## 3 Data and Methods
### Methodology
#### 3.2.2 Estimation of Change Detection Threshold, \(f_{t}\)
\[\underset{w,b}{min}\frac{1}{2}\parallel w\parallel^{2}\text{ subjectto}\,y_{i}(x_{i}^{T}w+b)\geq 1,\;\forall i \tag{5}\]  
where:  
o \(w\) is the weight vector (coefficients for each feature)  
o \(b\) is the bias term,  
o \(x_{i}\) is the input feature vector  
o \(y_{i}\) is the class label The feature importance is given by the absolute values of the components of the weight vector:  
\[Importance_{f}=\mid w_{f}\mid\]  
KNN, lacking a parametric model structure, did not provide inherent feature importance [64]; instead, its utility was evaluated based on test set accuracy using Equation (6). To facilitate comparison across models, min-max normalization was applied to all raw feature importance scores. _Accuracy_ is defined as the proportion of correct predictions:  
\[Accuracy=\frac{True\ positive+True\ negative}{True\ positive+False\ positive+True\ negative+False\ negative} \tag{6}\]  
Model Evaluation  
Subsequently, a precision-recall analysis was performed to explore the trade-off between detecting true flood events and minimizing false alarms. _Precision_, _Recall_, and the _F1_ score were calculated for different classification thresholds. This analysis helped identify the optimal threshold that balanced sensitivity and specificity, enhancing the reliability of flood detection. _Precision_, _Recall_, and _F1_ score were calculated using Equations (7)-(9). \[Precision=\frac{True\ positive}{True\ positive+False\ positive}=\frac{True\ positive}{Total\ actual\ positive} \tag{7}\]  
Figure 3:0  
Figure 3:1  
To further verify the optimal threshold, receiver operating characteristic (_ROC_) curves and the area under the curve (_AUC_) metric were employed. The _ROC_ curve illustrated the relationship between the true positive rate (_TPR_) and the false positive rate (_FPR_) across different thresholds [65; 66]. The _TPR_ is also known as sensitivity or recall and is defined with the same equation as _Recall_ as shown in Equation (9). The _FPR_ is defined according to Equation (10). Figure 3:2  
where:  
* \(FP\) is the number of false positives
* \(TN\) is the number of true negatives  
_AUC_ was computed as the integral of the _ROC_ curve, with values close to 1 indicating excellent model performance in distinguishing between flooded and non-flooded classes. Mathematically, the _AUC_ can be expressed according to Equation (11). Figure 3:3  
Statistical Significance Evaluation  
Finally, the statistical significance of model performance was evaluated using the paired _t_-test and [PERSON]'s test. The paired _t_-test assessed whether the observed per formance differences across models were statistically significant over multiple validation folds [67]. It is defined by Equation (12). Figure 3:4  
where \(\overline{d}\) is the mean of the differences between the performance measures of the two classifiers, \(s_{d}\) is the standard deviation of these differences, and \(n\) is the total number of observations. [PERSON]'s test [67] was used to compare the prediction outcomes of two classifiers on the same dataset, particularly focusing on cases where the models disagreed. The test statistic (\(\chi^{2}\)), with Yates' continuity correction, is calculated using Equation (13)  
Figure 3:5  
where \(n_{01}\) represents the number of instances where the classifier demonstrates significant predictive capability in one scenario, and \(n_{10}\) represents the number of instances where the classifier shows significant performance in another scenario. Together, these tests ensured the robustness and reliability of the comparative model evaluation.