1. 1 Applications of space-time point processes in wildfire forecasting 1.Background 2.Problems with existing models (BI) 3.A separable point process model 4.Testing separability 5.Alarm rates & other basic assessment techniques Thanks to: Herb Spitzer, Frank Vidales, Mike Takeshida, James Woods, Roger Peng, Haiyong Xu, Maria Chang.

2. 2 Los Angeles County wildfires, 1960-2000

3. 3 Background  Brief History. 1907: LA County Fire Dept. 1953: Serious wildfire suppression. 1972/1978: National Fire Danger Rating System. (Deeming et al. 1972, Rothermel 1972, Bradshaw et al. 1983) 1976: Remote Access Weather Stations (RAWS).  Damages. 2003: 738,000 acres; 3600 homes; 26 lives. (Oct 24 - Nov 2: 700,000 acres; 3300 homes; 20 lives) Bel Air 1961: 6,000 acres; $30 million. Clampitt 1970: 107,000 acres; $7.4 million.

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6. 6 NFDRS’s Burning Index (BI): Uses daily weather variables, drought index, and vegetation info. Human interactions excluded.

7. 7 Some BI equations : (From Pyne et al., 1996:) Rate of spread: R = I R  (1 +  w  +  s ) / (  b  Q ig ).Oven-dry bulk density:  b = w 0 / . Reaction Intensity: I R =  ’ w n h  M  s.Effective heating number:  = exp(-138/  ). Optimum reaction velocity:  ’ =  ’ max (  /  op ) A exp[A(1-  /  op )]. Maximum reaction velocity:  ’ max =  1.5 (495 + 0.0594  1.5 ) -1. Optimum packing ratios:  op = 3.348  -0.8189. A = 133  -0.7913. Moisture damping coef.:  M = 1 - 259 M f /M x + 5.11 (M f /M x ) 2 - 3.52 (M f /M x ) 3. Mineral damping coef.:  s = 0.174 S e -0.19 (max = 1.0). Propagating flux ratio:  = (192 + 0.2595  ) -1 exp[(0.792 + 0.681  0.5 )(  + 0.1)]. Wind factors:  w = CU B (  /  op ) -E. C = 7.47 exp(-0.133  0.55 ). B = 0.02526  0.54. E = 0.715 exp(-3.59 x 10 -4  ). Net fuel loading: w n = w 0 (1 - S T ).Heat of preignition: Q ig = 250 + 1116 M f. Slope factor:  s = 5.275  -0.3 (tan  2.Packing ratio:  =  b /  p.

8. 8 On the Predictive Value of Fire Danger Indices: From Day 1 (05/24/05) of Toronto workshop: Robert McAlpine: “[DFOSS] works very well.” David Martell: “To me, they work like a charm.” Mike Wotton: “The Indices are well-correlated with fuel moisture and fire activity over a wide variety of fuel types.” Larry Bradshaw: “[BI is a] good characterization of fire season.” Evidence? FPI: Haines et al. 1983 Simard 1987 Preisler 2005 Mandallaz and Ye 1997 (Eur/Can), Viegas et al. 1999 (Eur/Can), Garcia Diez et al. 1999 (DFR), Cruz et al. 2003 (Can). Spread: Rothermel (1991), Turner and Romme (1994), and others.

9. 9 Some obvious problems with BI: Too additive: too low when all variables are med/high risk. Low correlation with wildfire.  Corr(BI, area burned) = 0.09  Corr(BI, # of fires) = 0.13  Corr(BI, area per fire) = 0.076 !Corr(date, area burned) = 0.06 !Corr(windspeed, area burned) = 0.159 Too high in Winter (esp Dec and Jan) Too low in Fall (esp Sept and Oct)

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14. 14 Some obvious problems with BI: Too additive: too high for low wind/medium RH, Misses high RH/medium wind. (same for temp/wind). Low correlation with wildfire.  Corr(BI, area burned) = 0.09  Corr(BI, # of fires) = 0.13  Corr(BI, area per fire) = 0.076 !Corr(date, area burned) = 0.06 !Corr(windspeed, area burned) = 0.159 Too high in Winter (esp Dec and Jan) Too low in Fall (esp Sept and Oct)

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16. 16 More problems with BI: Low correlation with wildfire.  Corr(BI, area burned) = 0.09  Corr(BI, # of fires) = 0.13  Corr(BI, area per fire) = 0.076 !Corr(date, area burned) = 0.06 !Corr(windspeed, area burned) = 0.159 Too high in Winter (esp Dec and Jan) Too low in Fall (esp Sept and Oct)

17. 17 r = 0.16 (sq m)

18. 18 More problems with BI: Low correlation with wildfire.  Corr(BI, area burned) = 0.09  Corr(BI, # of fires) = 0.13  Corr(BI, area per fire) = 0.076 !Corr(date, area burned) = 0.06 !Corr(windspeed, area burned) = 0.159 Too high in Winter (esp Dec and Jan) Too low in Fall (esp Sept and Oct)

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21. 21 Constructing a Point Process Model as an Alternative to BI…. Definition: A point process N is a Z + -valued random measure N(A) = Number of points in the set A.

22. 22 More Definitions: Simple: N({x}) = 0 or 1 for all x, almost surely. (No overlapping pts.) Orderly: N(t, t+  )/  ----> p 0, for each t. Stationary: The joint distribution of {N(A 1 +u), …, N(A k +u)} does not depend on u.

23. 23 Intensities (rates) and Compensators -------------x-x-----------x----------- ----------x---x--------------x------ 0t- t t+ T Consider the case where the points are observed in time only. N[t,u] = # of pts between times t and u. Overall rate:  (t) = lim  t -> 0 E{N[t, t+  t)} /  t. Conditional intensity: (t) = lim  t -> 0 E{N[t, t+  t) | H t } /  t, where H t = history of N for all times before t. If N is orderly, then (t) = lim  t -> 0 P{N[t, t+  t) > 0 | H t } /  t.

24. 24 Intensities (rates) and Compensators -------------x-x-----------x----------- ----------x---x--------------x------ 0t- t t+ T These definitions extend to space and space-time: Conditional intensity: (t,x) = lim  t,  x -> 0 E{N[t, t+  t) x B x,  x | H t } /  t  x, where H t = history of N for all times before t, and B x,  x is a ball around x of size  x. The conditional intensity uniquely characterizes the distribution of a simple process. Suggests modeling  e.g. Model (t,x) as a function of BI on day t, interpolating between stations at x 1, x 2, …, x k using kernel smoothing:

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27. 27 Model Construction -- Some Important Variables: Relative Humidity, Windspeed, Precipitation, Aggregated rainfall over previous 60 days, Temperature, Date. -- Tapered Pareto size distribution g, smooth spatial background . (t,x,a) =  1 exp{  2 R(t) +  3 W(t) +  4 P(t)+  5 A(t;60) +  6 T(t) +  7 [  8 - D(t)] 2 }  (x) g(a). Two immediate questions: a) How do we fit a model like this? b) How can we test whether a separable model like this is appropriate for this dataset?

28. 28 Conditional intensity (t, x 1, …, x k ;  ): [e.g. x 1 =location, x 2 = size.] Separability for Point Processes: Say is multiplicative in mark x j if (t, x 1, …, x k ;  ) =  0 j (t, x j ;  j ) -j (t, x -j ;  -j ), where x -j = (x 1,…,x j-1, x j+1,…,x k ), same for  -j and -j If  ~ is multiplicative in x j ^ and if one of these holds, then  j, the partial MLE,  =  j, the MLE: S -j (t, x -j ;  -j ) d  -j = , for all  -j. S j (t, x j ;  j ) d  j = , for all  j. ^ ~ S j (t, x;  ) d  = S j (t, x j ;  j ) d  j = , for all .

29. 29 Individual Covariates: Suppose  is multiplicative, and j (t,x j ;  j ) = f 1 [X(t,x j );  1 ] f 2 [Y(t,x j );  2 ]. If H(x,y) = H 1 (x) H 2 (y), where for empirical d.f.s H,H 1,H 2, and if the log-likelihood is differentiable w.r.t.  1, then the partial MLE of  1 = MLE of  1. (Note: not true for additive models!) Suppose is multiplicative and the jth component is additive: j (t,x j ;  j ) = f 1 [X(t,x j );  1 ] + f 2 [Y(t,x j );  2 ]. If f 1 and f 2 are continuous and f 2 is small: S f 2 (Y;  2 ) 2 / f 1 (X; ~  1 ) d  p 0], then the partial MLE  1 is consistent.

30. 30 Impact Model building. Model evaluation / dimension reduction. Excluded variables.

31. 31 Model Construction (t,x,a) =  1 exp{  2 R(t) +  3 W(t) +  4 P(t)+  5 A(t;60) +  6 T(t) +  7 [  8 - D(t)] 2 }  (x) g(a). Estimating each of these components separately might be somewhat reasonable, as a first attempt at least, if the interactions are not too extreme.

32. 32 r = 0.16 (sq m)

33. 33 Testing separability in marked point processes: Construct non-separable and separable kernel estimates of by smoothing over all coordinates simultaneously or separately. Then compare these two estimates: (Schoenberg 2004) May also consider: S 5 = mean absolute difference at the observed points. S 6 = maximum absolute difference at observed points.

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35. 35 S 3 seems to be most powerful for large-scale non-separability:

36. 36 Testing Separability for Los Angeles County Wildfires:

37. 37 r = 0.16 (sq m)

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39. 39 (F) (sq m)

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42. 42 Model Construction Wildfire incidence seems roughly separable. (only area/date significant in separability test) Tapered Pareto size distribution f, smooth spatial background .  (t,x,a) =  1 exp{  2 R(t) +  3 W(t) +  4 P(t)+  5 A(t;60) +  6 T(t) +  7 [  8 - D(t)] 2 }  (x) g(a). Compare with:  (t,x,a) =  1 exp{  2 B(t)}  (x) g(a), where B = RH or BI. Relative AICs (Poisson - Model, so higher is better): PoissonRHBIModel  0262.9302.7601.1

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44. 44 Comparison of Predictive Efficacy False alarms per year % of fires correctly alarmed BI 150:3222.3 Model  :3234.1 BI 200:138.2 Model  :1315.1

45. 45 One possible problem: human interactions. …. but BI has been justified for decades based on its correlation with observed large wildfires (Mees & Chase, 1993; Andrews and Bradshaw, 1997). Towards improved modeling Time-since-fire (fuel age)

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47. 47 Towards improved modeling Time-since-fire (fuel age) Wind direction

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50. 50 Towards improved modeling Time-since-fire (fuel age) Wind direction Land use, greenness, vegetation

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52. 52 Greenness (UCLA IoE)

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54. 54 Towards improved modeling Time-since-fire (fuel age) Wind direction Land use, greenness, vegetation Precip over previous 40+ days, lagged variables

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58. 58 Conclusions: (For Los Angeles County data, Jan 1976- Dec 2000:) BI is positively associated with fire incidence and burn area, though its predictive value seems limited. Windspeed has a higher correlation with burn area, and a simple model using RH, windspeed, precipitation, aggregated rainfall over previous 60 days, temperature, & date outperforms BI. For multiplicative models (and sometimes for additive models), can estimate different components separately.