This document describes a survival analysis project modeling the lifetimes of forest fires in Ontario. It defines two lifetimes - time until a fire is declared out and time until it is declared being held. The analysis uses data on fires from 1976-2004 including when they were reported, held, and out. Survival models are used to explore how factors like ignition source, suppression success, and difficulty impact lifetimes. The Weibull and loglogistic accelerated failure time models best fit the data. The analysis finds ignition source, suppression success, and difficulty significantly impact lifetimes across compartments, with lightning-caused, escaped fires having longer lifetimes. Compartments in extensive and measured zones see longer lifetimes than intensive zones.

1. Modeling the “Lifetimes” of
Forest Fires in Ontario:
A Survival Analysis Project
By: Dhriti Chakraborty

2. Two definitions of “lifetime”:
 Lifetime1: The time elapsed from when a fire is reported, until it is
declared “out”
 Lifetime2: The time elapsed from when a fire is reported, until it is
declared “being held”
 Why Model these “Lifetimes” of Fires?

3. Data
 All fires recorded in Ontario from year 1976 to 2004: records included
temporal information about when a fire is reported, being held and declared
out.
 “Fire management compartments”
were to be used as the basic unit
of analysis for the data;
areas of relatively
homogeneous weather,
fuel and level of fire
management protection
(Martell and Sun, 2008)

4. Survival Analysis
 Used to model lifetimes of people or mechanical devices, or more
generally “time to events”
 Non-parametric models can be used for exploratory analysis
 Two types of models are generally used to show the impact of factors
that affect lifetimes:
 The Cox Proportional Hazards model: Semi parametric model
 The Accelerated Failure Time model: Fully parametric model
 Can use one of several log-location-scale distributions for lifetimes:
e.g. loglogistic, weibull, lognormal
 T = u(x) + bZ; −∞ < u(x) < ∞, b>0.
 Parameters estimated using the maximum likelihood method, and
are distributed approximately normally for large sample sizes

5. Components of Survival Model
 The survival function:
 The hazard function:
 Mean residual
lifetime:

6. Exploratory analysis using the
Kaplan-Meier estimator
 The KM estimator of the survival function is a non-
parametric estimator that is based on an estimate of the
(discrete) hazard:
(h = hazard, d = number of events that occurred at a particular time,
and Y = number of events that could have occurred)
 The Kaplan Meier estimator of the survival function at time t is:

7. 1.0 Survival curves as estimated by the KM estimator
1.0
Lifetime1 Lifetime2
0.8
0.8
S(t) estimate
S(t) estimate
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0 1000 2000 3000 4000 0 200 400 600 800 1000
t (hours) t (hours)

8. What model should be used?
 Decision to use an AFT model for each “lifetime”, as well as each
fire management compartment was based on its parametric form,
as well as the ability to calculate estimated expected lifetime
 Appropriateness of AFT model was assessed through plots of
S0-1(KM) against log(time), varying S0-1 to check for each possible
model
 Linearity showed appropriateness
 Weibull AFT model was chosen to model lifetime1, Loglogistic AFT
model for lifetime2

9. Lifetime1: Plot to check for Weibull
Model Appropriateness
2
FMZ: I
FMZ:M
S0inverse_weibull(KMhat)
0
log(−log (KM(t))
FMZ: E
-2
-4
-6
-2 0 2 4 6 8
log(time)
log t

10. Lifetime2: Plot to check for Loglogistic
Model Appropriateness
10
log(1− (KM(t))/KM(t))
5
FMZ:M
loglogistic
FMZ: I
0
-5
-4 -2 0 2 4 6
logt
log t

11. Covariates
 Ignition source: People/lightning caused fire
 Success of initial attack: Whether fire is declared “being held” by the next
day noon after it is reported– ecape/no escape
 Difficulty of suppression: “Flame index” = Sqrt(FWI)*Sqrt(area at initial
attack in squared metres)
 Chosen because the regression parameters for these covariates were significant
across FMCS and both definitions of lifetimes, and tended to give higher
likelihoods for the models that they were in.

12. Showing the impact of covariates and fire management compartment
on lifetime1
No escape, people caused, Flame index 125 Escape, lightning caused, Flame index 125
1.0
1.0
FMC9 FMC9
0.8
0.8
FMC11 FMC11
FMC27 FMC27
0.6
0.6
S(t) estimate
S(t) estimate
FMC9
FMC9
0.4
0.4
0.2
0.2
0.0
0.0
0 200 400 600 800 1000 0 200 400 600 800 1000
t t
t (hours) t (hours)

13. cont...
FMC9, no escape, varying flame index FMC9, escape, varying flame index
1.0
1.0
FI:125 FI:125
0.8
0.8
FI:250 FI:250
FI:500 FI:500
FMC9_125
FMC9_125
0.6
0.6
S(t) estimate
S(t) estimate
0.4
0.4
0.2
0.2
0.0
0.0
0 200 400 600 800 1000 0 200 400 600 800 1000
t t
t (hours) t (hours)

14. Expected Value of Lifetimes
Estimated Expected value of
lifetime1 for compartment 9
FMC Int. Att. Success Flame Index Expected
Lifetime (hours)
9 No escape 125 56
9 No escape 250 62
9 No escape 500 76
9 Escape 125 182
9 Escape 250 202
9 escape 500 249

15. Showing the impact of covariates and fire management compartment
on lifetime2
No escape, people caused, Flame index 125 Escape, lightning caused, Flame index 125
1.0
1.0
FMC9 FMC9
0.8
0.8
FMC11 FMC11
0.6
0.6
S(t) estimate
S(t) estimate
FMC9
FMC9
0.4
0.4
0.2
0.2
0.0
0.0
0 200 400 600 800 1000 0 200 400 600 800 1000
t t
t (hours) t (hours)

16. cont...
FMC9, people caused, no escape, varying FMC9, people caused, escape, varying
flame index flame index
1.0
1.0
FI:125 FI:125
0.8
0.8
FI:250 FI:250
FI:500 FI:500
FMC9_125
FMC9_125
0.6
0.6
S(t) estimate
S(t) estimate
0.4
0.4
0.2
0.2
0.0
0.0
0 200 400 600 800 1000 0 200 400 600 800 1000
t t
t (hours) t (hours)

17. Expected Value of Lifetimes
Estimated Expected Value of
lifetime2 for compartment 9
FMC Int. Att. Cause Flame Index Expected
Success Lifetime (hours)
9 No escape Lightning 125 10
9 No escape Lightning 250 11
9 No escape Lightning 500 12
9 Escape Lightning 125 35
9 Escape Lightning 250 37
9 Escape Lightning 500 41

18. Residual Analysis Lifetime1
KM estimates of residuals for FMC9
1.0
KM estimates of residuals for FMC11
0.8
1.0
0.6
0.8
0.4
0.6
KM estimates of residuals for FMC27
0.2
0.4
1.0
0.0
0.2
0 5 10 15 20
0.8
0.0
0.6
0 5 10 15 20
0.4
0.2
0.0
0 5 10 15 20

19. Residual Analysis Lifetime2
KM estimates of residuals for FMC9
1.0
0.8
KM estimates of residuals for FMC11
0.6
1.0
0.4
0.8
0.2
0.6
0.0
0 2 4 6 8 10
0.4
0.2
0.0
0 2 4 6 8 10

20. Conclusions
 Ignition source, Success of initial attack, Difficulty of suppression are all
important variables in predicting the “lifetimes” of fires
 These covariates affect lifetimes of fires in similar way across most fire
management compartments, and both definitions of lifetime
 Lightning caused fires are longer
 Fires that escape are longer
 A higher “flame index” makes a fire longer
 The factors affect lifetimes to different degrees in different fire management
compartments (that are relatively homogenous with respect to weather,
fuels, fire management)
 Fire Management Compartments in the Extensive and Measured Fire
Management Zones tend to experience longer fire lifetimes than those in the
Intensive Zone

21. Questions? Suggestions?