# Use of Logistic Regression for Forecasting Short-Term Volcanic Activity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Eruption Forecasting Algorithm

**Figure 1.**Schematic representation of the event tree, where the clone label indicates the tree structure at that point is identical to that below. The color code represents the USGS ground-based hazard declarations and are superimposed over their respective event tree branch.

_{1}is the summation of several explanatory variables that are defined in Section 2.3.

^{th}location is calculated from the product, P(Θ

_{1}) P(Θ

_{2}) P(Θ

_{3}) P(Θ

_{4}) , whereas the probability of an eruption occurring at a given volcano is estimated by P(Θ

_{1}) P(Θ

_{2}) P(Θ

_{3}).

Episode | Response Var. | Independent Variable | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

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 | Eyjafjallajökull^{(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 | Iliamna^{(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 | Grimsv"{o]tn^{(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 | Katla^{(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] |

2000 | Hekla^{(2,3,4)} | 3 | 1 | 1 | 1 | 196 | 3.4e + 20 | 15 | 9 | [30,31] |

2000 | Eyjafjallajökull^{(2,3,4)} | 0 | 0 | 1 | 1 | 170 | 4.1e + 20 | 365 | 2 | [32] |

2001 | Etna^{(2,3,4)} | 2 | 1 | 1 | 1 | 414 | 1.0e + 23 | 28 | 115 | [33] |

2001 | Okmok^{(4)} | 0 | 0 | 0 | 1 | 19 | 8.1e + 21 | 2 | 16 | [34] |

2001 | Aniakchak^{(2,3,4)} | 0 | 0 | 0 | 1 | 13 | 5.8e + 20 | 64 | 1 | [35] |

2002 | Hood^{(2,3,4)} | 0 | 0 | 0 | 0 | 86 | 7.8e + 22 | 60 | 2 | [36] |

2002 | Etna^{(2,3,4)} | 3 | 1 | 1 | 1 | 353 | 2.1e + 23 | 94 | 115 | [37] |

2003 | Veniaminof^{(2)} | 2 | 1 | 1 | 1 | 103 | 6.2e + 20 | 1050 | 22 | [38] |

2004 | Grimsvötn^{(2,3,4)} | 3 | 1 | 0 | 0 | 920 | 6.3e + 21.9 | 490 | 29 | [39] |

2004 | Spurr^{(3,4)} | 0 | 0 | 0 | 0 | 2743 | 5.1e + 20 | 239 | 2 | [40,41] |

2004 | Etna^{(2,3,4)} | 1 | 1 | 1 | 1 | 156 | 4.8e + 21 | 186 | 115 | [42] |

2004 | Saint Helens^{(2,3,4)} | 2 | 1 | 1 | 1 | 1094 | 1.5e + 23 | 21 | 14 | [43] |

2005 | Augustine^{(2,3,4)} | 3 | 1 | 1 | 1 | 2007 | 3.1e + 20 | 80 | 9 | [44] |

2006 | Korovin^{(2,3,4)} | 1 | 1 | 0 | 1 | 377 | 1.4e + 21 | 329 | 7 | [45,46] |

2007 | Pavlof^{(2)} | 2 | 1 | 1 | 0 | 2 | 8.8e + 18 | 30 | 39 | [47] |

2007 | Upptyppingar^{(2,3,4)} | 0 | 0 | 1 | 0 | 3124 | 6.5e + 20 | 133 | 0 | [48] |

2008 | Yellowstone^{(2,3)} | 0 | 0 | 1 | 1 | 2594 | 6.9e + 22 | 49 | 0 | [49] |

2008 | Paricutin^{(2,3,4)} | 0 | 0 | 0 | 0 | 0 | 0 | 21900 | 1 | [50] |

2008 | Hengill | 0 | 0 | 0 | 0 | 3309 | 1.1e + 24 | 10 | 0 | [51] |

2008 | Okmok^{(2,3,4)} | 4 | 1 | 1 | 1 | 464 | 4.9e + 21 | 100 | 16 | [34,52] |

2008 | Kasatochi^{(2,3,4)} | 4 | 1 | 1 | 0 | 1489 | 7.4e + 24 | 22 | 1 | [53] |

2009 | Redoubt^{(2,3,4)} | 3 | 1 | 1 | 1 | 4219 | 3.9e + 21 | 365 | 6 | [31] |

2010 | Eyjafjallajökull^{(2,3,4)} | 4 | 1 | 1 | 1 | 4019 | 1.1e + 22 | 100 | 2 | [54] |

#### 2.1. Logistic Regression

_{0}is a constant (intercept), β

_{m}are the weighting (regression) coefficients, and X

_{m}are the explanatory variables. Values of β

_{0}and β

_{m}are determined through maximum likelihood estimation via a GLMR. This requires a training dataset consisting of a collection of observations that relate the outcome of a particular event to the explanatory variables.

^{th}event tree node. These models are defined as the conditional probability

_{n}is the outcome (i.e., 1 or 0) of the event represented by the n

^{th}node and z

_{n}is its respective logistic model. Weighting coefficients for each logistic model are described in Section 2.4.

_{n}), of the null and full models, where the null model is simply a constant with no associated explanatory variables. The deviance is similar to the residual sums of squares metric from ordinary least squares regression in the sense that it attempts to estimate the discrepancy between the modeled and observed data. Thus, the quality of the regression model decreases with increasing deviance. Here G is estimated via the ratio

_{null}and L

_{full}are the likelihoods of the null and full models. The test statistic is x

^{2}distributed, where its degrees of freedom, df, are equivalent to the number of constrained predictors (explanatory variables) in the logistic model. Statistical significance of the difference is estimated using a hypothesis test which compares the p-value of the test statistic to a predefined significance level (0.05), where the null hypothesis states the null model fits the data better than the full model. If there is strong evidence against the null hypothesis (e.g., p < 0.05), then it can be rejected in favor of the full model.

#### 2.2. Training Data

#### 2.3. Node 1: Detection of Volcanic Unrest

_{1}and Q

_{3}are the first and third quartiles of the sample distribution, and (Q

_{3}−Q

_{1}) is the interquartile range [64]. Unique thresholds are derived empirically on a volcano-by-volcano and monitoring discipline basis using Equation 15, where c is set empirically for each monitoring discipline.

_{m}are weighting coefficients and X

_{m}are binary variables that are toggled from 0 to 1 when measurements from their respective monitoring discipline exceeds its outlier detection threshold. Here the functional form of Equation 16 is expressed as

_{m}= 0.25) to allow activation of the forecasting algorithm upon the detection of unrest by one or more monitoring techniques. Severity declarations are based on the number of simultaneous cross-disciplinary detections, where ν

_{1}values of 0.25, 0.50, 0.75, and 1.0 represent low, moderate, heightened, and extreme levels of unrest. This triggering mechanism differs from those described in previously published event tree implementations (e.g., [7]) through its use of outlier analysis for detection threshold estimation and the incorporation of source modeling information.

Explanatory Variable | Description | Value |
---|---|---|

X_{sr} | Seismicity Rate | 0/1 |

X_{df} | Surface Deformation | 0/1 |

X_{lm} | Large Magnitude | 0/1 |

X_{md} | Model Indicates Intrusion | 0/1 |

#### 2.4. Nodes 2-4: Intrusion, Eruption, and Intensity

Explanatory Variable | Description | Value |
---|---|---|

X_{MM} | Unrest consistent with intrusion model | 0 or 1 |

X_{NE} | Average Number of Earthquakes Per Day | 0 – ∞ |

X_{CSM} | Average Normalized Cumulative Seismic Moment Per Day | 0 – ∞ |

X_{DAYS} | Episode Duration in Days | 0 – ∞ |

X_{ERH} | Average Eruption History | 0 – ∞ |

_{MM}) on each logistic function is examined by setting all other explanatory variables equal to each other and plotting the model response for values ≥ 0.

**Figure 2.**Logistic functions derived from bootstrapping process for the intrusion node, where the black and red curves represent X

_{MM}=0 and X

_{MM}=1. The transition of X

_{MM}from 0 to 1 increases the probability of occurrence by 0.60, which is the difference between the black and red curves when all continuous input variables are equal to 0.

**Figure 3.**Logistic functions derived from bootstrapping process for the Eruption node, where the black and red curves represent X

_{MM}=0 and X

_{MM}=1. The transition of X

_{MM}from 0 to 1 increases the probability of occurrence by 0.17, which is the difference between the black and red curves when all continuous input variables are equal to 0.

**Figure 4.**Logistic functions derived from bootstrapping process for the intensity node, where the black and red curves represent X

_{MM}=0 and X

_{MM}=1. The transition of X

_{MM}from 0 to 1 increases the probability of occurrence by 0.13, which is the difference between the black and red curves when all continuous input variables are equal to 0.

#### 2.5. Node 5: Vent Location

^{th}location is derived from the data used for monitoring a particular volcanic center. Here the PDF is constructed from deformation data (X

_{def}) and the spatiotemporal distribution of seismic epicenters (X

_{seis}). Combining this information yields the expression

#### 2.6. Cross Validation

- • True Positive (TP): Predicted and actual results are 1 (Valid Detection)
- • False Positive (FP): Predicted result is 1, but actual result is 0 (False Alarm)
- • False Negative (FN): Predicted result is 0, but actual result is 1 (Missed Detection)
- • True Negative (TN): Predicted and actual results are 0 (Valid Non-detection)

**Figure 5.**Receiver Operating Characteristics for the intrusion event tree node. The AUROC value of approximately 0.78 suggests this node will have fair to good predictive capabilities. TPR, FPR, accuracy, and precision estimates of 71%, 29%, 71% and 85% are obtained at the optimum detection threshold, (0.91).

**Figure 6.**Receiver Operating Characteristics for the eruption event tree node. The AUROC value of approximately 0.81 suggests this node will have fair to good predictive capabilities. TPR, FPR, accuracy, and precision estimates of 75%, 21%, 78%, and 74% are obtained at the optimum detection threshold, (0.47).

**Figure 7.**Receiver Operating Characteristics for the intensity event tree node. The AUROC value of approximately 0.80 suggests this node will have fair to good predictive capabilities. TPR, FPR, accuracy, and precision estimates of 73%, 19%, 78%, and 70% are obtained at the optimum detection threshold, (0.21).

#### 2.7. Algorithm Implementation

_{MM}) is set to 1 if the unrest is the result of a magmatic intrusion or 0 otherwise. In addition, the algorithm provides a mechanism for a human analyst to override any of the automated results. This allows the analyst to introduce information that may not have been accounted for in the algorithm’s decision making process (e.g., personal experience or information not included in the training data).

**Figure 9.**Internal functionality of the forecasting algorithm, where gray indicates processes internal to the algorithm, blue represents external data sources, and green identifies products.

## 3. Results and Discussion

#### 3.1. Grimsvötn Volcano, Iceland

**Table 4.**Estimated Mogi source parameters derived from GFUM 240 days before and 30 days after the 2011 Grimsvötn Eruption, where positive or negative values of C indicate uplift or subsidence.

Sample | Δ h | Δ r | d | C |
---|---|---|---|---|

Pre-eruption | 40 mm | 36 mm | 3.33 km | 0.0011 km^{3} |

Post-eruption | 250 mm | 468 mm | 1.60 km | –0.0061 km^{3} |

#### 3.2. Forecasts Preceding Grimsvötn’s 2011 Eruption

_{l}) of 2.27.

**Figure 12.**Boxplots highlighting the distribution of seismicity beneath the Grimsvötn caldera between 2005 and 2011, where the events per day and magnitude whiskers are set to 1.5 time the interquartile range. Monitoring thresholds are 8.0 events per day and a m

_{l}of 2.27.

_{L}3.54) on 25 November 2010. Unrest severity and trigger state vectors are shown in Figure 13 A and B. The rapidly fluctuating unrest severity is caused by elevated periods of anomalous seismicity throughout the episode (See Figure 14 A and B). On algorithm day 134, positive modeling results were injected into the algorithm. This injection was done to simulate potential delays associated with manual source modeling exercises that may be initiated upon the detection of unrest.

**Figure 13.**Algorithm state as a function of processing day. (

**A**) Unrest severity estimates per day; (

**B**) Trigger state per day.

**Figure 14.**VEFA input parameters. (

**A**) Count and average (X

_{NE}) number of earthquakes per episode day (X

_{DAYS}), shown in red and blue; (

**B**) Raw and average (X

_{CSM}) normalized seismic moment per episode day (X

_{DAYS}), shown in red and blue.

**Figure 15.**Forecasts of volcanic activity preceding Grimsvötn 2011 eruption, where the intrusion, eruption, and intensity probabilities and thresholds are shown by the red, black, and blue, solid and dotted lines. Introduction of positive modeling results occurs on episode day 134. (

**A**) Probability of occurrence for event tree nodes; (

**B**) Volcanic hazard color code declaration.

^{2}, J = 7,094,495, and all = 0. Modeling results shown in Figure 11 suggest that the deformation source is located approximately 3.33 km below the caldera center and has an isotropic radiation pattern with a 17 km radius. This information provides the justification to quantitatively constrain the PDF to a square search area spanning 1156 km

^{2}, reduce J to 2,788,920 probable locations, and set all = 1. As seismic activity accumulates over time, the spatial PDF begins to highlight a preferred vent location. Maps computed for day 134 and day 177 show that the most likely area for vent formation is in the south and southeastern portions of the caldera. Grimsvötn’s 2011 eruption occurred in the southern section of the caldera with a VEI greater than 1.0 [71].

**Figure 16.**Spatial Probability Density Maps for volcanic activity preceding Grimsvötn’s 2011 eruption, where the black ellipse outlines the approximate perimeter of the caldera. The plot for day 177 has been enlarged to emphasize regions with a higher probability of vent formation within the quantitatively constrained area.

## 4. Comparisons Between Algorithms

_{MM}) were included in the BETEF training set to determine their influence on its forecasting capability. Since eruption history is built directly into the BETEF model during the design process, the X

_{ERH}parameter is not required as a real time input. Weighting coefficients for all BETEF input parameters (X

_{NE}, X

_{CSM}, X

_{MM}, and

_{X}

_{DAYS}) were set to 1. All VEFA and BETEF forecasts are based on the same input data to ensure the comparability of results. The median value of the BETEF forecast is used for comparison purposes in all examples shown below. Error estimates were excluded from the comparisons since they are large and typically span the entire probability range.

**Table 5.**Test event outcome for each algorithm stage and associated seismicity level, where 1 or 0 indicates whether the event occurred or not and if seismicity levels were high or low.

Volcano | Intrusion | Eruption | Intensity (VEI>1) | High Seismicity |
---|---|---|---|---|

Grimsvötn | 1 | 1 | 1 | 1 |

Mount Saint Helens | 0 | 0 | 0 | 0 |

Okmok | 1 | 1 | 1 | 0 |

Yellowstone | 1 | 0 | 0 | 1 |

**Figure 17.**Intrusion probability comparisons for selected episode days, where each time sample is highlighted with a circle and the VEFA and BETEF results are shown in blue and red. (

**A**) Grimsvötn 2011; (

**B**) Mount Saint Helens 2011; (

**C**) Okmok 2008; (

**D**) Yellowstone 2010.

**Figure 18.**Eruption probability comparisons for selected episode days, where each time sample is highlighted with a circle and the VEFA and BETEF results are shown in blue and red. (

**A**) Grimsvötn 2011; (

**B**) Mount Saint Helens 2011; (

**C**) Okmok 2008; (

**D**) Yellowstone 2010.

**Figure 19.**Intensity probability comparisons for selected episode days, where each time sample is highlighted with a circle and the VEFA and BETEF results are shown in blue and red. (

**A**) Grimsvötn 2011; (

**B**) Mount Saint Helens 2011; (

**C**) Okmok 2008; (

**D**) Yellowstone 2010.

## 5. Summary and Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Junek, W.N.; Jones, L.W.; Woods, M.T.
Use of Logistic Regression for Forecasting Short-Term Volcanic Activity. *Algorithms* **2012**, *5*, 330-363.
https://doi.org/10.3390/a5030330

**AMA Style**

Junek WN, Jones LW, Woods MT.
Use of Logistic Regression for Forecasting Short-Term Volcanic Activity. *Algorithms*. 2012; 5(3):330-363.
https://doi.org/10.3390/a5030330

**Chicago/Turabian Style**

Junek, William N., Linwood W. Jones, and Mark T. Woods.
2012. "Use of Logistic Regression for Forecasting Short-Term Volcanic Activity" *Algorithms* 5, no. 3: 330-363.
https://doi.org/10.3390/a5030330