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 Los Angeles County wildfires, 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. 4 5 6 NFDRSs Burning Index (BI): Uses daily weather variables, drought index, and vegetation info. Human interactions excluded. 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 ( 1.5 ) -1. Optimum packing ratios: op = A = 133 Moisture damping coef.: M = M f /M x (M f /M x ) (M f /M x ) 3. Mineral damping coef.: s = S e (max = 1.0). Propagating flux ratio: = ( ) -1 exp[( 0.5 )( + 0.1)]. Wind factors: w = CU B ( / op ) -E. C = 7.47 exp( 0.55 ). B = E = exp(-3.59 x ). Net fuel loading: w n = w 0 (1 - S T ).Heat of preignition: Q ig = M f. Slope factor: s = -0.3 (tan 2.Packing ratio: = b / p. 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 Simard 1987 Preisler 2005 Mandallaz and Ye 1997 (Eur/Can), Viegas et al (Eur/Can), Garcia Diez et al (DFR), Cruz et al (Can). Spread: Rothermel (1991), Turner and Romme (1994), and others. 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) = !Corr(date, area burned) = 0.06 !Corr(windspeed, area burned) = Too high in Winter (esp Dec and Jan) Too low in Fall (esp Sept and Oct) 10 11 12 13 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) = !Corr(date, area burned) = 0.06 !Corr(windspeed, area burned) = Too high in Winter (esp Dec and Jan) Too low in Fall (esp Sept and Oct) 15 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) = !Corr(date, area burned) = 0.06 !Corr(windspeed, area burned) = Too high in Winter (esp Dec and Jan) Too low in Fall (esp Sept and Oct) 17 r = 0.16 (sq m) 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) = !Corr(date, area burned) = 0.06 !Corr(windspeed, area burned) = Too high in Winter (esp Dec and Jan) Too low in Fall (esp Sept and Oct) 19 20 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 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 Intensities (rates) and Compensators x-x x x---x x t- 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 Intensities (rates) and Compensators x-x x x---x x t- 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: 25 26 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 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 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 Impact Model building. Model evaluation / dimension reduction. Excluded variables. 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 r = 0.16 (sq m) 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. 34 35 S 3 seems to be most powerful for large-scale non-separability: 36 Testing Separability for Los Angeles County Wildfires: 37 r = 0.16 (sq m) 38 39 (F) (sq m) 40 41 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 43 44 Comparison of Predictive Efficacy False alarms per year % of fires correctly alarmed BI 150: Model : BI 200:138.2 Model :1315.1 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) 46 (years) 47 Towards improved modeling Time-since-fire (fuel age) Wind direction 48 49 50 Towards improved modeling Time-since-fire (fuel age) Wind direction Land use, greenness, vegetation 51 52 Greenness (UCLA IoE) 53 (IoE) 54 Towards improved modeling Time-since-fire (fuel age) Wind direction Land use, greenness, vegetation Precip over previous 40+ days, lagged variables 55 (cm) 56 57 58 Conclusions: (For Los Angeles County data, Jan 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.