Bark Beetle Infestation - Preisler - Equation 2

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 Yi=1 if a fire at location (utmxi, utmyi ) and time ti burns an area less than or equal to 1 ha; Yi=2 if area burned is 1,100 ha; and Yi=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[Yi=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 where the functions g(utmxi, utmyi ) and g(X) are as defined in Bark Beetle Infestation section. We obtained estimates of the multinomial probabilities from the conditional probabilities using the relationships, i1 =pi1; i2 = (1- pi1)pi2; and i3 = (1- pi1- pi2). As in Bark Beetle Infestation section, the model goodness-of-fit was appraised by comparing observed and predicted frequencies of fires in the three different size classes.