# Metaheuristics and Support Vector Data Description for Fault Detection in Industrial Processes
## 4 Industrial Application

Table 1: Variables description. Furthermore, this approach presents the highest F1 score values in 80% of the experiments while SSA, KH, and PSO achieve similar performance only in 16, 13, and 0%, respectively. On the other hand, SHO presents the least computational time in the 100% of the runs achieving execution times two and even three times less than SSA and PSO. In addition, SHO presents the minimum variability with respect to the remaining approaches. In order to confirm the statistical significance of the results, a Mardia test is first performed to determine if the data present multivariate normal distribution. Since the data set is not normally distributed, the non-parametric ANOVA test of the Kruskal-Wallis is used. Figures 2 and 3 show the corresponding box plots for the implemented approaches. As it can be seen in Figure 2, SHO reaches the highest median F1 score value with low variability. On the other hand, the remaining approaches present the equal median F1 score values. However, KH shows the highest variability on its results. Although SSA sometimes yields similar results than those of SHO, they are so sporadic throughout the runs which is the reason for the boxplot to show these values as outliers. Figure 3 clearly depicts that SHO achieves the best computational times since the highest outliers of SHO are below of the lowest valued reported by KH. Tables 4 and 5 present the numerical results of the ANOVA test where for both cases (F1 score and computational time) the \(p\)-value is much lower than the significance level of 0.05. Thus, there is a statistical significance of the difference between the means of the results obtained by the four approaches. On the other hand, the high error in the sum of the squares (SS) in the F1 score analysis shows that there is a great variability which can not be explained by the predictors. Once considerable differences have been detected when using the Kruskal-Wallis ANOVA, a post hoc Tukey test is performed to determine the mean with the most significant difference. This analysis compares the means of all treatments with the mean of every other treatment and the best available method is considered in cases when confidence intervals are desired (for details, refer to [51]). Figures 4 and 5 show the graph of the estimates and comparison interval. Each group mean is represented by a little circle and the interval is represented by a line extending out from the circle. Two group means are significantly different if their intervals are disjoint and they are not significantly different if their intervals overlap. As it can be seen, the SHO approach does not overlap with any other in both cases (F1 score and computational times), i.e., the Tukey test concludes that there is a significant difference between the SHO algorithm and the remaining approaches. This shows that SHO-SVDD offers a higher performance for fault detection in the plastic injection machine (F1 score is equal to 0.9702 in average) with a shorter computational training time. Finally, the overfitting effect is analyzed to test the SHO-SVDD generalization performance. The complete data set (normal and mixed operation samples) is used to implement a K-fold cross validation procedure. In addition, 5-fold (test 1) and 10-fold (test 2) cross validation are computed 30 times, and F1 score mean value is calculated i.e., a total of 150 and 300 training-test experiments are computed, respectively. The obtained F1 score mean values are 0.9701 for test 1 and 0.9714 for test 2, both are very close to F1 score mean value of 0.9702 obtained in the previous analysis. Considering this, it is concluded that the SHO-SVDD approach for fault detection presents a good generalization for this particular data set.