## 1 Introduction
As far as the chemical processes are concerned, PCA reveals difficulties in fault detection since it is based on the assumption of linearity of data [22]. However, an interesting limitation that can be found in KPCA is that of data uncertainty management due to, for example, sensor problems or data noise [21]. Besides, KPCA is still critical of incipient failures, i.e., those with small fluctuations and very slow changes [23]. By analyzing the literature on the subject, it is possible to observe that the KPCA has often been modified or used in combination with other techniques to monitor industrial systems, such as Kullback Leibler divergence [23], or more recently combined with deep learning algorithms for degradation assessment and subsequent prognostication of rolling bearings [24]. Moreover, KPCA has been very successfully used both in the field of fault detection and diagnosis for rotating machines given their intrinsic nonlinearity of the process [24]. In fact, KPCA has been applied not only in the field of fault identification but also in the field of fault diagnosis [25]. Furthermore, the application of the KPCA technique does not necessarily require a defined priori of the number of components to be extracted [26]. Moreover, due to its ability to use different kernels, this analysis technique can handle a wide range of nonlinearities [26]. For this reason, KPCA is considered highly user-friendly and appealing in the context of process monitoring. Taking into consideration the nature and behavior of the process, it is possible to observe a clear temporal correlation of the process variables--typical of chemical process--which should be somehow taken into account as presented, for example, in other contributions and with other techniques, e.g., canonical variate analysis [27], long short-term memory [28], recurrent neural networks [29], or canonical variate dissimilarity analysis [30]. Future research developments in this regard are discussed in the conclusion section. Finally, another feature that makes this process complex to monitor is the way of recording data. In fact, the monitoring of the examined process is not continuous in time nor at regular time intervals. The available data have been obtained with a sample monitoring of the parameters without the regularity of time intervals for this sampling. Based on the above-mentioned process characteristics, the paper proposes a KPCA-based reverse osmosis process monitoring approach. However, in order to consider the temporal correlation in the dataset, the entire dataset will not be given as input to the KPCA, neither in training nor in the testing phase. Instead, the past and future correlation matrices of the dataset were calculated and used as input for the application of the KPCA. The advantage of using KPCA in synergy with temporal matrices lies in the fact that KPCA is an established monitoring technique whose results in monitoring dynamic processes have been proven. Moreover, the temporal matrices allow its application to be extended by including information on the time evolution of the analyzed process. Considering the existing literature on the subject, the time dependency in this type of process, which also supports a technique for handling nonlinearities in the process, presents a novel approach that has yet to be investigated. In detail, the approach will be based on two steps:  
* Study of the temporal correlation of the process with itself to determine the time intervals to consider for the construction of the past and future matrices
* Application of the KPCA for fault detection using as an input only the past and future matrices constructed in the previous point, not the entire dataset