## 2 Eruption Forecasting Algorithm
_Intensity: Will the eruption intensity exceed a VEI of 1?_ The determination of the intensity will be driven by the training dataset defined in Table 1. The probability of the eruption's intensity exceeding a Volcanic Explosivity Index, \((VEI)\), of 1 is given by \[P(\Theta_{4})=P(4|3)=\frac{P(3|4)P(4)}{P(3|4)P(4)+P(3|4^{\prime})P(4^{\prime})}\] (7) where \(P(4)\) is defined as \[P(4)=P(VEI>1|X_{4,N})\] (8) and \(X_{4,N}\) is a node specific set of \(N\) independent explanatory variables described in Section 2.4. * Node 5. _Vent Location: Where is the eruption likely to occur?_ A vent location probability map is generated and the area with the highest probability of occurrence is identified as the likely eruption location. Given that the previous conditions are satisfied, the most probable vent location may be those with collocated surface deformation and seismicity. The probability of an eruption occurring at any location in a specified area is given by \[P(\Theta_{5}^{(j)})=P(5^{(j)}|4)=\sum_{j=1}^{J}\frac{P(4|5^{(j)})P(5^{(j)})}{P( 4|5^{(j)})P(5^{(j)})+P(4|5^{(j)})P(5^{(j)})}\] (9) where \(J\) is the total number of probable vent locations, \(j\) is the location under consideration, and \(P(5)\) is defined in Section 2.5. The probability of an event occurring at a particular event tree node is defined as  
\[P(\Theta)=\prod_{1}^{l}P(\Theta_{l}) \tag{10}\]  
where \(l\) is the event tree node for which the probability is being calculated. For example, the probability of an eruption occurring at the \(j^{th}\) location is calculated from the product, \(P(\Theta_{1})\)\(P(\Theta_{2})\)\(P(\Theta_{3})\)\(P(\Theta_{4})\)\(P(\Theta_{5}^{(j)})\), whereas the probability of an eruption occurring at a given volcano is estimated by \(P(\Theta_{1})\)\(P(\Theta_{2})\)\(P(\Theta_{3})\). \begin{table}
\begin{tabular}{l l|c c c c|c c c c|c} \hline \hline  & \multicolumn{2}{c}{**Episode**} & \multicolumn{2}{c}{**Response Variable**} & \multicolumn{6}{c}{**Independent Variable**} \\ \hline Year & Volcano & VEI & Er & In & MM & TNE & TCSM & Days & TEH & Ref. \\
1993 & Medicine Lake\({}^{(2,3,4)}\) & 0 & 0 & 0 & 0 & 115 & 6.0e + 21 & 2492 & 1 & [13] \\
1993 & Makushin\({}^{(3,4)}\) & 0 & 0 & 1 & 1 & 0 & 0 & 365 & 12 & [14] \\
1994 & Hengill\({}^{(2,3,4)}\) & 0 & 0 & 1 & 1 & 63450 & 7.7e + 23 & 1607 & 0 & [15] \\
1995 & Trident\({}^{(2,3,4)}\) & 0 & 0 & 1 & 1 & 69 & 3.2e + 19 & 137 & 13 & [16] \\
1996 & Lassen Peak\({}^{(2,3,4)}\) & 0 & 0 & 0 & 0 & 110 & 3.4e + 21 & 1460 & 1 & [17] \\
1996 & Eyjafjallajokull\({}^{(2,3,4)}\) & 0 & 0 & 1 & 1 & 144 & 5.2e + 19 & 114 & 2 & [18] \\
1996 & Akutan\({}^{(2)}\) & 0 & 0 & 1 & 1 & 1194 & 7.6e + 22 & 32 & 34 & [19] \\
1996 & Iliamma\({}^{(2,3,4)}\) & 0 & 0 & 1 & 1 & 1477 & 2.1e + 21 & 382 & 2 & [20] \\
1996 & Peulik\({}^{(2,3,4)}\) & 0 & 0 & 1 & 1 & 0 & 0 & 365 & 2 & [21,22] \\
1997 & Kilauea\({}^{(2,3)}\) & 1 & 1 & 1 & 0 & 1869 & 1.9e + 22 & 20 & 63 & [23] \\
1998 & Kiska\({}^{(2,3,4)}\) & 0 & 0 & 0 & 0 & 0 & 0 & 365 & 3 & [24] \\
1998 & Grimsvötn\({}^{(2,3,4)}\) & 3 & 1 & 1 & 1 & 31 & 9.4e + 20 & 10 & 29 & [25] \\
1999 & Shishaldin\({}^{(2,3,4)}\) & 3 & 1 & 1 & 0 & 688 & 9.0e + 22 & 42 & 34 & [26] \\
1999 & Fisher\({}^{(2,3,4)}\) & 0 & 0 & 0 & 1 & 0 & 0 & 365 & 1 & [27] \\
2000 & Kata\({}^{(2,3,4)}\) & 0 & 0 & 1 & 1 & 12460 & 1.2e + 23 & 2190 & 4 & [28] \\
2000 & Kilauea\({}^{(2,3,4)}\) & 0 & 0 & 0 & 0 & 48 & 5.8e + 20 & 13 & 63 & [29] \\
2000 & Three Sisters\({}^{(2,3,4)}\) & 0 & 0 & 1 & 1 & 0 & 0 & 1460 & 1 & [22] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Raw logistic regression training data. The event tree was adapted to issue warnings according to the United States Geological Survey (USGS) ground-based volcanic hazard system. Our system assigns colors ranging from green to red to a particular volcano according to its current eruption hazard condition, where green \(=\) normal, yellow \(=\) advisory, orange \(=\) watch, and red \(=\) warning. Figure 1 shows the color code assignments relative to each stage of the event tree.