## 11 **The amount of half-rate capacity usage**
### Identifying the Fault Cause
#### 2.6.4 Evaluation of Produced Clusters  
In this step, we evaluated the obtained clusters. Due to the lack of access to the gold data, we used intrinsic measures to evaluate the clustering results with the correct mode. Intrinsic measures assess the quality of obtained clusters without the need for external information. For example, the criteria that check the degree of adhesion or separability of the data in a cluster are among the internal criteria for evaluating clusters. On the other side are external measures. These criteria, which are supervised methods, are used when gold data are available, and the quality of clustering algorithms can be evaluated by comparing them with the target data [42]. In the following, we consider the internal methods that were compatible with the limitations of the work in this project and were used to evaluate the produced clusters. It should be noted that the same coefficients were used to obtain the optimal number of clusters, and the value of these coefficients was calculated for different numbers of clusters. The optimal value of these coefficients indicated the optimal number of clusters.  
The Davies-Bouldin method is one of the most common ways of internally evaluating clusters obtained in clustering algorithms. The problem with this algorithm is that its excellent evaluation value does not use all the hidden information in the data and does not recover all the information [43]. The Silhouette coefficient, another internal evaluation measure of clustering methods [44], was used in this research. Both of these methods are based on the stickiness of the data in one cluster and the separability of the data with respect to other clusters. In the following, we review the details of these two methods.  
Silhouette Coefficient MethodThe Silhouette coefficient is one of the most common ways of internally evaluating clusters obtained in clustering algorithms, and is based on calculating data adhesion and separability. Unlike the Davies-Bouldin method, which considers only the centers of the clusters, this method performs its calculations on all the points of the clusters. The general relationship below shows how to calculate the Silhouette coefficient.  
Equation (4): The general formula for calculating the Silhouette coefficient:  
\[S(o)=\frac{b(o)-a(o)}{max\{a(o),b(o)\}} \tag{4}\]  
where _a(o)_ indicates the cluster stickiness containing record o and is calculated through the following relationship.  
Equation (5):  
\[a(o)=\frac{\sum_{o^{\prime}\in C_{i}o^{\prime}\
eq o}\ dist(o,o^{\prime})}{|C_ {i}\mid-1} \tag{5}\]  
And  
Equation (6):  
\[b(o)=\min_{C_{j}:1\leq j\leq N}\left\{\frac{\sum_{o^{\prime}\in C_{i}}\ dist(o,o^{ \prime})}{\left|C_{j}\right.|}\right. \tag{6}\]  
where \(dis(o,o^{\prime})\) calculates the Euclidean distance between two points of \(o\) and \(o^{\prime}\). Therefore, \(a(o)\) calculates the average distance of point \(o\) with all the points that are in the same cluster with \(o\). The value _b(o)_ also calculates the average distance for all clusters from point \(o\) with all points that are not in the same cluster with \(o\). Finally, for each cluster, the Silhouette coefficient value is obtained by averaging the values of \(S_{i}\) in each cluster. The following figure shows an interpretation of Silhouette coefficient values for two clusters. In fact, the value of this coefficient is between \(-1\) and \(1\). If \(b_{i}\) which indicates the distance of each point from other points that are not in the same cluster, is smaller than \(a_{i}\), which indicates the distance of points within a cluster, the result of the Silhouette coefficient is smaller than zero and has negative values. The closer the value of this coefficient to \(1\), the more correct is the assignment of each point to the cluster similar to itself. If the value of this coefficient is zero, it means that the point is located on the border between two clusters (Figure 6).  
Figure 6: The Silhouette coefficient values’ interpretation.