## 2 Eruption Forecasting Algorithm
### Nodes 2-4: Intrusion, Eruption, and Intensity  
Ideally, the regression uses many combinations of random samples from the population we are attempting to model. After many iterations, a distribution of parameter estimates, such as the sample mean or regression coefficients, can be produced and their true value estimated. In this case, however, there are no additional data available. Therefore, a bootstrapping approach is used to estimate the distribution of logistic model coefficients for nodes 2, 3, and 4 using selected subsets of data from Table 1.  
Bootstrapping is a resampling technique that produces \(M\) datasets from a single set of random samples taken from a specific population. Each of the newly constructed datasets contains a random combination of samples that were drawn from the original dataset. Since a sample with replacement process is used, it is possible for each new dataset to contain multiple or no copies of any particular sample. Thus, the probability that a particular sample is selected for inclusion in a bootstrapped dataset each time a drawing is made is \(1/N\), where \(N\) is the number of samples in the original dataset. Model coefficients for each node were estimated from the distributions generated after 50,000 bootstrapped GLMR iterations. The distributions for each of the logistic model coefficient appears to be bimodal. The two distributions represent cases where the regression is either properly or ill constrained. Situations where the regression is ill constrained caused the GLMR to fail after reaching its maximum number of iterations, which results in a bogus set of model coefficients. Conversely, properly constrained cases represent runs where the GLMR converges to a solution within its maximum number of iterations. Therefore, the ill constrained results are rejected as outliers and the properly constrained distributions are retained and used to estimate logistic model information. In all cases, the median value of the truncated distributions are used to estimate the coefficients for each model. The logistic models for nodes 2, 3, and 4 are shown in Equations 18-20 and a description of each explanatory variable is listed in Table 3. Node 3 (Equation 19) required an additional explanatory variable to account for the relationship between historic eruption activity and monitoring data. The likelihood ratio test for each model yielded p-values of 0.001, 0.014, and 0.007, which provides strong evidence against the null hypothesis for each node. The influence of the source modeling results (\(X_{MM}\)) on each logistic function is examined by setting all other explanatory variables equal to each other and plotting the model response for values \(\geq\) 0. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Explanatory Variable** & **Description** & **Value** \\ \hline \(X_{MM}\) & Unrest consistent with intrusion model & \(0\) or \(1\) \\ \(X_{NE}\) & Average Number of Earthquakes Per Day & \(0-\infty\) \\ \(X_{CSM}\) & Average Normalized Cumulative Seismic Moment Per Day & \(0-\infty\) \\ \(X_{DAYS}\) & Episode Duration in Days & \(0-\infty\) \\ \(X_{ERH}\) & Average Eruption History & \(0-\infty\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Logistic model explanatory variable names, descriptions, and possible range of values. Figures 2-4 show that positive modeling results increase the probability of occurrence for each node. In addition, each plot shows that even in the absence of positive modeling results, episodes producing large numbers of earthquakes and/or releasing large amounts of seismic energy have a high probability of being generated by a magmatic intrusion. Moreover, they also show that the probability of an eruption is non-zero when all explanatory variables are equal to zero. This result is reassuring, since eruptions have been known to occur without any precursory activity or warning.