Introduction
Statement of the problem
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The class of Quite (Q) state, labeled also as 0, indicates quiet, i.e. no relevant activity or unknown activity, characterized by low amplitudes of all physical features that can be related with the volcanic activity. Representing classes with a double symbol, numerical and literal, make easier on one hand the visualization as a timeseries of classes (numeric representation) and the readability (literal representation).
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the class of Strombolian (S) activity, labeled also as 1, indicates Strombolian activity, essentially characterized by a mildly explosive activity, with medium amplitude of seismic tremor RMS, shallower source of the seismic tremor, presence of clustered infrasonic events, no eruption column, but possible ash emissions.
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the class of Paroxysm (P) activity, labeled also as 2, indicates paroxysm, i.e. an energetic activity with lava fountains, characterized by high amplitude of seismic tremor RMS, presence of clustered infrasonic events, shallower source of the seismic tremor, eruption column formation and ash emissions.
Dataset and preliminary analysis
Volcanic seismic tremor
StationID | Latitude | Longitude | Elevation | Location |
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ECNE | 37.7653 | 15.0018 | 2946 | Etna Cratere Nord Est |
ECBD | 37.7802 | 15.0865 | 1465 | Etna Case Bada |
ECZM | 37.7313 | 14.9041 | 1391 | Etna Case Zampini |
ECPN | 37.7437 | 14.9865 | 3038 | Etna Cratere del Piano |
EMCN | 37.7912 | 15.0336 | 1916 | Monte Conca |
EMFO | 37.7357 | 15.0902 | 1209 | Etna Monte Fontane |
EMFS | 37.7196 | 14.9979 | 2552 | Etna Monte Frumento |
EMNR | 37.8168 | 15.026 | 1845 | Etna Monte Nero |
EMPL | 37.679 | 14.9703 | 1484 | Etna Monte Parmentelli |
EMSG | 37.8208 | 4.9498 | 1435 | Etna Monte Spagnolo |
EPDN | 37.7659 | 15.0168 | 2862 | Etna Pizzi Deneri |
EPIT | 37.8113 | 15.0567 | 1657 | Etna Pozzo Pitarrone, |
ESLN | 37.6934 | 14.9744 | 1787 | Etna Serra La Nave |
ESPC | 37.6925 | 15.0274 | 1655 | Etna Serra Pizzuta Calvarina |
ESVO | 37.7731 | 14.9469 | 1736 | Etna Monte Scavo |
Regression analysis of tremor time series
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x(t) is the tremor value measured at time t in a given reference station, assumed as input;
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y(t) is the tremor estimated by the linear model, into another station, assumed as output;
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m and b are the slope and the offset coefficients, respectively.
Station | R | slope | offset | Missing % |
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EBCN | 0.91 | 1.78 | 2.42 | 38.86 |
ECBD | 0.78 | 0.55 | 0.19 | 21.96 |
ECNE | 0.07 | 2.07 | 3.75 | 30.03 |
ECZM | 0.90 | 2.71 | 2.50 | 27.15 |
ECPN | 0.33 | 0.41 | 0.41 | 22.81 |
EMCN | 0.68 | 1.07 | 0.55 | 29.78 |
EMFO | 0.89 | 0.56 | 0.17 | 31.83 |
EMFS | 0.89 | 1.36 | 0.52 | 19.27 |
EMNR | 0.52 | 0.44 | 0.24 | 25.26 |
EMPL | 0.6 | 0.77 | 0.24 | 19.14 |
EMSG | 0.69 | 0.41 | 0.29 | 25.29 |
EPDN | 0.14 | 2.27 | 1.84 | 32.91 |
EPIT | 0.32 | 0.57 | 0.62 | 32.72 |
EPLC | 0.77 | 1.5 | 1.79 | 32.16 |
ESLN | 1 | 1 | 0 | 24.90 |
ESPC | 0.99 | 0.68 | 0.17 | 23.48 |
ESVO | 0.94 | 0.69 | 0.59 | 27.51 |
Supervised classification of volcanic activity
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A Fisher Discriminant model, here simply referred to as DISC, implemented by using the fitcdiscr function of the Matlab Statistical and Machine and Learning Toolbox, choosing a linear kernel. More details about this kind of classifiers are reported in Section A.
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A Multiclass error-correcting output model. In particular, as ECOC, in this paper we refer to an Ensemble multiclass classifier implemented by using the fitcecoc Matlab function, using a Support Vector Machine (SVM) algorithm, with linear kernel and One-Versus-One (OVO) coding scheme. More details are reported in Section A.
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An Ensemble model, here referred to as ENSE, implemented through the fitcensemble function of the Matlab Statistical Toolbox. We have adopted as basic learner a Adaboost2 algorithm, building a medium coarse tree with a maximum number of splits set to 20, a maximum number of cycles up to 30 and a learning rate set to 0.1. A short description of Ensemble models is reported in Appendix A.
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A K-Nearest Neighbor model, here referred to as KNN, implemented through the fitcknn function of the Matlab library, setting Euclidean distance metrics and 10 samples as the maximum number of neighbors. More details in Section A.
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A Naive-Bayes model, here referred to as NBYE, based on the fitcnb function of the Matlab library, with normal kernel distribution and unbounded support. More details in Section A.
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A Decision Tree for multiclass classification, here simply referred to as TREE, implemented through the Matlab function fitctree, growing a medium coarse tree, with a maximum number of splits up to 20. The Gini’s index was adopted as a split criterion. More details in Section A.
Reducing the dimensionality of the dataset
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The time series recorded at these stations are not affected by very high tremor levels, as for instance observed at stations ECNE, EPDN and EPIT.
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From trials performed, it seems that this couple of stations allow a good classification of class P. However, it is to be stressed, that other choices would be possible, without significantly affecting the classifier performances.
Classification metrics
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The TPR(i) expresses the proportion of actual positives that are correctly classified by the model as belonging to the ith class. Best values of TPR approach to 1, while in the worst case TPR approach 0. The TPR is referred to also as specificity or Recall (r).
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The TNR(i) expresses the proportion of actual negatives that are correctly classified as not belonging to the ith class. As for the TPR, best values of TNR approach 1, while worst values approaches 0. The TNR is referred to also as specificity of selectivity.
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The FNR(i) expresses the proportion of false negatives in the ith class, with respect to all actual positives in the same class. Of course in the best case FNR approaches 0, while in the worst case approaches 1.
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The FPR(i) expresses the proportion of false positives in the ith class with respect to the total number of actual negatives in the same class. Similar to the FNR in the best case FNR approaches 0, while in the worst case approaches 1.
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I(g) is a function that returns 1 if g is true and 0 otherwise,
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C(xn) the class label assigned by the classifier to the sample xn
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yn the true class label of the sample xn
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N is the number of samples in the testing set.
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m is the dimension of the confusion matrix CM, i.e. the number of classes.
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N is the number of samples in the testing set.
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CMii,i = 1,…,m the entries of the CM main diagonal.
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Citrue the true number of labels of class i.
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Cipred the predicted number of labels of class i.
Supervised classification
Preliminary considerations
Assessing the generalization capabilities of classifiers
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A group is held out and used for testing.
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The remaining groups are used for training a model.
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The fitted model is evaluated on the testing set.
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The obtained score is then retained while the model is discarded.
Numerical results
Supervised classification by using a more balanced data set
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The f1 indices for the Q class are almost equal for the two intercompared cases, except that the ECOC model trained with the subset was not able to classify samples of this class.
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As concerning the S class, the f1 index is lower than 0.6 for all classifiers and in particular DISC and ECOC perform very poorly. It is to be stressed that the ECOC model trained with the Subset was not able to correctly classify any samples of the Full dataset assigned to the S class.
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For the P class there are not substantial differences for the two inter-compared cases.
Numerical results concerning unsupervised classification
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k-means.
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Fuzzy c-means.
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Gaussian mixture models.
Results by using the k-means and the 2-D dataset
Class | μ1 | μ2 |
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Q | 0.8697 | 2.9700 |
S | 11.9562 | 26.9495 |
P | 52.4986 | 89.0279 |
Results by using the FCM and 2-D dataset
Class | μ1 | μ2 |
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Q | 0.6598 | 2.0578 |
S | 1.8531 | 6.5577 |
P | 39.7895 | 69.0634 |
Results by using the GMM
Component | πi | μ1 | μ2 |
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1 | 0.982276 | 0.8756 | 3.0013 |
2 | 0.015458 | 12.9308 | 28.3762 |
3 | 0.002266 | 54.5323 | 88.9329 |
Comparison among the unsupervised classifiers for the 2-D data
Models in 1-D and comparison with 2-D
Class | Q |
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Q | 0.2272 |
S | 17.5085 |
P | 65.8976 |
Class | μ |
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Q | 0.1791 |
S | 16.2816 |
P | 62.1765 |
Component | πi | μ |
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1 | 0.981812 | 0.8756 |
2 | 0.015824 | 18.8321 |
3 | 0.002364 | 54.5323 |
Measuring the performance of the FCM classifier assuming as true the output of the ENSE supervised classifier
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Supervised and unsupervised algorithms almost fully agree in classifying samples of class Q, being TPR% = 100 and PPV % = 91.6.
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The two kinds of classifiers almost fully agree also for the P class, being TPR% = 100 and PPV % = 97.6.
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Supervised and unsupervised algorithms do not fully agree with samples of S class, exhibiting PPV = 100% but TPR = 18.5%.
Comparing the true and the unsupervised time series of class
Discussion and Conclusions
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All the considered classification algorithms, both supervised and non, agree that by using the RMS of tremor as the only feature, only events belonging to class P can be classified with a reliability that, measured in terms of the f1 index, is about 0.7 and in terms of AuC about 0.8.
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Almost all the supervised classifiers considered are able to perform this task of classifying the class P events, but a limited superiority can be assigned to ENSE, KNN and TREE models. Similarly, among the considered unsupervised classifiers a limited superiority can be assigned to the FCM one.
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The greatest uncertainties of classifiers concern the attribution of events belonging to class S. In this regard, the RMS level of tremor alone is not enough to safely discriminate whether these samples are to be attributed to class Q or to class S.