## 4 Performance Measure  
The performances of the proposed models were evaluated by a number of metrics, identified using a confusion matrix for proper comparison (Table 4).  
The definitions of key terminology used to formulate the elementary metrics while considering the non-occurrence of liquefaction is actually negative class are:  
* True positive (TP) and true negative (TN) indicate that the samples are predicted correctly.
* False positive (FP) represents the number of non-liquefied samples that are predicted incorrectly as positive.
* False negative (FN) denotes the number of liquefied samples that are predicted incorrectly as negative.
* Precision refers to the accuracy of the predictions for a single class (positive or negative).
* Recall measures the accuracy of predictions, considering only the predicted value.  
Based on the confusion matrix, the following metrics were used to evaluate and compare the prediction models:  
\[OA=\frac{TP+TN}{TP+FN+FP+TN} \tag{7}\]  
\[Precision=\frac{TP}{TP+FP}\text{ {or} }\frac{TN}{FN+TN} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & & \multicolumn{2}{c}{**Predicated Class**} & \\ \hline  & & Yes & No \\ \hline \multirow{2}{*}{**Actual Class**} & Yes & True Positive (TP) & False Negative (FN) \\  & No & False Positive (FP) & True Negative (TN) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Confusion matrix.  
Figure 4: Decision tree of seismic soil liquefaction generated by a C4.5 algorithm. \({}^{a}\) Number of cases in this partition; \({}^{b}\) Number of cases misclassified.  
\[Recall=\frac{TP}{TP+FN}\text{ \ or \ }\frac{TN}{FP+TN} \tag{9}\]  
\[F-measure=\frac{2\times Precision\times Recall}{Precision+Recall} \tag{10}\]  
\[MCC=\frac{TP\times TN-FN\times FP}{\sqrt{(TP+FN)(TN+FP)(TP+FP)(TN+FN)}} \tag{11}\]  
Data on seismic soil liquefaction case histories usually exhibit class imbalances, wherein one event i.e., liquefaction, in this case, is delineated by a large number of events, while non-liquefaction event is presented by scarcely any. Therefore, overall accuracy (OA) may be deceptive--it has high scores when the liquefied samples are in the majority class. In this study, one of the standard measures used is the Matthews correlation coefficient (MCC) [43]. MCC presents the degree of correlation between the actual and predicted classes of liquefied and non-liquefied instances. MCC takes values in the interval [-1,1], with \"1\" showing complete agreement, \"-1\" complete disagreement, and \"0\" presenting that the prediction was uncorrelated with the ground truth. The MCC value is regarded to be the best evaluation measure for overall performance of a classifier method [44]. F-measure combines precision and recall values to attain a harmonic mean. F-measure ranges from 0 (worst value) to 1 (best value). The receiver operating characteristic (ROC) curve is a graphic plot of the sensitivity or recall against false positive rate (FPR). The FPR is given by:  
\[FPR=\frac{FP}{FP+TN} \tag{12}\]  
The ROC curve gives a comprehensive scalar value that describes the expected performance of the model. The area under curve (AUC) is employed to summarize the ROC curve. The ROC curve has five degrees of rating [45]: excellent (0.9-1), good (0.8-0.9), fair (0.7-0.8), poor (0.6-0.7), and not discriminating (0.5-0.6).  
Concisely, a model with good OA, larger AUC, high F-measure, and high MCC is an ideal one.