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# Modeling of lifetimes of forest fires

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### Modeling of lifetimes of forest fires

1. 1. Modeling the “Lifetimes” of Forest Fires in Ontario:A Survival Analysis Project By: Dhriti Chakraborty
2. 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. 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. 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. 5. Components of Survival Model The survival function: The hazard function: Mean residual lifetime:
6. 6. Exploratory analysis using theKaplan-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. 7. 1.0 Survival curves as estimated by the KM estimator 1.0 Lifetime1 Lifetime2 0.8 0.8 S(t) estimateS(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. 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. 9. Lifetime1: Plot to check for Weibull Model Appropriateness 2 FMZ: I FMZ:M S0inverse_weibull(KMhat) 0log(−log (KM(t)) FMZ: E -2 -4 -6 -2 0 2 4 6 8 log(time) log t
10. 10. Lifetime2: Plot to check for Loglogistic Model Appropriateness 10log(1− (KM(t))/KM(t)) 5 FMZ:M loglogistic FMZ: I 0 -5 -4 -2 0 2 4 6 logt log t
11. 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. 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) estimateS(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. 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:500FMC9_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. 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. 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 0.6S(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. 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 0.6S(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. 17. Expected Value of Lifetimes Estimated Expected Value of lifetime2 for compartment 9FMC Int. Att. Cause Flame Index Expected Success Lifetime (hours)9 No escape Lightning 125 109 No escape Lightning 250 119 No escape Lightning 500 129 Escape Lightning 125 359 Escape Lightning 250 379 Escape Lightning 500 41
18. 18. Residual Analysis Lifetime1 KM estimates of residuals for FMC91.0 KM estimates of residuals for FMC110.8 1.00.6 0.80.4 0.6 KM estimates of residuals for FMC270.2 0.4 1.00.0 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. 19. Residual Analysis Lifetime2 KM estimates of residuals for FMC91.00.8 KM estimates of residuals for FMC110.6 1.00.4 0.80.2 0.60.0 0 2 4 6 8 10 0.4 0.2 0.0 0 2 4 6 8 10
20. 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. 21. Questions? Suggestions?