"Did your model account for earthworms?" Rick Paik Schoenberg, UCLA 1. Missing covariates, confounding, and misspecification. 2. Missing covariate results involving point process models. 3. Examples involving eq weather and alternatives to BI.

1. Missing covariates and confounding. Some models for wildfire occurrence use a function of vegetation type, rel. greenness, fuel moisture, precipitation, temperature, windspeed, and perhaps other variables. Missing other confounding variables is inevitable. Even if your model incorporated many other variables, you would likely have a major mis-specification problem. Also, data on some of the variables would be missing or impractical to obtain.

Burning Index (BI) NFDRS: Spread Component (SC) and Energy Release Component (ERC), each based on dozens of equations. BI = [10.96 x SC x ERC] 0.46 Uses daily weather variables, drought index, and vegetation info. Human interactions excluded.

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.

In practice, most of these variables are not recorded, and the BI for instance is estimated using just a few variables like RH, windspeed, temp, veg. type, and precipitation. For a given BI range, area burned varies dramatically based on month. This may be due to missing covariates or to mis-specification.

In practice, most of these variables are not recorded, and the BI for instance is estimated using just a few variables like RH, windspeed, temp, veg. type, and precipitation. For a given BI range, area burned varies dramatically based on month. This may be due to missing covariates or to mis-specification. In general, a model might not be completely invalidated just because of a missing variable. Under some circumstances, the parameters in the model can be estimated consistently in the absence of missing variables. This is well known but has been difficult to find a citation for.

Point process modeling. Consider l(t, x1, …, xk; q). [For fires, x1=location, x2 = area.]

Separable Estimation for Point Processes Consider l(t, x1, …, xk; q). [For fires, x1=location, x2 = area.] 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), and q-j and l-j are defined similarly. If l ~is multiplicative in xj and if one of these holds, then qj, the partial MLE, is consistent (Ogata 1978, Schoenberg 2016): S l-j(t, x-j; q-j) dm-j = g, for all q-j. 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.

Individual Covariates: If l is multiplicative, lj(t,xj; qj) = f1[X(t,xj); b1] f2[Y(t,xj); b2], X and Y are independent, and the log-likelihood is differentiable w.r.t. b1, then the partial MLE of b1 is consistent. If the missing component j is additive: lj(t,xj; qj) = f1[X(t,xj); b1] + f2[Y(t,xj); b2], and f2 is small (S f2(Y; b2)2 / f1(X;~b1) dm / T ->p 0), ~ then the partial MLE b1 is consistent.

Impact 1. If a variable is missing but it has a small additive effect on the rate, then you can still have consistent parameter estimates. An example is earthquake weather.

Earthquake weather example. l(t,x,y,m) = mr(x,y)exp(n Temp(t) + Kg(t-ti, x-xi, y-yi ; mi), with g(u,x,y ; mi) = (u+c)-p exp{a(mi-M0)} (||x+y||2 + d)-q. model with temp model without temp

Impact 1. If a variable is missing but it has a small additive effect on the rate, then you can still have consistent parameter estimates. An example is earthquake weather. 2. Model building. When building a model with many variables, such as BI, misspecification is a major problem. One way to help avoid misspecification is to look at the effect of one explanatory variable at a time on your response variable, using e.g. kernel smoothing. For instance, one might model the rate as exponential in RH and windspeed, linear in temp, and linear in precip. with a threshold. A simple model built this way, using the same variables as BI, fits much better than BI.

r = 0.16 (sq m)

(sq m) (F)

Relative AICs (Poisson - Model, so higher is better): Model Construction Wildfire incidence seems roughly multiplicative. (only marginally significant in separability test) Windspeed. RH, Temp, Precip. Tapered Pareto size distribution f, smooth spatial background m. [*] l(t,x,a) = f(a) m(x) b1exp(b2RH + b3WS) (b4 + b5Temp)(max{b6 - b7Prec,b8}) Relative AICs (Poisson - Model, so higher is better): Poisson RH BI Model [*] 262.9 302.7 601.1

% of fires correctly alarmed Comparison of Predictive Efficacy False alarms per year % of fires correctly alarmed BI 150: 32 22.3 Model [*]: 34.1 BI 200: 13 8.2 15.1

Conclusions: Build initial attempts at point process models by examining each covariate individually. Surely important variables will be missing from the model. If the missing variables have a small additive effect on the response, or if the missing variables have a purely multiplicative effect, then parameter estimates will be consistent despite the missing variables. References: Ogata Y (1978). The asymptotic behaviour of maximum likelihood estimators for stationary point processes. Annals of the Institute of Statistical Mathematics 30, 243-261. Schoenberg FP (2016). A note on the consistent estimation of spatial-temporal point process parameters. Statistica Sinica 26, 861-879.