## 3 Results and Discussion
### Receiver Operating Characteristic Curves  
A receiver operating characteristic (ROC) curve is a graphical analysis tool made up of a false-positive rate (FPR) on the x-axis and a true-positive rate (TPR) on the y-axis. Clarity and sensitivity are terms used to describe FPR and TPR, respectively. A binary classification model derives the coordinate points from the observed values and expected values of all samples. The ROC is situated in a square between 0 and 1 if the coordinate points are drawn on the x-y plane and connected with straight lines; during the judgment process, the space is divided into upper-left and lower-right blocks using the diagonal line from (0, 0) to (1, 1) as a reference line. As illustrated in Figure 17, an ROC curve's classification result is better if it is above the diagonal line and travels to the upper left. Conversely, its classification result is worse if it is below the diagonal line. \begin{table}
\begin{tabular}{c c c} \hline
**Factor** & **Importance** & **Importance of Normalization** \\ \hline Total area of the catchment area & 0.228 & 100.0\% \\ Average slope of the catchment area & 0.104 & 45.5\% \\ Average slope of the stream bed & 0.043 & 18.7\% \\ Ahape coefficient & 0.227 & 99.8\% \\ Type of stratum & 0.065 & 28.6\% \\ Distance from the fault & 0.052 & 22.9\% \\ Maximum daily rainfall & 0.169 & 74.3\% \\ Maximum hourly rainfall & 0.112 & 48.9\% \\ \hline \end{tabular}
\end{table} 
Table 15: Results of the importance of the factors of the back-propagation neural network. Figure 17: Schematic diagram of the ROC curve. The main output of ROC analysis is a binary classification model, such as occurred/not occurred. A threshold value is needed to define a continuous data output result. The following equations were used for calculating TPR and FPR:  
\[\text{TPR}=\frac{\text{TP}}{\text{TP}+\text{FN}}, \tag{13}\]  
\[\text{FPR}=\frac{\text{FP}}{\text{FP}+\text{TN}}, \tag{14}\]  
where, as indicated in Table 16, TP and TN represent the correctness and error of interpretation, respectively. The area under the ROC curve, known as AUC, is a measure of how accurately landslides can be interpreted based on values that are observed and predicted. AUC's value ranges from 0 to 1. The more AUC points there are, the more accurate the judgment. Perfect categorization is indicated by an AUC of 1. A model is considered to have no predictive value if its AUC is lower than 0.5, a fair predictive value if it is at least 0.5 but less than 0.7, a good predictive value if it is at least 0.7 but less than 0.9, and an excellent predictive value if 0.9 or higher. The ROC curves for each type of analysis are shown in Figures 18-20. Specifically, the AUC of the neural network was computed as 0.93, indicating this technique's excellent ability to predict when a debris-flow disaster will occur. The AUC of logit regression analysis was 0.79, i.e., good, and that of the instability index method was 0.64, i.e., fair. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Predictive Value**} & \multirow{2}{*}{**Observed Value**} & \multirow{2}{*}{**Unstable**} & \multirow{2}{*}{**Stable**} \\ \hline Unstable & True Positive (TP) & False Positive (FP) \\ Stable & False Negative (FN) & True Negative (TN) \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Receiver operating characteristic curve evaluation form. Figure 18: Receiver operating characteristic curves for neural network analysis. As can be observed from the AUC area, all three models reached above 0.7, representing the ability to effectively predict the occurrence of a landslide in the Ching-Shuei River using the instability index, Rogers regression, and artificial neural network. By comparing the three methods of evaluating model accuracy, it could be found that the accuracy of the ROC curve was higher than that of the categorical error matrix.