Transcript chs2007 6882

Applications of space-time point processes
in wildfire forecasting
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Background
Problems with existing models (BI)
A separable point process model
Testing separability
Alarm rates & other basic assessment techniques
Thanks to: Herb Spitzer, Frank Vidales, Mike Takeshida, James Woods,
Roger Peng, Haiyong Xu, Maria Chang.
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Los Angeles County wildfires, 1960-2000
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Background
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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).
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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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NFDRS’s Burning Index (BI):
Uses daily weather variables, drought index, and
vegetation info. Human interactions excluded.
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Some BI equations: (From Pyne et al., 1996:)
Rate of spread: R = IR x (1 + fw + fs) / (rbe Qig).
Oven-dry bulk density: rb = w0/d.
Reaction Intensity: IR = G’ wn h hMhs.
Effective heating number: e = exp(-138/s).
Optimum reaction velocity: G’ = G’max (b / bop)A exp[A(1- b / bop)].
Maximum reaction velocity: G’max = s1.5 (495 + 0.0594 s1.5) -1.
Optimum packing ratios: bop = 3.348 s -0.8189. A = 133 s -0.7913.
Moisture damping coef.: hM = 1 - 259 Mf /Mx + 5.11 (Mf /Mx)2 - 3.52 (Mf /Mx)3.
Mineral damping coef.: hs = 0.174 Se-0.19 (max = 1.0).
Propagating flux ratio: x = (192 + 0.2595 s)-1 exp[(0.792 + 0.681 s0.5)(b + 0.1)].
Wind factors: sw = CUB (b/bop)-E. C = 7.47 exp(-0.133 s0.55). B = 0.02526 s0.54.
E = 0.715 exp(-3.59 x 10-4 s).
Net fuel loading: wn = w0 (1 - ST).
Heat of preignition: Qig = 250 + 1116 Mf.
Slope factor: fs = 5.275 b -0.3 (tan f)2.
Packing ratio: b = rb / rp.
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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?
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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).
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Spread: Rothermel (1991), Turner and Romme (1994), and others.
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Some obvious problems with BI:
• Too additive: too low when all variables are med/high risk.
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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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Some obvious problems with BI:
• Too additive: too high for low wind/medium RH,
Misses high RH/medium wind. (same for temp/wind).
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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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More problems with BI:
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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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(sq m)
r = 0.16
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More problems with BI:
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•
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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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.
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More Definitions:
• Simple: N({x}) = 0 or 1 for all x, almost surely.
(No overlapping pts.)
• Orderly: N(t, t+ D)/D ---->p 0, for each t.
• Stationary: The joint distribution of {N(A1+u), …,
N(Ak+u)} does not depend on u.
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Intensities (rates) and Compensators
-------------x-x-----------x----------- ----------x---x--------------x-----0
tt
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: m(t) = limDt -> 0 E{N[t, t+Dt)} / Dt.
•Conditional intensity: l(t) = limDt -> 0 E{N[t, t+Dt) | Ht} / Dt,
where Ht = history of N for all times before t.
•If N is orderly, then l(t) = limDt -> 0 P{N[t, t+Dt) > 0 | Ht} / Dt.
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Intensities (rates) and Compensators
-------------x-x-----------x----------- ----------x---x--------------x-----0
tt
t+
T
These definitions extend to space and space-time:
Conditional intensity:
l(t,x) = limDt,Dx -> 0 E{N[t, t+Dt) x Bx,Dx | Ht} / DtDx,
where Ht = history of N for all times before t,
and Bx,Dx is a ball around x of size Dx.
The conditional intensity uniquely characterizes the distribution of a
simple process. Suggests modeling l.
e.g. Model l(t,x) as a function of BI on day t, interpolating between
stations at x1, x2, …, xk using kernel smoothing:
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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 m.
l(t,x,a) = b1exp{b2R(t) + b3W(t) + b4P(t)+ b5A(t;60)
+ b6T(t) + b7[b8 - D(t)]2} m(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?
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Conditional intensity l(t, x1, …, xk; q): [e.g. x1=location, x2 = size.]
Separability for Point Processes:
• Say l is multiplicative in mark xj if
l(t, x1, …, xk; q) = q0 lj(t, xj; qj) l-j(t, x-j; q-j),
where x-j = (x1,…,xj-1, xj+1,…,xk), same for q-j and l-j
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If l ~is multiplicative in xj ^ and if one of these holds, then
qj, the partial MLE, = qj, the MLE:
• S l-j(t, x-j; q-j) dm-j = g, for all q-j.
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S lj(t, xj; qj) dmj = g, for all qj.
^
~
S lj(t, x; q) dm = S lj(t, xj; qj) dmj = g, for all q.
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Individual Covariates:
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Suppose l is multiplicative, and
lj(t,xj; qj) = f1[X(t,xj); b1] f2[Y(t,xj); b2].
If H(x,y) = H1(x) H2(y), where for empirical d.f.s H,H1,H2,
and if the log-likelihood is differentiable w.r.t. b1,
then the partial MLE of b1 = MLE of b1.
(Note: not true for additive models!)
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Suppose l is multiplicative and the jth component is additive:
lj(t,xj; qj) = f1[X(t,xj); b1] + f2[Y(t,xj); b2].
If f1 and f2 are continuous and f2 is small:
S f2(Y; b2)2 / f1(X;~b1) dm / T ->p 0],
then the partial MLE b1 is consistent.
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Impact
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Model building.
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Model evaluation / dimension reduction.
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Excluded variables.
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Model Construction
l(t,x,a) = b1exp{b2R(t) + b3W(t) + b4P(t)+ b5A(t;60)
+ b6T(t) + b7[b8 - D(t)]2} m(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.
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(sq m)
r = 0.16
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Testing separability in marked point processes:
Construct non-separable and separable kernel estimates of l by
smoothing over all coordinates simultaneously or separately.
Then compare these two estimates: (Schoenberg 2004)
May also consider:
S5 = mean absolute difference at the observed points.
S6 = maximum absolute difference at observed points.
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S3 seems to be most powerful for large-scale non-separability:
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Testing Separability for Los Angeles County Wildfires:
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(sq m)
r = 0.16
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(F)
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(sq m)
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Model Construction
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Wildfire incidence seems roughly separable.
(only area/date significant in separability test)
• Tapered Pareto size distribution f, smooth spatial background m.
[*] l(t,x,a) = b1exp{b2R(t) + b3W(t) + b4P(t)+ b5A(t;60)
+ b6T(t) + b7[b8 - D(t)]2} m(x) g(a).
Compare with:
[**] l(t,x,a) = b1exp{b2B(t)} m(x) g(a), where B = RH or BI.
Relative AICs (Poisson - Model, so higher is better):
Poisson
0
RH
262.9
BI
302.7
Model [*]
601.1
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Comparison of Predictive Efficacy
False alarms
per year
% of fires
correctly
alarmed
BI 150:
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22.3
Model [*]:
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34.1
BI 200:
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8.2
Model [*]:
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15.1
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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
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Time-since-fire (fuel age)
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(years)
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Towards improved modeling
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Time-since-fire (fuel age)
Wind direction
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Towards improved modeling
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Time-since-fire (fuel age)
Wind direction
Land use, greenness, vegetation
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Greenness (UCLA IoE)
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(IoE)
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Towards improved modeling
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Time-since-fire (fuel age)
Wind direction
Land use, greenness, vegetation
Precip over previous 40+ days, lagged variables
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(cm)
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Conclusions:
(For Los Angeles County data, Jan 1976- Dec 2000:)
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BI is positively associated with fire incidence and burn area,
though its predictive value seems limited.
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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.
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For multiplicative models (and sometimes for additive models),
can estimate different components separately.
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