# A Novel Adaptive FCM with Cooperative Multi-Population Differential Evolution Optimization
## 4 Proposed Co-Evolutionary Framework
### Evaluation of Clustering Results
The data is generated using three normal distributions centered at (3,2), (5,5,5,5) and (8,5,6). The intercluster variance is greater than that in **Data1**. The dataset is shown in Figure 5. FCM-AO, PSO-V, GAKFCM and EwFCM are run for \(c=2\) to \(c=45\). For ADEFC and CDE-FCM, the vector representation is of length 90 with \(d=2\) and \(c_{max}=45\). FCM-AO, GAKFCM, and EwFCM identify \(c=5\) as the most-optimal cluster. On the other hand, the evolutionary algorithm-based approaches, viz. PSO-V, ADEFC and the proposed CDE-FCM correctly identify the \(c=3\) partition. In fact, the terminating elite population in CDE-FCM has 22% individuals that encode for \(c=3\) and 10% that encode for \(c\) = 5. The best subset of cardinality 200 as identified by Algorithm 2 is shown in Figure 6 and the initial cluster prototypes evolved by the CDE (Algorithm 3) are shown in Figure 7. The \(c=5\) partition as identified by FCM-AO as the most optimal partition is shown in Figure 8 and the \(c=3\) partition as identified by FCM-AO is shown in Figure 9. The correct \(c=3\) partition as identified by CDE-FCM with evolved initial prototypes is shown in Figure 10. By simple visual inspection, it can be said that the quality of the partition in Figure 10 is better than that in Figure 9 (same \(c=3\)). The fact that FCM-AO with randomly initialized cluster prototypes identifies \(c=5\) as better than \(c=3\) on the basis of all cluster validity measures shows the promise of the proposed method. After the initial cluster prototypes are identified by Algorithm 3, the FCM-AO converged on an average in 4.5 iterations over 10 implementations. This is in comparison to the naive FCM-AO with random initialization which took at least 25 iterations for an average of 28 iterations with the same convergence criterion of \(\varepsilon=0.001\). Figure 6: Best subset of **Data2** using sparse sampling (\(n=200\), \(H=0.9025\)). Figure 7: Initial cluster prototype as calculated by CDE on best subset of **Data2**. The Xs mark the initial cluster prototype location. Table 3.0 Best partition by FCM-AO, \(c=5\) with randomly initialized cluster prototypes. Table 3.1 Partition by FCM-AO, \(c=3\) with randomly initialized cluster prototypes. **Iris** data consists of \(n=150\) datapoints divided into three types of Iris flowers-Setosa, Virginica, and Versicolor with 50 samples in each class. Each sample has four associated features: sepal length, petal length, sepal width, and petal width. FCM-AO, PSO-V, GAKFCM and EwFCM are run for \(c=2\) to \(c=12\). For ADEFC and CDE-FCM the vector representation is of length 48 with \(d=4\) and \(c_{mxx}=12\). The performances of different algorithms as measured by the cluster validity criteria are listed in Table 3.2 Two classes (Versicolor and Virginica) are known to be linearly inseparable from each other although Setosa is linearly separable from the other classes ([PERSON], 2016) and therefore many algorithms identify the \(c=2\) solution as the most optimal one. Almost 20% of the candidate vectors in the terminating elite population of CDE-FCM encoded for \(c=2\) while 18% encoded for \(c=3\) with an equal number encoding for \(c=9\). The cluster prototypes identified by CDE-FCM in almost all candidates (>85%) in the terminating elite population are almost optimal, and a further application of FCM-AO converges on average in 4.1 iterations over 10 independent runs of the algorithm. On the other hand, the best-performing instance of the naive FCM-AO with random initializations took 22 iterations.