Fuzzy-C-Means (FCM) clustering algorithm 

It's a prototype based method. An object represents a cluster. It's a probabilistic method and the sum of the membership values of an object to all clusters is 1. The idea for cluster computing is to minimize the distances of all the data points from all cluster centers. The cluster number c must be specified. And it is also the C in Fuzzy-C-Means.

$J_{\mathit{FCM}}=\underset{k=1}{\overset{N}{\sum }}\underset{i=1}{\overset{C}{\sum }}{(\mathit{mik})}^m\cdot {\mathit{dik}}^2$

It is a heuristic method and I use it as follows:

Step 1: Initialization

I choose the Euclidean distance norm for distance measure. The initial distribution of the fuzzy partition matrix U variable. You can choose random values or one cluster with 1.0 and the other with 0.0. The termination condition for termination threshold is normally 1.0e-7. The object length and the number of clusters has to be determined.

Step 2: Determination of the cluster centers

$\mathit{vi}[k][]=\frac{\underset{i=0}{\overset{\mathit{mik.length}}{\sum }}\mathit{mik}[i]{[k]}^{m=2}\cdot \mathit{object}[i][]}{\underset{k=1}{\overset{\mathit{mik.length}}{\sum }}\mathit{mik}[i]{[k]}^{m=2}}$

Step 3: Calculate the new partition matrix

$\mathit{mik}[i][k]=\frac{{(\frac{1}{\mathit{dik}=\mathit{object}[i][]-\mathit{vi}[k][]})}^{\frac{1}{m=2-1}}}{\underset{j=0}{\overset{\mathit{vi.length}}{\sum }}{(\frac{1}{\mathit{dik}=\mathit{object}[i][]-\mathit{vi}[j][]})}^{\frac{1}{m=2-1}}}$

Step 4: Termination or repetition

Step 5: optional - Repeat calculation (steps 2 to 4)