# A Novel Adaptive FCM with Cooperative Multi-Population Differential Evolution Optimization
## 3 Subset Selection  
A co-evolutionary algorithm recognizes the diversity of the possible candidate solutions in the population. It emphasizes the correlation between the domain of the problem being solved and the candidate solutions. Evolutionary algorithms are prone to get trapped in local optima, and performance depends on fine tuning a few or many parameters. Co-evolutionary systems, on the other hand, have been shown to have the ability to avoid local optima by dividing the solution space and also use different evolutionary strategies to support either a competitive system or a cooperative system. Their strength lies in the divide-and-conquer decomposition strategy and in their implicit parallelism. These algorithms can be divided into competitive co-evolutionary systems and cooperative co-evolutionary systems and include systems such as co-evolution of predator and prey systems, competing species, among others [38; 39; 40]. These systems can balance the exploration and exploitation capability by utilizing cooperative or competitive mechanisms among different subpopulations using different strategies for different subpopulations. They can also improve convergence properties (not necessarily speed) by facilitating information exchange among the subpopulations. There are two important reasons for using coevolutionary algorithms for clustering problems: (1) evolution of candidate solutions on a subset of the larger dataset improves the convergence properties of the individual subpopulation. This is because a judiciously selected subset of a larger dataset can capture the structure of the larger dataset and since it is smaller than the original dataset, an evolutionary algorithm can converge faster. (2) Coevolution, both competitive and cooperative, ensures that excellent candidates which in a single-strategy one-population evolution may not be able to evolve for several generations which is an implicit strain on resources, now have a better chance of getting selected for evolution. The proposed co-evolutionary framework is described here. The dataset of size \(n\) to be clustered is first divided into SP distinct subsets such that in each subset there are \(n/\)SP objects. If \(n\) is not evenly divisible by SP, then the largest number \(<n\) evenly divisible by SP is used. For example, if \(n=528\) and SP = 10, we select 520 objects at random from the dataset and divide them into 10 subsets with 52 objects in each. Every subset is mutually exclusive so that a data object is available to only one subset. This ensures an even coverage of the entire dataset. The algorithm to create mutually exclusive subsets of the data is shown as Algorithm 1. ```
Input: Dataset data of size n and dimensions d: X = {x\({}_{1}\), x\({}_{2}\),... x\({}_{n}\)}, where x\({}_{i}\) = {x\({}_{1}\), x\({}_{2}\),... x\({}_{n}\)}, SP
1. Calculate rem = remainder(n/SP). If rem! = 0, discard rem datapoints randomly from data and retain (n - rem) datapoints as data
2. For k = 1 to SP
3. Select (n - rem)/SP datapoints at random from data to populate subset P\({}_{k}\)
4. Remove the same (n - rem)/SP points from data: data = data - P\({}_{k}\)
5. End Output: SP Subsets P = {P\({}_{1}\), P\({}_{2}\),... P\({}_{SP}\)}, rem
```  
**Algorithm 1**Dividing dataset into subsets  
Of these distinct datasets, only the one with a similar underlying structure as the original dataset is then selected. This is performed by employing a sparse sampling statistic based on a test of spatial randomness. Sparse sampling tests are based on sampling origins randomly assigned in a sampling window and the underlying structure of the data can be quantified by nearest neighbor measurement of sampling origins to points and comparing them to the random nearest neighbor measurement of paired points within the sampling window. Several tests involving sampling origins have been proposed in the literature, mostly from the field of biological statistics, based on a multitude of tests such as the Hopkins [41; 42], Holgate [43], T-square [44], Eberhardt [45] and Cox-Lewis [46].