# Fault Detection, Diagnosis, and Prognosis of a Process Operating under Time-Varying Conditions
## 4 Case Study
After completion of the contribution plots analysis, six variables were selected for the prognostics model: \(x_{2}\), \(x_{6}\), \(x_{8}\), \(x_{10}\), \(x_{12}\), and \(x_{13}\). The selection of the relevant variables started from the analysis of the contribution plot results, and it was also based on several considerations made about the process and about the relationships between the variables. Clearly, both the \(x\)-axis and the \(y\)-axis move on discrete values. The \(y\)-axis can only take on integer values representing the variables, and the \(x\)-axis represents a time sequence at discrete intervals. What makes the view of the graph as if it were completely active is that it represents a 2D view of a matrix of three values, thus actually representing 3D space. By considering a 2D view of the isolines and choosing a \"full\" representation of the graph, i.e., one in which the inside of the areas delimited by the isolines are colored, it is possible to obtain the above-described view of a completely full graph. To facilitate the reading of the graphs, the authors have inserted horizontal lines in Figure 2 which represent the active zones of the graph for each of the considered variables. For convenience, in this paper such lines have only been included in Figure 2, but the considerations made so far apply to the contribution plot of \(Q\) (Figure 3) as well. Going into the details of the contribution plot analysis, it is necessary to state that the authors started by analyzing only the contribution plot for \(T^{2}\). Figure 3: Contribution plot of Q in faulty states. Figure 2: Contribution plot of \(T^{2}\) in faulty states. During this analysis, the authors identified the areas where the lines added to the graph, i.e., the ones representing active areas of the graph for the considered process variables, intercepted one or more dark red areas. The colored bar shows that this is the color representing the greatest influence of a variable on the value of \(T^{2}\), i.e., the first color from the top of the colored bar. However, in case of the absence of dark red-colored areas, the second level of intensity would have been used. The output of this first round of analysis selected five variables, specifically \(x_{2},\ x_{8},\ x_{10},\ x_{12}\), and \(x_{13}\). The circles added within the graph in correspondence with the dark-red zones represent areas of high influence of variables on the \(T^{2}\) metric. It is interesting to note that two variables comprising several areas with considerable intensity of influence on the \(T^{2}\) value, but not the greatest intensity defined, are closely correlated with two of the selected variables, precisely \(x_{9}\) and \(x_{7}\). The process variable \(x_{9}\) is linearly correlated with \(x_{10}\), and \(x_{7}\) is linearly correlated with \(x_{13}\). Following this analysis of \(T^{2}\)'s contribution plot, considerations were made about \(Q\)'s role. Considering the insights extracted from the previous work [5], we proceeded in the following way:  
1. The variables showing the strongest relationship with \(Q\) were identified. 2. We analyzed which of these variables were activated in a disjointed manner with respect to variables already identified with the \(T^{2}\) analysis. 3. As a result of this analysis, variable \(x_{6}\) was further selected. As can be noticed from the circled areas in Figure 2, in no less than two areas variable \(x_{6}\) proves to be the most associated variable with the value of \(Q\), while none of the variables selected with the analysis of \(T^{2}\) were activated in the same way. The considerations made for the analysis of \(T^{2}\) apply for the analysis of the contribution plot of \(Q\) as well. The only difference lays in the chosen color bar. For the analysis of the contribution plot of \(Q\), the color representing a greater influence of the variable on the metrics is yellow.