# Fuzzy Clustering Approaches Based on Numerical Optimizations of Modified Objective Functions
## 4 Experimental Studies
The similarity is assessed by comparing the fuzzy partition matrix \(U\) with the real class labels \(C\), based on mutual agreements of memberships weighted by the membership degrees. While adjusted Rand index measures are based on hard assignments, Fuzzy ARI employs partial memberships, enabling the capture of soft boundaries present in the fuzzy clustering approaches [32]. If the two compared partitions are \(U=\left[\mu_{ij}\right]\) and \(V=\left[v_{ij}\right]\), for any pair of instances \(x_{k}\) and \(x_{l}\), their similarity is evaluated according to partition \(U\) and their similarity according to partition \(V\), respectively, as  
\[s_{kl}^{(U)}=\sum_{r=1}^{n}\mu_{kr}\cdot\mu_{lr}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & **\# Features** & **\# Instances** & **\# Clusters** \\ \hline Banknotes & 5 & 1372 & 2 \\ Liver & 10 & 583 & 2 \\ Yeast & 8 & 1484 & 10 \\ Dermatology & 34 & 366 & 6 \\ Vehicle Silhouettes & 18 & 846 & 4 \\ LANDSAT Satellite & 36 & 6435 & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the employed datasets. \[s_{kl}^{(V)}=\sum_{r=1}^{n}v_{kr}\cdot v_{lr}\]  
Based on these mutual similarity values, the adjusted fuzzy Rand index (F-ARI) is evaluated as  
\[F-ARI=\frac{\sum_{i<j}s_{kl}^{(U)}\cdot s_{kl}^{(V)}-\left(\sum_{i<j}s_{kl}^{(U)} \right)\left(\sum_{i<j}s_{kl}^{(V)}\right)/\binom{n}{2}}{\frac{1}{2 \left(\sum_{i<j}s_{kl}^{(U)}+\sum_{i<j}s_{kl}^{(V)}\right)-\left(\sum_{i<j}s_{ kl}^{(U)}\right)\left(\sum_{i<j}s_{kl}^{(V)}\right)/\binom{n}{2}}}\]  
The evaluation score varies in the interval \([-1,\,1]\), with value 1 indicating perfect matching, value 0 indicating independence (random labelling) and negative values indicating worse than random scenarios. Normalized mutual information (NMI) is a metric used to assess the quality of clustering in cases when the intrinsic structures are available (but not used during the clustering procedure). It evaluates the shared information between the generated clusters and the true classes of the dataset. If the two compared partitions are \(U=\left[\mu_{ij}\right]\) and \(V=\left[v_{ij}\right]\), then their fuzzy mutual information is evaluated as  
\[F-MI=\sum_{i=1}^{n}\sum_{j=1}^{c}\sum_{k=1}^{c}\mu_{ij}\cdot v_{jk}\cdot\log \left(\frac{\mu_{ij}\cdot v_{jk}}{p_{j}^{(U)}\cdot p_{k}^{(V)}}\right)\]  
Here, \(p_{j}^{(U)}\) and \(p_{k}^{(V)}\) represent respectively the average membership of cluster \(j\) in partition \(U\) and the average membership of cluster \(k\) in partition \(V\), thus:  
\[p_{j}^{(U)}=\frac{1}{n}\sum_{i=1}^{n}\mu_{ij}\]  
\[p_{k}^{(V)}=\frac{1}{n}\sum_{i=1}^{n}v_{ik}\]  
On the other hand, the Fuzzy Normalized Fuzzy Mutual Information guarantees that the value is between 0 and 1 by dividing the fuzzy mutual information by the sum of the entropies of the two respective partitions:  
\[F-NMI=-\frac{F-MI}{\sum_{l=1}^{n}\sum_{j=1}^{c}\mu_{ij}+\sum_{l=1}^{n}\sum_{j =1}^{c}v_{ij}}\]  
NMI is a broad coverage measure as it comprises both how well each cluster corresponds to a real class in the dataset (homogeneity) and how well the instances of a real class in a dataset correspond to a single cluster (completeness) [33]. The final score is normalized, guaranteeing the result to be in the interval [0, 1], making the clustering results more comparable. The value 1 is an indicator of perfect clustering, while the value 0 indicates independence between the real classes and the clustering results. Fuzzy normalized mutual information is a generalized version of NMI, which considers the membership values (treating the clusters as fuzzy sets) instead of hard assignments into clusters [32]. The evaluation proceeds by calculating the joint entropy for the fuzzy membership values. Table 2, given below, summarizes the performance evaluation of classical Fuzzy C-means based on the fuzzy adjusted Rand index (F-ARI) and the fuzzy normalized mutual information (F-NMI). The FCM algorithm is applied 20 times, and the results shown in the table are the averages of these evaluations.