# A Novel Adaptive FCM with Cooperative Multi-Population Differential Evolution Optimization
## 3 Subset Selection
Thesehave been used as a test of clusterability, clustering tendency, and cluster validation [47; 48]. In this paper, we use the Hopkins statistic which is by far the easiest of the statistical measures to implement. For a dataset \(X\) of size \(n\) in \(d\)-dimensions, \(n_{0}\) sampling origins \(Y\) are placed at random in a sampling window such that \(n_{0}<<n\). The sampling origins \(Y\) are \(d\) dimensional as well. Two types of distances are defined: \(u_{k}\) is the distance of the sampling origin \(y_{k}\) to its nearest datapoint in \(X\), and \(w_{k}\) is the distance of a randomly chosen datapoint \(x_{k}\) to its nearest neighbor in \(X\). The Hopkins statistic [41] is defined as,  
\[H=\frac{\sum\limits_{k=1}^{n_{0}}u_{k}^{d}}{\sum\limits_{k=1}^{n_{0}}u_{k}^{d }+\sum\limits_{k=1}^{n_{0}}w_{k}^{d}} \tag{5}\]  
The statistic, therefore, compares the nearest-neighbor distribution of randomly selected locations in the dataset, to the nearest-neighbor distribution among objects in the data. If the underlying structure of dataset \(X\) is random, then on an average the sum of \(u_{k}\) would be equal to the sum of \(w_{k}\) over \(d\) dimensions and therefore \(H\) will be close to 0.5. However, if dataset \(X\) has distinct separable clusters (without any notion of how many clusters there are), then on an average the sum of \(u_{k}\) will greatly exceed the sum of \(w_{k}\) and therefore \(H\) will be very close to unity. In most cases, real-world datasets will fall somewhere within this spectrum of complete underlying randomness and distinct separable clusters--as a result, Hopkins statistic \(H\) will range between 0.5 and 1.0 for most datasets. If it is close to 0.5, the dataset is more random than clustered, and if close to 1.0 then there are distinct separable clusters. It is easy to see why the Hopkins statistic has been so popular for use as a measure of clustering tendency (the question of whether a dataset has an underlying cluster structure). In this paper, we use the statistic simply as a quantification of the underlying structure and compare it to the underlying structure of the subsets. The subset that under repeated spare sampling has a Hopkins statistic value close to that of the entire dataset most likely captures the underlying structure of the dataset. This is shown as Algorithm 2 below. ```
Input: Dataset data of size (n \(-\) rem), SP subsets P = [P\({}_{1}\), P\({}_{2}\),... P\({}_{\text{SP}}\)] each of size (n \(-\) rem)/SP, number of sampling origins n\({}_{0}\) for data, convergence threshold \(\varepsilon\)
1. Populate n\({}_{0}\) datapoints at random in data
2. Calculate [PERSON] statistic H for data
3. diff = \(\varepsilon\), best_subset = \(\O\)
4. Repeat
5. For k = 1 to SP
6. Populate n\({}_{0}\)/SP datapoints at random in P\({}_{k}\)
7. Calculate [PERSON] statistic H\({}_{k}\) for subset P\({}_{k}\)
8. If \(\mid\)H \(-\) H\({}_{k}\)\(\mid\) < diff, diff = \(\mid\)H \(-\) H\({}_{k}\)\(\mid\) and best_subset = k
9. End
10. Until (best_subset! = \(\O\)) Output: Subset P\({}_{\text{best\_subset}}\)
```  
**Algorithm 2**Identifying subset with similar underlying structure to the original dataset