## 3 Results and Discussion
### Fault Diagnosis
Figure 7 shows the graphical representation of the fixed value fault, which was correctly and promptly identified after 6 h of the fault appearance. The complete failure minimum is correctly diagnosed after 6.75 h of its intervention, as it is revealed in Figure 8.  
The complete failure maximum diagnosis is diagnosed successfully after 9.5 h, as is shown in Figure 9.  
Figure 3:0 Complete failure minimum diagnosis: graphs of the FDA discriminant functions \(g_{i}(x)\) for each of the normal and seven fault classes, along the 24 h of the 140 th day and the detailed representation revealing the class of the observations and the time moment of the firm fault type identification. Figure 3:1 Fixed value diagnosis: graphs of the FDA discriminant functions \(g_{i}(x)\) for each of the normal and seven fault classes, along the 24 h of the 140 th day and the detailed representation revealing the class of the observations and the time moment of the firm fault type identification. Figure 3:2 Complete failure maximum diagnosis: graphs of the FDA discriminant functions \(g_{i}(x)\) for each of the normal and seven fault classes, along the 24 h of the 140 th day and the detailed representation revealing the class of the observations and the time moment of the firm fault type identification. The FDA-based identification methodology for the type of DO sensor fault proved to be successful for all of the investigated faults except the bias one (which has very similar effects with complete failure maximum failure). This means an accuracy of 87.5% correct identification of the eight considered cases. The time needed for obtaining the consolidated diagnosis decision varied from 2.5 h to 16.5 h.  
From the rare previous studies that addressed fault diagnosis in WWTPs sensors, one showed good detection of bias, drift and precision degradation (loss of accuracy) of sensors used in a WWTP with classic and dynamic PCA-based methods but showed some limitations regarding the fault identification with variable reconstruction-based methods [42]. These limitations included the fact that a reconstruction-based method cannot identify the faulty sensor which causes process transition, so the approach is inappropriate for identifying the fault type connected to a control loop. Additionally, it did not study the fault identification time in detail. The present study investigated several fault types, and the identification time was determined for each of them. Another work proposed a complex-valued slow independent component analysis (CSICA) based method for fault detection and diagnosis with applications to wastewater treatment processes. Despite the fact that the method had good performance in detecting and diagnosing incipient faults, it was inadequate for scenarios of multi-fault and large-scale nonlinear systems [23]. The sub-period division strategies combined with multiway principal component analysis for only two faults diagnosis showed that the used methods could manifest false identification results during normal operation periods and the inability to detect the fault during some time intervals [35]. The present investigations considered six different fault types that implied increased difficulty in the promptitude and accuracy of the fault diagnosis. The efficiency of the proposed FDA identification is substantiated by the fact that discrimination was achieved by the FDA discrimination model that was trained with faulty data from the five days subsequent to the first one, when the fault actually started to act, and testing was performed for this very first day.