Cmeans

C-means, or C-means clustering, is a family of partitioning clustering methods that organize data into a predefined number of clusters based on proximity to cluster centers. The core idea is to represent each cluster by a center, typically the mean of its members, and to assign data points to clusters in a way that minimizes a chosen within-cluster measure. In many sources, hard C-means refers to the traditional k-means algorithm, where each point is assigned to exactly one cluster, while fuzzy C-means (FCM) is a related soft-clustering variant that permits partial membership in multiple clusters.

Hard C-means (k-means) operates by iteratively performing two steps: assignment and update. In the assignment step, each data point is placed in the cluster with the nearest center. In the update step, each cluster center is recomputed as the mean of all points assigned to that cluster. The process repeats until convergence, typically when changes in assignments or centers fall below a threshold. The algorithm is simple and scalable but converges to a local optimum and can be sensitive to initialization and outliers. It also presumes roughly spherical clusters of similar size and requires the number of clusters to be specified in advance.

Fuzzy C-means extends the approach by allowing degrees of membership for each point to every cluster, controlled by a fuzziness parameter. This yields softer boundaries and can handle overlapping clusters, but requires choosing the membership exponent and can be more computationally intensive.

Applications of C-means methods include image segmentation, document clustering, market research, and exploratory data analysis. Limitations include sensitivity to initialization, the need to predefine the number of clusters, and assumptions about cluster shape and density. See also k-means and fuzzy c-means.