Clustering > Fuzzy Clustering

## What is Fuzzy Clustering?

**Fuzzy clustering **is a clustering method where **data points can belong in more than one group (“cluster”)**. Clustering divides data points into groups based in similarity between items and looks to find patterns or similarity between items in a set; Items in clusters should be as similar as possible to each other and as dissimilar as possible to items in other groups [1]. Computationally,** it’s much easier to create fuzzy boundaries than it is to settle on one cluster for one point.**

In “hard” clustering, each data point can only be in one cluster. In “soft” or “fuzzy” clustering, data points can belong to more than one group. Fuzzy clustering uses least-squares solutions to find the optimal location for any data point.

**This optimal location may be in a probability space between two (or more) clusters.**

Fuzzy clustering is very similar to atomic orbitals and electron behavior: an electron isn’t in a single location but only has a *probability *of being in a particular orbital shell. If you think of orbital shells as “clusters” and electrons as “data points” (where each data point is assigned a probability for being located in a particular cluster), then you’ve got a basic grasp of the fundamentals of fuzzy clustering.

## Algorithms

Fuzzy clustering algorithms are divided into two areas:** classical fuzzy clustering **and **shape-based fuzzy clustering.**

### Classical fuzzy clustering algorithms.

**Fuzzy C-Means algorithm (FCM)**. This widely-used algorithm is practically identical to the K-Means algorithm. A data point can theoretically belong to all groups, with a membership function (also called a membership grade) between 0 and 1, where: 0 is where the data point is at the farthest possible point from a cluster’s center and 1 is where the data point is closest to the center. Subtypes include Possibilistic C-Means (PCM), Fuzzy Possibilistic C-Means (FPCM) and Possibilistic Fuzzy C-Means (PFCM).**Gustafson-Kessel (GK) algorithm**: associates a data point with a cluster and a matrix. While C-means assumes the clusters are spherical, GK has elliptical-shaped clusters.**Gath-Geva algorithm**(also called Gaussian Mixture Decomposition): similar to FCM, but clusters can have*any*shape.

### Shape-based fuzzy clustering algorithms.

**Circular shaped**: circular-shaped (CS) algorithms are what constrains data point to a circular shape. When this algorithm is incorporated into Fuzzy C-Means it’s called CS-FCM.**Elliptical shaped**: an algorithm that constrains points to elliptical shapes. Used in the GK algorithm.**Generic shaped**: most real life objects are neither circular not elliptical; the generic alorithm allows for clusters of any shape.

## References:

[1] Suganya, R. & Shanthi, R. Fuzzy C-Means Algorithm — A Review. International Journal of Scientific and Research Publications, Volume 2, Issue 11, November 2012 1

[2] By Danielcschossow – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=58428026