# A Novel Adaptive FCM with Cooperative Multi-Population Differential Evolution Optimization
## 2 Background
### Determination of Optimal Number of Clusters  
For two-dimensional datasets, the number of clusters can be determined by simple visualization. For higher dimensional data, the dataset can be mapped to a two-dimensional plane using dimensionality reduction techniques such as principal component analysis (PCA), kernel, locally linear embedding (LLE), diffusion maps, and sparse dictionary representations. These techniques do not, however, provide a mapping that conserves the internal partitioned structure of the dataset. More popular are cluster validity methods that quantify the goodness of a partition after the clustering method produces the partition. Different partitions for different values of \(c\) are created and then the goodness of partition is compared using a measurable index. The value of the index reaching a maximum or minimum or an inflection point is often a good indicator of the goodness of partition and the corresponding number of clusters is the optimal number of clusters. We use several of these cluster validity measures in this paper. However, these are used not to determine an optimal value for \(c\), but instead to compare the performance of different algorithms.  
The first cluster validity index was proposed by [PERSON] called the degree of separation which was later refined as the concept of partition coefficient and the closely related partition entropy by [PERSON] [6]. [PERSON] proposed a fuzzy clustering validity index using the distinguishableness of clusters measured by the object proximities [7]. Based on Shannon entropy and fuzzy variation theory, [PERSON] and [PERSON] proposed a fuzzy clustering validity index taking into account of the geometry structure of the dataset [8]. [PERSON] et al. presented an algorithm based on differential evolution for automatic cluster detection, which evaluated the validity of the clustering result [9]. [PERSON] et al. partitioned the original data space into a grid-based structure and proposed a cluster separation measure based on grid distances [10]. Based on the idea that a natural partition is one that creates well-separated compact clusters, measures of compactness and separation were proposed as part of several cluster validity measures. These include Fukuyama-Sugeno index _FS_[11], Xie-Beni index _XB_[12], Bensaid index _SC_[13], Tang index _V\({}_{T}\)_[14], Kwon index _V\({}_{K}\)_[15], _PBMF_ index [16], _PCAES_ index [17], _WL_ index [18], Wang index _V\({}_{W}\)_[19], _CWB_ index [20], and Zhu index _V\({}_{Z}\)_[21].  
Since these measures compare and quantify the goodness of the partition after the partition is generated, they are not practical for problems involving large datasets (high cardinality \(n\)) or high number of dimensions \(d\). Lately, some new clustering algorithms have been proposed that adjust the number of clusters while the partition is being generated. Competitive agglomeration [22] and split hierarchical clustering [23] have been used to guide the optimization of the number of clusters in a partitional clustering process. A similarity-based clustering method that combines single-point iteration with hierarchical clustering to determine the number of clusters is proposed in [24]. A Mercer kernel-based clustering [25] estimated the number of clusters by the eigenvectors of a kernel matrix. A clustering algorithm based on maximal \(\theta\)-distant subtrees [26] detected any number of well-separated clusters of any shape.