Overview
The potential effect of climate change on forest fire risk is of significant concern, especially
in determining whether fire ignition risk is increasing, as measured by annual trends and/or
the lengthening of the “fire season” within each year. Public concern has been heightened by
high profile wildfire events, such as the many fires that burned across the southern interior
of the province of British Columbia in 2003, including the Okanagan Mountain Park forest
fire that burned parts of the city of Kelowna. These fires occurred during the driest summer
in over 100 years of weather records. Could such events be attributed to climate change?
We use logistic generalized additive models to investigate for annual trends and for
changes in fire season lengths. Our preliminary study suggests the need for a mixturemodel framework. An extreme value distributed component appears to be necessary since
the smooth terms do not adequately describe the extreme events. It is these events that are
of primary interest in the context of climate change. Methodology for fitting and comparing
these types of models is being developed, along with tests for change-point shifts in the proportion of extreme ignition events. These methods are being applied to historical records of
forest fires in Ontario, Canada in collaboration with individuals at Simon Fraser University,
the Fire Systems Management Laboratory at the University of Toronto, and the Ontario
Ministry of Natural Resources.
Generalized Additive Modelling
Define a fire day as a day during which one or more fires are reported in an area. Let
twy denote week w of year y, and let Z(twy ) be the number of fire days during this period.
Assume the random variable Z(twy ) is distributed Binomial(n, p(twy )), where n = 7 and
p(twy ) is the corresponding probability of ignition of a forest fire. Climate change effects of
interest include an annual trend and changes in the duration of the fire season over time.
We model these using generalized additive models of the form
logit(p(twy )) = f1 (w) + f2 (y) + f3 (w, y) ,
where the univariate smooth functions f1 (w) and f2 (y) represent the marginal seasonal and
year effects, respectively, while the bivariate smoother f3 (w, y) accounts for any interaction
between these two partial effects.
The functions f1 (w) and f2 (y) are estimated as linear combinations of basis functions:
f1 (w) =
Kw
X
ck φk (w) , f2 (y) =
k=1
Ky
X
bk ψk (y) ,
k=1
where a cyclic cubic regression spline is employed in the basis functions φk (w) to model the
seasonal partial effect and the basis functions for the annual trend term ψk (y) are specified as cubic B-splines. The number of knots, Kw and Ky , are chosen by minimizing the
generalized cross-validations for each of the respective basis systems above. The bivariate
smoother f3 (w, y) is estimated as a linear combination of thin plate regression splines. All
basis coefficients are estimated by maximizing the likelihood function.
1
The standardized residuals of a fitted generalized additive model are defined as
z(twy ) − µ̂wy
²wy = p
,
V (µ̂wy )
where z(twy ) is the observed number of fire days during week w of year y, µ̂wy = 7 p̂(twy ) and
V (µ̂wy ) = µ̂wy (1−µ̂wy /7) are the expected number of fire days and the corresponding variance
for this period; these follow from the assumption that the response variable is binomially
distributed. In our analyses, we have observed that the distribution of ²̂wy has a heavy
right tail, and hence we need to accommodate this feature in the model. In this preliminary
analysis, we define a threshold u > 0, u large, representing an upper limit on residuals which
are not extreme. Then ²wy , given ²wy ≥ u, is assumed to follow an distribution which models
extremes. Hence the errors ²wy are modelled as a mixture distribution:
²wy ∼ (1 − δwy )ηwy + δwy τwy ,
(1)
where ηwy is in the truncated normal distribution with mean 0, variance σ 2 on support
set [−u, u] and we use here the generalized Pareto distribution GP(ξ, σ) for τwy , with the
following probability density function:
f (τwy |τwy ≥ u) =
ξ(τwy − u) (− 1ξ −1)
1
(1 +
)
.
σ
σ
The probability of extreme residuals δwy = P (²wy > u) may be modelled as a smooth function
of week, or year, or both, but is kept as a constant proportion in this initial exploratory study.
The mixture distribution (1) follows from the law of total probability. That is,
P (²wy ) = P (²wy |²wy < u) · P (²wy < u) + P (²wy |²wy ≥ u) · P (²wy ≥ u) .
Preliminary Results
We illustrate the use of the above methodology by performing an exploratory analysis of
forest fire ignitions in a section of Canada’s boreal forest. Our data consist of records from
1963 through 2004 on all detected lightning-caused forest fires that occurred in a 9,884,943
hectare ecologically homogeneous region in the northwest of Ontario’s Extensive fire management zone. Data were provided by the Aviation and Forest Fire Management Branch or
the Ontario Ministry of Natural Resources (OMNR). The historical fire management strategy for this zone has been to allow detected fires to burn unsuppressed, unless they posed
an immediate threat to settlements or other values at risk. 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.
The resulting fit is illustrated in Figures 1 and 2. Figure 1 illustrates the location of the
study region and visualizes the fitted model in three dimensions, while Figure 2 displays the
marginal seasonality and annual trend effects. Both of these figures plot the fitted values
on the response scale (i.e., as estimated probabilities). As expected, the seasonal nature
of forest fire ignitions is evident: the ignition risk is zero during the late fall through early
2
spring and then it increases, peaking in the early summer, while there is an overall increasing
trend in ignition risk across years. Moreover, the heterogeneity of the surface across years
suggests 1) an increasing risk of lightning ignitions across years, and 2) a possible increase
in the length of the fire season each year.
A commonly employed technique to assess the goodness of fit of a logistic model is to
compare the observed number of events to those expected under the model. The latter
quantity is readily obtained by first transforming the model’s fitted values to the response
scale via the inverse logit function, and then summing these fitted probabilities over any
period of interest. Based on such a comparison, the obtained model appears to fit quite
well: 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
in Figure 3, where the observed and expected number of fire days each year are compared for
fortnight 12. From this plot it is clear that, although the model does a good job at capturing the overall general trend, the more extreme events are being missed. Similar behaviour
was observed for other fortnights during the fire season. Setting a threshold for extreme
events during the fire season, we fit the generalized Pareto distribution to the corresponding residuals via maximization likelihood, obtaining estimates of ξˆ = −0.25 and σ̂ = 3.25.
ˆ = 0.10 and SE(σ̂)
c ξ)
c
The corresponding estimated standard errors are SE(
= 0.57. A visual
goodness of fit check is presented in Figure 4, which illustrates that the pdf of the generalized
Pareto distribution with these estimated parameters is close to the empirical distribution of
the extreme residuals.
Discussion
Although this preliminary model fits the data well, and suggests climate change impacts,
there are other confounding factors that need to be address. For example, given the reduced
need for early detection in the Extensive protection zone and the presence of commercial
aircraft, the OMNR does not route their aerial detection patrol aircraft (the “organized”
detection system) to look for fires in that area on a regular basis. Instead, they rely on
commercial and charter aircraft pilots as well as local residents (the “unorganized” detection
system) to report fires in that area. It is quite possible that OMNR initiatives to encourage
the reporting of fires and heightened awareness of the need to do so might have increased
the effectiveness of the unorganized detection system over time in our study area. Changes
in unorganized detection patterns (e.g., increased amount of recreational fly-in fishing) are
significant outstanding confounding factors that need to be addressed before one can assess
the extent to which the observed changes in fire ignitions can or cannot be attributed to
climate change. Methods for quantifying this are being developed for future studies.
3
Figure 1: Visualization of the fitted model in three dimensions, plotted on the response scale.
4
10
20
30
40
50
0.00 0.01 0.02 0.03 0.04
ignition probability
0.08
0.04
0.00
ignition probability
0
1970
week
1990
year
Figure 2: Marginal seasonal (left panel) and annual trend (right panel) effects of the fitted
model, plotted on the response scale.
5
12
10
8
6
0
2
4
observed
1970
1980
1990
2000
year
Figure 3: Observed (points) versus expected (line) for fortnight 12, aggregated year-by-year.
6
probability density function
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
2
3
4
5
6
7 8 9 10 11 12
Residuals > 2
Figure 4: The pdf of the generalized Pareto distribution with the estimated parameter values
ξˆ = −0.25 and σ̂ = 3.25, fitted to the extreme residuals.
7
Related Links
• Alberta Sustainable Resource Development
<http://www.srd.gov.ab.ca/wildfires/default.aspx>
• Aviation and Forest Fire Management Branch, Ontario Ministry of Natural Resources
<http://www.mnr.gov.on.ca/en/Business/AFFM/>
• Canadian Forest Service
<http://cfs.nrcan.gc.ca/>
• Fire Management Systems Laboratory at the University of Toronto
<http://www.firelab.utoronto.ca/>
• Forest, Fires and Stochastic Modelling, GEOIDE Phase IV Initiative and NICDS
<http://www.stat.sfu.ca/ dean/forestry/index.php>
• Pacific Forestry Centre
<http://cfs.nrcan.gc.ca/regions/pfc>
Supporting Agencies
• Geomatics for Informed Decisions (GEOIDE)
<http://www.geoide.ulaval.ca/>
• National Institute for Complex Data Structures (NICDS)
<http://www.fields.utoronto.ca/programs/scientific/NICDS/>
• Natural Sciences and Engineering Research Council of Canada (NSERC)
<http://www.nserc-crsng.gc.ca/>
8