## 4 Results and Discussions
### Failure-Detection Performance  
Based on the previous optimization results, the numbers of hidden layers and neurons are three and 200, and the selected activation function and optimizer are Sigmoid and Adamax, respectively. The epoch is set as 2000. These parameters and functions are further utilized to check the failure-detection accuracy of the present algorithm. Again, the algorithm randomly selects 80% of the entire dataset for training from the first eight environmental conditions given in Table 3 and uses the remaining 20% of the dataset for testing by checking the corresponding accuracy and loss. Figure 10 shows the correlation between numerical sensors for lateral and vertical motions. Correlation analysis is a technique that analyzes the linear relationship between two variables measured continuously. For example, in Figure 10a, the figure in the first row and the second column shows the relationship between the sensor No. 2 and sensor No. 3 (see Figure 1 for sensor numbers). We can see their strong relationship since the figure looks like a very slender ellipse along the diagonal. Figure 8: Performance test for the number of hyperparameters. Figure 9: Performance test for the selection of the activation function and optimizer. For the combination of first row and last column representing sensors 2 and 8, a less strong relationship can be observed with a wider ellipse. In other words, the longer the distance between the sensors, the less the correlation. The relationship is less strong in vertical displacements under different failure scenarios since vertical motions are smaller having 60-degree mooring toe angle and relatively large BWR (=1.3), and thus less sensitive to certain mooring-line failure. If the BWR is decreased, the vertical displacements are increased [7; 17], then they are to play a more important role. As a result, the distance between sensors should be short enough to acquire high detection accuracy but large enough to minimize cost and complexity. Therefore, the optimization of the sensor interval and number in real monitoring applications is important. Figure 10: Correlation of numerical sensor data for lateral (**a**) and vertical (**b**) motions. Figures 11-16 show the confusion matrices at different sensor combinations. The confusion matrix represents the performance of the classifier in a series of test data of which actual values are known [35]. The confusion matrices are a good option for reporting the performance results in multi-class classification problems because it is possible to observe the relations between the classifier outputs and the true ones. Table 5 summarizes the accuracy of classification with different numbers of sensors. The elements \(n_{ij}\) in the confusion matrix, where \(i\) and \(j\) are row and column identifiers, indicate the cases belonging to \(i\) that is classified as \(j\). Hence, the elements in the diagonal (\(n_{ii}\)) are the elements correctly classified, while the off-diagonal elements in the matrix are misclassified. The total number of cases is expressed as:  
\[N=\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{M}n_{ij} \tag{6}\]  
where \(M\) is the number of failure/intact scenarios. The confusion matrices show the ratio of detected cases to total cases in a range [0,1]. An asymmetric confusion matrix can reveal a biased classifier. The accuracy is the probability of performing the correct classification. The accuracy can be calculated from the confusion matrix as:  
\[\text{Accuracy}=\frac{\sum\limits_{i=1}^{M}n_{ii}}{N} \tag{7}\]  
Figure 11: Failure detected location when there are 4 sensors at points 2, 5, and 8 (Accuracy: 84.1%). Figure 12: Failure detected location when there are 4 sensors at points 2, 4, 6, and 8 (Accuracy: 90.4%). Figure 14: Failure detected location when there are 5 sensors at points 2, 3, 5, 7, and 8 (Accuracy: 94.2%). Figure 13: Failure detected location when there are 5 sensors at points 3, 4, 5, 6, and 7 (Accuracy: 92.9%).