# Anomaly Detection in Automotive Industry Using Clustering Methods--A Case Study
## 4 Case Study, Results, and Discussions
### Computational Results
This means that there is no significant reduction in the value of the index starting from this point. This index uses a ratio between SSW and SSB, which may indicate the groups' formed quality, as much as the cohesion of the groups (SSW) and the distinction between the groups (SSB). Note that for the other metrics, this point is not easy to identify. As mentioned, the multi-functional team of the company indicated the number of groups among the interval 200 to 621. The best performances were reached by the Hierarchical model, followed by K-Medoids and K-Means. Figure 11 shows the SI index. Considering the Silhouette metric, the value is between \([-1,1]\), and the closer to 1, the more precise the formation of the groups. Then, we observe an advantage of the Hierarchical algorithm and a worse performance for FCM. The other algorithms have a more stabilized behavior concerning this metric, obtaining intermediate values. There is a significant difference in results and an inconclusive definition of the ideal number of centroids. For each metric and algorithm application was checked an overall performance (Table 2), considering each result. The first place was awarded 9 points, the second 8, and so on, until the last place winner received 1 point. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Algorithm** & **SSE** & **SSW** & **SSB** & **WB** & **CH** & **SI** & **Total** \\ \hline
**Hierarchical** & 8 & 8 & 7 & 8 & 8 & 9 & 48 \\ \hline
**K-Medoids** & 9 & 9 & 8 & 9 & 9 & 3 & 47 \\ \hline
**K-Means** & 7 & 7 & 6 & 7 & 7 & 8 & 42 \\ \hline
**GA** & 6 & 6 & 5 & 6 & 6 & 7 & 36 \\ \hline
**PSO** & 5 & 5 & 4 & 5 & 5 & 6 & 30 \\ \hline
**DE** & **4** & 4 & 3 & 4 & 4 & 4 & 23 \\ \hline
**DBSCAN** & 1 & 1 & 9 & 1 & 1 & 5 & 18 \\ \hline
**SOM** & 3 & 3 & 2 & 3 & 3 & 2 & 16 \\ \hline
**FCM** & 2 & 2 & 1 & 2 & 2 & 1 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Overall ranking using Borda count method considering different indexes and algorithm results. Figure 11: SI index evaluation over different group amounts considering nine clustering methods. On overall performance, the Hierarchal, was the general winner, slightly superior to K-Medoids, even though K-Medoids was the first in 4 of 6 different indexes when using the Borda count method [72]. It is an important finding since the literature strongly indicates that these classic methods tend to reach an inferior performance to more recent and sophisticated models, such as DBSCAN, FCM, and SOM. We also highlight that K-Medoids presented a lousy performance for SI, which led to a worse position in the ranking than the Hierarchical approach. Moreover, it seems to be clear that the advantages in their use are in the initialization and as actual data instead of a randomly generated centroid, as the ranking position of the K-Medoids is higher than the K-Means. In this sense, we can resort to the no-free lunch theorem, since this investigation presents some particular premises, which are not usually addressed in the benchmark or other real-world problems. We expected that the FCM and SOM might achieve the best performances due to their complex clustering schemes. Once again, we can state that more than one algorithm should be used for clustering tasks. The bio-inspired algorithm (GA, PSO, and DE) presented similar performances, highlighting the GA. Despite showing global search potential, they could not find the optimum global point in this study, and probably fell in local minimum points. Other bio-inspired approaches can be found in the literature, which can improve and expand the results reached in this investigation [73; 74; 75; 76; 77; 78]. Another remark found in the literature is the computational cost regarding the hierarchical clustering method. We highlight that the Hierarchical approach is a deterministic one that does not depend on initialization. Therefore, a stochastic method like K-Medoids may request more aggregate computational effort, depending on the database, since it needs 30 independent runs.