Probability Framework

We are interested in obtaining estimates of disturbance probabilities in the presence of multiple stressors. For example, we would like to quantify: Prob {damage from disturbance 1 given size of damage from disturbance 2 and values of other predictors}. Two specific disturbances, bark beetle infestation and fire size, were analyzed in the present case study. Fires of all sizes were used in the analysis.

Bark Beetle Infestation

Let Y be a random variable such that Y_i=1 if an infestation is present in a km² grid at location (utmx_i,utmy_i) and year yr_i and Y_i=0 otherwise. The random variable Y is assumed to follow a Bernoulli distribution with probability of response

Equation 1

and where the matrix X of predictors includes history of fire and insect infestation in previous years and within 1 km of infestation. The functions g_k(X_k) in equation [1] are transformations of the variable X_k using linear-basis splines (Hastie and others 2001). For the spatial component, g(utmx utmy), we utilized the 2-dimensional version of the basis function, specifically, the thin plate spline function. All estimations were done within the statistical package R (R Development Core Team 2004). The required modules for fitting thin plate splines within R were downloaded from the web (Geophysical Statistical Project 2002). By including variables such as location and history of insect infestation in the model, we were able to study the effect of fire (occurrence and size), on the probability of an infestation, in the presence of confounding factors, (e.g., presence of infestations in previous years). The spatial component in the model can be viewed as a surrogate for other predictors, (e.g., elevation, fuel type, and other variables) that do not change substantially over time and are not part of the model for a variety of reasons (Hobert and others 1997). Because the number of cells with no insect damage was very large (2.4 million), only a random sample of insect-free locations was included in the analysis. The final probabilities were then adjusted accordingly (Preisler and others 2004, Maddala 1992).

Results of fitting equation [1] were expressed as maps of estimated probabilities (and estimated standard errors) that are spatially explicit on 1-km² grid cells. The maps may be used to forecast probabilities of occurrence of an insect infestation for a succeeding year, given the history of fire and insect infestation up to that year. Outputs can be appraised directly by comparing observed frequencies of occurrences with predicted probabilities using cross-validation.

Using this model, we were able to quantify the interactions between different disturbances by studying the significance and shape of the relationships between the probability of an infestation and the occurrence and size of fire in previous years. In particular, we produced estimates of the odds of an infestation as a function of fire size relative to the odds when no fire is present. Odds were defined as: , where is the probability of a response of interest, in this case, bark beetle attack.

Fire Size

To study the association between insect disturbances and the risk of wildfire becoming large at a given location, we divided fires into three size classes. Consider the ordinal random variable Y where Y_i=1 if a fire at location (utmx_i, utmy_i ) and time t_i burns an area less than or equal to 1 ha; Y_i=2 if area burned is 1,100 ha; and Y_i=3 if area burned is greater than 100 ha. Next, assume that the random variable, Y, follows a multinomial distribution with probability of response at level k (k=1, 2, 3), given by _ik =Pr[Y_i=k | X] and _i1 + _i2 + _i3=1. The list of predictors, X, included average monthly temperature and PDSI, size of bark beetle infestation (trees killed per km²), and size of spruce budworm infestation (area defoliated per km²) in the previous 6 years.

We obtained estimates for _ik by fitting a complementary log-log function to the conditional probabilities of response. Specifically, we used the model

Equation 2

where the functions g(utmx_i, utmy_i ) and g(X) are as defined in (2.2.1). We obtained estimates of the multinomial probabilities from the conditional probabilities using the relationships, _i1 =p_{i1; i2} = (1- p_i1)p_i2; and _i3 = (1- p_i1- p_i2). As in section (2.2.1) the model goodness-of-fit was appraised by comparing observed and predicted frequencies of fires in the three different size classes.

We produced maps of estimated probabilities displaying the spatial patterns of fires of different size classes. We also studied association between insect infestations and fire size by producing graphs of estimated odds of a fire of a given size class—plotted against infestation size—relative to the odds when no infestation is present.