## 1 Introduction
MSPC methodologies are most frequently used and are superior to statistical process control (SPC) methods since they straightforwardly consider and use idle factors, prompting effective issue identification [34]. Another study combined sub-period division strategies with multiway principal component analysis for the fault diagnosis on a sequence batch reactor of the wastewater treatment process in a paper mill [35]. FDA is a widely used pattern classification technique [36], and its application to chemical process data analysis has continuously increased in the last two decades [37; 38; 39; 40]. FDA application to faulty sensors was studied for air handling units [41]. The target of the FDA method is to determine the Fisher optimal discriminant vector that maximizes the Fisher criterion function. Fisher discriminant analysis uses a linear combination of features to distinguish between two or more classes in an optimal way. It is an empirical method based on observed characteristics over a large number of cases. FDA for fault identification provides the best lower dimensionality representation in terms of a discriminant between data classes, where each class corresponds to data acquired during a specific and known fault. Unlike PCA, which is looking for directions that are effective for representation, FDA is looking for directions that are effective for discrimination. From a theoretical standpoint, FDA has advantages for fault visualization and diagnostics [39]. In general, the economic efficiency and environmental friendliness of sewage treatment plants are primarily based on the process of removing nitrogen from pollutants by biologically catalyzed oxidation using aeration control. This control is primarily sustained by measuring the content of dissolved oxygen (DO) with dedicated sensors. The effectiveness of an aeration control system can be significantly impacted by wrong information that emerges from the dissolved oxygen sensors, resulting in unexpected degradation of the system functionality and problems throughout the wastewater treatment process. Negative consequences in terms of deterioration consist of a decrease in the quality of the effluent, an increase in energy consumption, a decrease in environmentally sustainable performance or even a temporary shutdown of the plant. Few studies have addressed the problem of DO sensor failure. In typical research, the PCA approach was used to detect the set of three categories of faults [42]. Another study examined single-type DO and level sensor failures [28]. The wrong output signal from the sensor was analyzed in another work [19], and a study was devoted to detecting the clogging bias of the sensor based on PCA [20]. The variety of the detection tools was extended to element recognition, neural networks with radial transfer functions [43], binary classification instruments [44], approaches based on impulse response [45] or deep dropout neural networks (MC-DDNN) to identify incipient faults of sensors installed in wastewater treatment plants [46]. In a previous investigation, the current authors developed a study based on the PCA-based methodology that proposed detection solutions for determining the presence of inappropriate functioning of the dissolved oxygen sensor [47]. The motivation and contributions of the present work originate from several reasons. The reported fault identification studies referring to the DO sensor defects were typically focused on a single or on a limited number of fault types. Extending the set of fault types embedded in a single diagnosing tool hinders the efficiency of the fault categorization but makes it most appreciated. The DO sensor fault diagnosing in a controlled A\({}^{2}\)O configured WWTP, where the sensor is implied in the automatic operation associated with the nitrites and nitrates concentration control loop, was not specifically reflected in the literature.