# A Novel Adaptive FCM with Cooperative Multi-Population Differential Evolution Optimization
## 4 Proposed Co-Evolutionary Framework
### Evaluation of Clustering Results  
After the initial cluster prototypes are evolved, they are then used as input to the FCM-AO algorithm for clustering. In Section 5, we present results from simulations conducted using both synthetic and benchmark datasets as well as a case study with exploratory data analysis from experimental data in Section 6. We compare the performance of various algorithms with our proposed method using several cluster validity indices from the literature. Cluster validity measures have been used in the literature for accessing the correctness of a partition. Since the Xie-Beni index \(XB\) and the \(\mathbf{\Psi}\) index are used as a combined measure of fitness, other cluster validity indices are used as a quantitative measure of comparison of various clustering techniques. The Bensaid Index \(V_{B}\) is insensitive to the number of data points in a cluster [13]. The index is defined by the ratio of compactness of a cluster to the separation between clusters. \[V_{B}=\sum\limits_{i=1}^{c}\frac{\sum\limits_{i=1}^{n}u_{i}^{2}||x_{i}-v_{i} ||^{2}}{\sum\limits_{i=1}^{n}\left\|(v_{i}-v_{i})\right\|^{2}} \tag{21}\]  
where \(n_{k}\) is the fuzzy cardinality of cluster \(k\) defined by \(n_{k}=\sum\limits_{i=1}^{n}u_{i}\). The final cluster prototypes are denoted by \(\Upsilon=(v_{1},\,v_{2},\ldots,\,v_{c})\). Note that the initial cluster centers are denoted by \(v\) while the final cluster centers are denoted by \(v\) to distinguish the two. The Bensaid index for a good partition is lower when compared to an inferior partition (one with an incorrect value of \(c\) or incorrect cluster prototypes \(v\)). [PERSON] et al. [14] introduced a punishing function which is the average distance between cluster centers-second term in the numerator in equation (22). This is used to counter the decreasing tendency of any cluster validity index as \(c\to n\). The second term in the denominator is also a punishing function. The Tang index is denoted by \(V_{T}\). \[V_{T}=\sum\limits_{i=1}^{c}\frac{\sum\limits_{i=1}^{c}\frac{\sum \limits_{i=1}^{n}u_{i}^{2}||x_{i}-v_{i}||^{2}+\frac{1}{\sigma(c-1)}\sum\limits _{i=1}^{c}\frac{\sum\limits_{i=1}^{c}||v_{i}-v_{i}||^{2}}{||v_{i}-v_{i}||^{2} }}}{\min\limits_{i\
eq k}||v_{i}-v_{i}||^{2}+\frac{1}{c}} \tag{22}\]  
The modified Kwon index \(V_{K2}\) proposed in [59] uses two terms in the numerator as punishing functions applied to eliminate the decreasing tendency as \(c\to n\) and a dummy term for the stability of the index when \(c\to n\) and \(m\rightarrow\infty\), respectively. The second term of the denominator is an ad hoc punishing function used to strengthen the numerical stability as \(m\rightarrow\infty\). The last term of the denominator is another ad hoc punishing function used to strengthen the numerical stability as \(c\to n\). \[V_{K2}=\frac{\frac{\alpha-\alpha+4}{2}\left[\left(\frac{c-1}{c-1}\right)^{ \alpha}\frac{C}{c-1}\sum\limits_{k=1}^{c}u_{k}^{2\alpha/\alpha/2}||x_{i}-v_{k} ||^{2}+\frac{\frac{\sum\limits_{i=1}^{c}||v_{i}-v_{i}||^{2}}{||v_{i}-v_{i}||^{2 }}}{\frac{\sum\limits_{i=1}^{c}||v_{i}-v_{i}||^{2}}{||v_{i}-v_{i}||^{2}}}+ \frac{\alpha}{(\alpha-\alpha+1)^{2}}\right]}{\min\limits_{i\
eq k}||v_{i}-v_{k} ||^{2}+\frac{1}{c}\frac{\sum\limits_{i=1}^{c}||v_{i}-v_{i}||^{2}}{||v_{i}-v_{ k}||^{2}}} \tag{23}\]where, \(\overline{v}=\frac{1}{n}\sum\limits_{j=1}^{n}x_{j}\) is the mean cluster prototype when \(c=1\). Ren index \(V_{R}\)[60] is an improvement on the Bensaid index \(V_{B}\). The numerator represents the compactness of the cluster \(C_{k}\), where \(n_{k}\) is its fuzzy cardinality. Its second item, an introduced punishing function, denotes the distance from the cluster prototype of the \(k\)th cluster to the average of all cluster prototypes, which can eliminate the monotonically decreasing tendency as the number of clusters increases to \(n\). The denominator represents the mean distance from the \(k\)th cluster prototype to all other cluster prototypes, which is a measure of intercluster separation.