# Fuzzy C-Means Clustering Algorithm with Multiple Fuzzification Coefficients
## 4 Evaluation of the Proposed MC-FCM Algorithm  
In order to evaluate the performance of the proposed MC-FCM algorithm, we tested it using several UCI-benchmark datasets [13], described in Table 1 and compared it with FCM and FCM72I. The algorithm was built and tested using C#. To make our comparison objective, we adopted the following performance indices outlined in [14]:  
* The Davies-Bouldin (DB) index is based on a ratio involving within-group and between-group distances. \(DB=\frac{1}{C}\sum\limits_{j=1}^{C}D_{j}\), where \(D_{j}=\max\limits_{k\
eq j}\left\{D_{j,k}\right\}\), \(D_{j,k}\) is the within-to-between cluster spread for the \(j\)-th and \(k\)-th clusters, i.e., \(D_{j,k}=\left(\overline{d}_{j}+\overline{d}_{k}\right)/d_{j,k}\), where \(\overline{d}_{j}\) and \(\overline{d}_{k}\) are the average within-group distances for the \(j\)-th and \(k\)-th clusters, respectively and \(d_{j,k}\) is the inter-group distance between these clusters. These distances are defined as \(\overline{d}_{j}=(1/N_{j})\sum_{X_{i}\in C_{j}}\|X_{i}-\overline{X}_{j}\|\) and \(d_{j,k}=\|\overline{X}_{j}-\overline{X}_{k}\|\). Here, \(D_{j}\) represents the worst-case within-to-between cluster spread involving the \(j\)-th cluster. Minimizing \(D_{j}\) for all clusters minimizes the DB index. Hence, good partitions, which are comprised of compact and separated clusters, are distinguished by small values of DB. * The Alternative Silhouette Width Criterion (ASWC) is the ratio between the inter-group distance and the intra-group distance. \(ASWC=\frac{1}{N}\sum\limits_{i=1}^{N}S_{X_{i}}\), where \(S_{X_{i}}=\frac{b_{k,j}}{a_{k,j}+\varepsilon}\). Let us consider that the \(i\)-th element of the dataset, \(X_{i}\) belongs to a given cluster \(k\in\{1,2,\ldots,C\}\), then \(a_{k,i}\) is the average distance of \(X_{i}\) to all other elements in this cluster, \(d_{q,i}\) is the average distance of \(X_{i}\) to all elements in another cluster \(q\), with \(q\
eq k\), \(b_{k,i}\) is the minimum of \(d_{q,i}\) computed over \(q=1,2,\ldots,C\), with \(q\
eq k\) and \(\varepsilon\) is a small constant (e.g., \(10^{-6}\) for normalized data) used to avoid division by zero when \(a_{k,j}=0\). Large ASWC values indicate good partitions. * The PBM index is also based on the within-group and between-group distances. \(PBM=\left(\frac{1}{C}\frac{E_{1}}{E_{C}}D_{C}\right)^{2}\), where \(E_{1}=\sum\limits_{i=1}^{N}\|X_{i}-\overline{X}\|\) denotes the sum of distances between the elements and the grand mean of the data, \(E_{C}=\sum\limits_{j=1}^{C}\sum\limits_{X_{i}\in C_{j}}\|X_{i}-\overline{X}_{j}\|\) represents the sum of within-group distances and \(D_{C}=\max\limits_{j=k-1,\ldots,C}\|\overline{X}_{j}-\overline{X}_{k}\|\) is the maximum distance between group centroids. The best partition is indicated when PBM is maximized. * The Rand index (RI) can be seen as an absolute criterion that allows the use of properly labeled datasets for performance assessment of clustering results. This simple and intuitive index handles two hard partition matrices (\(R\) and \(Q\)) of the same dataset. The reference partition, \(R\), encodes the class labels, while the partition \(Q\) partitions the data into \(C\) clusters and is the one to be evaluated. We have \(RI=\frac{f_{11}+f_{00}}{N(N-1)/2}\), where \(f_{11}\) denotes the number of pairs of data elements belonging to the same class in \(R\) and to the same cluster in \(Q\), \(f_{00}\) denotes the number of pairs of data elements belonging to different classes in \(R\) and to different clusters in \(Q\). Large RI values indicate compatible clustering with the given class labels. * Mean accuracy (MA), \(MA=\min\limits_{j=1,\ldots,C}\left(\frac{a_{j}}{b_{j}}\right)\), where \(a_{j}\) is the number of elements in the cluster \(j\) after clustering and \(b_{j}\) is the actual number of elements in cluster \(j\). Large MA values often indicate good clustering.