## 1 Introduction
According to the same contribution, it is important to take into consideration the different quality levels of input water at the beginning of a process. However, this procedure makes the monitoring process more complex and delicate. In fact, different characteristics of water entering the system generate different output parameters that must be then contextualized in terms of the time dependence of the parameters. Therefore, according to the characteristics of the input  
Figure 1: Reverse osmosis process. water, the obtained parameters could represent either a healthy or an abnormal state of the machinery. Furthermore, an additional hindering aspect for the analysis is the absence of correct contextualization of the monitored data, which can invalidate the obtained results. For this reason, it is crucial to conduct further research on these types of processes [7]. Many plants use the reverse osmosis process to demineralize water [8]. As previously mentioned, this process is of paramount importance for the quality of the products processed, particularly in pharmaceutical contexts, such as the one under analysis. Consequently, it is crucial to ensure process continuity and a rapid resolution of maintenance problems. Taking into account the level of automation of these systems, reverse osmosis plants can be exposed to faults in several process components that can consequently affect plant operation [9]. The existing literature indicates that the most significant costs in many industrial plants are the maintenance, operating, and energy costs [10]. For this reason, the interest in reverse osmosis has become more and more important since reverse osmosis can guarantee relatively low operating and maintenance costs [11]. As previously mentioned, the reverse osmosis process has a crucial role in pharmaceutical processes, and it is therefore a critical aspect that should be carefully managed in pharmaceutical manufacturing, especially when considering semisolid and liquid products [12]. The issue of maintenance of this installation category is therefore an important and highly debated matter for which several approaches have been suggested over the years [13; 14; 15]. Nevertheless, the existing literature on the subject of reverse osmosis process monitoring is limited. Recently, a new approach was proposed for fault detection based on a structural analysis approach for an industrial seawater reverse osmosis plant [16]. Another approach proposed hybrid minimal structurally overdetermined sets for hybrid systems formulated as a binary integer linear programming optimization problem [17]. Furthermore, the analysis of the most similar approaches to the one proposed in this paper revealed that the unfold principal component analysis (UPCA) and principal component analysis (PCA) [13; 14] were applied for monitoring reverse osmosis processes. However, in this case the application of these two techniques was not considered appropriate since PCA does not allow for the management of process nonlinearities and UPCA is more suitable for batch production, which is absent in the process under analysis. The review of the existing literature on the subject indicates that the reverse osmosis process is nonlinear, as it is described as a set of differential equations [9; 18]. This point is particularly interesting because the literature highlights the need to investigate contexts that are increasingly realistic and complex and less ideal and simple, with an eye to data-driven approaches and to the management of data deficiency or incompleteness [19]. This particular feature of the reverse osmosis process motivated the application of a well-known technique in the scenario of fault identification, i.e., the kernel principal component analysis (KPCA). Unlike PCA, KPCA does not neglect the nonlinearities that can be detected in the behavior of the process under examination [20]. In fact, it is interesting to note that the KPCA technique allows for the extraction of nonlinear information from data [21].