## 2 Materials and Methods
### Methodology  
By analyzing the volume of contributions concerning the KPCA, it was considered appropriate to present the steps undertaken to structure the methodology rather than a detailed explanation of how the KPCA works, which can be found in other contributions [26; 31]. Figure 2 illustrates the steps followed in the approach presented in this paper. A fundamental step for the application of the KPCA is the choice of the kernel. Based on the literature review, several approaches can be implemented for this purpose. In the presented paper, the iterative approach was applied since it allows a prompt comparison of the obtained results and it ensures an easily applicable and replicable process of data analysis. The iterative approach is a process of continuous improvement of the model. This means that, in a cyclical process, some steps are carried out more than once with the aim to improve the outcome at each completion. Therefore, by observing the improvements or decline of the results achieved with the model, it is possible to identify the positive or negative changes in the applied kernel. As a result, different kernels were tested for each of the proposed approaches. Moreover, for each approach a parameter optimization process was applied in order to find the kernel and the related parameters that guarantee the best performance. The different approaches tested refers to the various combinations of past and future matrices that will be subsequently presented. Specifically, the most popular kernels in the existing literature have been chosen and tested, namely:  
* Linear kernel
* Gaussian RBF kernel
* Polynomial kernel
* Exponential kernel  
Figure 2: The logical path followed for the KPCA monitoring. * Sigmoid kernel
* Laplacian kernel  
However, considering the peculiarities of each process, digressions on the different outputs obtained with the different combinations of kernels are not discussed in the present paper. In fact, each monitored process is characterized by specific features, and this means that each process responds differently to different kernels applied. In line with this consideration, and according to the applicative nature of the proposed paper, it was not considered appropriate to deepen the theme of the application and the consequent performance obtained with different kernels. The results presented below refer to the best combination, identified through the performed analysis, i.e., the combination that guaranteed the achievement of the best values for the performance evaluation indices taken into consideration. In this case, the best configuration was the polynomial kernel configuration. The procedures presented hereafter were performed to confirm the need to apply the above-mentioned methodology. Firstly, an optimization of the application of fault detection with KPCA to the same dataset was performed without using the past and future matrices but using the original dataset with pure monitoring data. The indices used for monitoring were \(T^{2}\) and \(Q\), as presented in [26]. These indices account for two different aspects of the performance of the structured model. \(T^{2}\) represents the variation within the model. It is, in a word, the sum of the normalized squared errors. Referring to what is proposed in [26], \(T^{2}\) is defined as follows (1):  
\[T^{2}=\left[t_{1},\,\ldots,\,t_{p}\right]\Lambda^{-1}\left[t_{1},\,\ldots,\,t _{p}\right]^{T} \tag{1}\]  
In (1) the principal components \(t\) are obtained by projecting \(\phi\)(x) onto the eigenvectors \(v_{k}\) in the feature space, where \(\phi\)(x) is a nonlinear mapping function used to project the input vector to the feature space. \(Q\), on the other hand, provides another insight into model performance, as it represents how well a new sample fits the constructed model as the squared prediction error. For an in-depth detail of the calculation of both indicators please refer to [26].