1. Forecasting occurrences of wildfires & earthquakes using point processes with directional covariates Frederic Paik Schoenberg, UCLA Statistics Collaborators: Haiyong Xu, Ka Wong. Also thanks to: Yan Kagan, James Woods, USGS, SCEC, NCEC, & Harvard catalogs.

2. 1)Background 2)Existing point process models for wildfires & earthquakes 3)Problems, esp. wind & moment tensors 4)Directional kernel direction & wind 5)Using focal mechanisms in ETAS

3. Los Angeles County wildfire centroids, 1960-2000

4. 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)  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.

8. Global Earthquake Data: Local e.q. catalogs tend to have problems, esp. missing data. 1977: Harvard (global) catalog created. Considered the most complete. Errors best understood. Harvard Catalog, 1/1/84 to 4/1/07 Shallow events only (depth < 70km) Mw 3.0+ Only focal mechanism estimates of high or medium quality

11. 2. Existing models for forecasting eqs & fires NFDRS’s Burning Index (BI): Uses daily weather variables, drought index, and vegetation info. Human interactions excluded.

12. 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.

13. 1

14. Point Process Models Conditional rate (t, x 1, …, x k ;  ): [e.g. x 1 =location, x 2 = area.] a non-neg. predictable process such that ∫ (dN - d  ) is a martingale. Wildfire incidence seems roughly multiplicative. (only marginally significant in separability test) Roughly exponential in relative humidity (RH), windspeed (W), precipitation (P), avg precip over prior 60 days (A), temperature (T), and date (D). Tapered Pareto size distribution f, smooth spatial background . (t,x,a) = f(a)  (x)  1 exp{  2 R(t) +  3 W(t) +  4 P(t) +  5 A(t) +  6 T(t) +  7 [  8 - D(t)] 2 } Or, split by season: (t,x,a) = f(a)  (x)  1,i exp{  2,i R(t) +  3,i W(t) +  4,i P(t) +  5,i A(t) +  6,i T(t)}

17. Aftershock activity described by modified Omori Law: K/(t+c) p

19. 3. Some problems with existing models BI has low correlation with wildfire.  Corr(BI, area burned) = 0.09  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)

23. 3. Some problems with existing models, continued Wildfires: no use of wind direction. Santa Ana winds (from NE) typically hot & dry. ETAS: no use of focal mechanisms. Summary of principal direction of motion in an earthquake, as well as resulting stress changes and tension/pressure axes.

24. 4. Directional kernel regression and wind: ∑ i y i g  (  -  i ) / ∑ g  (  -  i ), using a circular kernel g, such as the von-Mises density g  (  ) = exp{  cos(  )}/2  I 0 (  ).

25. 4. Directional kernel regression and wind: f(  ) estimated via ∑ i y i g  (  -  i ) / ∑ g  (  -  i ), using a circular kernel g, such as the von-Mises density g  (  ) = exp{  cos(  )}/2  I 0 (  ).

26. RH< 15% 15% < RH < 30%

27. Improvement in forecasting

28. 5. Using focal mechanisms in ETAS

29. Distance to next event, in relation to nodal plane of prior event

30. In ETAS (Ogata 1998), (t,x,m) = f(m)[  (x) + ∑ i g(t-t i, x-x i, m i )], where f(m) is exponential,  (x) is estimated by kernel smoothing, i.e. the spatial triggering component, in polar coordinates, has the form: g(r,  ) = (r 2 + d) q. Looking at inter-event distances in Southern California, as a function of the direction   of the principal axis of the prior event, suggests: g(r,  ;   ) = g 1 (r) g 2 (  -   | r), where g 1 is the tapered Pareto distribution, and g 2 is the wrapped exponential. and

36. tapered Pareto / wrapped exp.  biv. normal (Ogata 1998)  Cauchy/ ellipsoidal (Kagan 1996) 

37. Thinned residuals: Data  tapered Pareto / wrapped exp.  Cauchy/ ellipsoidal (Kagan 1996)  biv. normal (Ogata 1998) 

38. Tapered pareto / wrapped exp. Cauchy / ellipsoidal

39. Cauchy/ ellipsoidal (Kagan 1996) biv. normal (Ogata 1998) tapered pareto / wrapped exp.

40. Conclusions: The impact of directional variables on a scalar response can readily be summarized using directional kernel regression. The resulting function can then be incorporated into point process models, to improve forecasting of the response variable. Wildfires: wind direction is very significant, and models incorporating wind direction and other weather variables forecast about twice as well as the BI (which uses these same variables). Earthquakes: focal mechanism estimates should be used to improve triggering functions in ETAS models.

45. Greenness (UCLA IoE)

46. (IoE)

49. On the Predictive Value of Fire Danger Indices: From Day 1 of Toronto workshop (05/24/05): Robert McAlpine: “[It] 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.