Fuzzy clustering

Fuzzy clustering (also referred to as soft clustering) is a form of clustering in which each data point can belong to more than one cluster.

Clustering or cluster analysis involves assigning data points to clusters (also called buckets, bins, or classes), or homogeneous classes, such that items in the same class or cluster are as similar as possible, while items belonging to different classes are as dissimilar as possible. Clusters are identified via similarity measures. These similarity measures include distance, connectivity, and intensity. Different similarity measures may be chosen based on the data or the application.

In non-fuzzy clustering (also known as hard clustering), data is divided into distinct clusters, where each data point can only belong to exactly one cluster. In fuzzy clustering, data points can potentially belong to multiple clusters.

Membership grades are assigned to each of the data points. These membership grades indicate the degree to which data points belong to each cluster. Thus, points on the edge of a cluster, with lower membership grades, may be in the cluster to a lesser degree than points in the center of cluster.

One of the most widely used fuzzy clustering algorithms is the Fuzzy C-means clustering (FCM) Algorithm.

Fuzzy c-means (FCM) clustering was developed by J.C. Dunn in 1973, and improved by J.C. Bezdek in 1981.

The fuzzy c-means algorithm is very similar to the k-means algorithm:

Any point x has a set of coefficients giving the degree of being in the kth cluster w_k(x). With fuzzy c-means, the centroid of a cluster is the mean of all points, weighted by their degree of belonging to the cluster:

${\displaystyle c_{k}={{\sum _{x}{w_{k}(x)}^{m}x} \over {\sum _{x}{w_{k}(x)}^{m}}}.}$

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