# Fault Detection, Diagnosis, and Prognosis of a Process Operating under Time-Varying Conditions
## 4 Case Study
This analysis allows circumscribing the variables that are relevant in the process monitoring, which are the focal point of the following step of the prognosis. This allows not only streamlining of the analysis by speeding up its execution but also elimination of unnecessary variables that can create noise in the machine learning model leading to the achievable results. This step can be attributed a double role. The prognostic analysis of the process allows timely prediction of the time to failure, thus ensuring the implementation of maintenance interventions with the aim of preventing the occurrence of a fault or limiting its extent. Additionally, this analysis is always accompanied by a continuous and real-time monitoring of the process to implement a possible precautionary fault detection, with the consequent compensation for any possible errors in prognostic prediction. The analysis of the variables representative of the fault ensures faster circumscription of the origin of the fault, and a consequent implementation of rapid maintenance interventions, thus limiting the duration of unwanted machine downtimes. In conclusion, it is possible to state that this phase clearly plays a key role in the maintenance decision-making process. In a state of emergency, it steers maintenance inspections, it predicts the machinery's failure time, and it identifies a subgroup of variables that are representative of the failure. For the case study under analysis, the logic analysis used in this phase differed from the one used in the fault detection phase. Despite the difficulties encountered in the latter due to the non-linearities of the process, the fault detection phase did not influence the application of the contribution plots. While the health thresholds and the relationship between health status and failure change in different production intervals, the logic inherent in the process remains unchanged. For this reason, the application of the contribution plots, aimed at skimming the variables to be input to the prognosis model, was performed only on a randomly selected production sub-interval, i.e., interval E. In addition, both the contribution plots of \(T^{2}\) and \(Q\) were considered. However, to extrapolate the representability of the variables, it was deemed sufficient to consider essentially only the contribution of \(T^{2}\), which represents the state space. As regards the evaluation of the contribution plot of \(Q\), it is dependent on the considerations extracted from that of \(T^{2}\). The results obtained from the application of the contribution plots are shown in Figures 2 and 3, which display only data related to a fault state. As can be seen in Figures 2 and 3, a contour plot was applied, i.e., a plot containing the isolines of a \(Z\) matrix, with \(Z\) containing height values on the \(x\)-\(y\) plane. The term isolines is widely used to describe images representing different levels of a certain value. Isolines are \"contour lines\", i.e., lines joining all the points presenting the same value of the chosen variable. In the case under analysis, the contour lines represent the value of the influence that a variable had on the value of the index considered in the specific graph. In this case, the \(x\)-\(y\) plane is structured as follows:  
* \(y\)-axis \(\rightarrow\) the process variables;
* \(x\)-axis \(\rightarrow\) dataset instances. Furthermore, \(Z\) represents the value of the contribution of a process variable to the value of the statistic, and like-colored lines represent an area with the same value of this contribution. It should be noted that, to improve the readability of the obtained results, the logarithm of the calculated contributions was graphed, thus helping to respond to skewness towards larger values. The colored bars on the right side of the graphs represent the scale of the magnitude of the influence of a variable on the \(T^{2}\) value and on the \(Q\) value, presented in Figures 2 and 3, respectively.