# Fuzzy C means

August 11, 2017

## Fuzzy C-means: Overview

the fuzzy c-means algorithm is one of the most popular clustering technique in data mining. that technique enable the data objects to be available in more than one cluster at the time. therefore that technique can be used for other clustering technique implementations. that is also a unsupervised technique of data mining. that technique directly input the data samples as input and produces the clusters of data according to user requirements.

### Functional Overview

basically that technique is works on the basis of optimization of the objective function. that helps to improve the cluster membership from the different clusters available. Clustering is a mathematical tool that attempts to discover structures or certain patterns in a dataset, where the objects inside each cluster show a certain degree of similarity. It can be achieved by various algorithms that differ significantly in their notion of what constitutes a cluster and how to efficiently find them. Cluster analysis is not an automatic task, but an iterative process of knowledge discovery or interactive multi-objective optimization. It will often necessary to modify pre-processing and parameter until the result achieves the desired properties.

### Fuzzy C-Means Clustering

Fuzzy clustering is a powerful unsupervised method for the analysis of data and construction of models. In many situations, fuzzy clustering is more natural than hard clustering. Objects on the boundaries between several classes are not forced to fully belong to one of the classes, but rather are assigned membership degrees between 0 and 1 indicating their partial membership.

Fuzzy c-means is a clustering method similar to K-means but the concept of fuzzy theory is incorporated to improve clustering results. Fuzzy c means (FCM) is a method of clustering which allows one piece of data to belong to two or more clusters. This method (developed by Dunn in 1973 and improved by Bezdek in 1981) is frequently used in pattern recognition. d. In FCM (Fuzzy C means), the goal is to minimize the criterion function, taking into account the similarity of elements and cluster centers. It is more useful for data sets that have highly overlapping groups It is based on minimization of the following objective function:

$J_m= ∑_{(i=1)}^N∑_{(j=1)}^Cu_{ij}^m ∥x_i-c_j∥ , 1≤m<∞$

Where m is any real number greater than 1, ​$$u_{ij}$$​ is the degree of membership ​$$x_i$$​ of  in the cluster j, ​$$x_i$$​ is the $$i^{th}$$ of d-dimensional measured data, ​$$c_j$$​ is the d-dimension center of the cluster, and ||*|| is any norm expressing the similarity between any measured data and the center.
Fuzzy partitioning is carried out through an iterative optimization of the objective function shown above, with the update of membership ​$$u_{ij}$$​ and the cluster centers ​$$c_j$$​ by

$u_{ij}= 1/(∑_{(k=1)}^C[(∥x_i-c_j∥)/(∥x_i-c_k∥)]^{(2/(m-1))})$

$c_j= (∑_{(i=1)}^Nu_{ij} .x_i)/(∑_{(i=1)}^Nu_{ij}^m )$

### FCM (Fuzzy C Means) algorithm

The FCM(Fuzzy C means) algorithm is composed of the following steps:

1: Initialize U=[uij] matrix, U(0)

2: At k-step: calculate the centers vectors C(k)=[cj] with U(k)

$c_j= (∑_{i=1}^Nu_{ij} .x_i)/(∑_{i=1}^Nu_{ij}^m )$

3: Update U(k) , U(k+1)

$u_{ij}= 1/(∑_{k=1}^C[(∥x_i-c_j∥)/(∥x_i-c_k∥)]^{(2/(m-1))} )$

4: If ||U(k+1) – U(k)||< ​$$ε$$​, then STOP; otherwise return to step 2.

#### References

[1] An-Xin Ye and Yong-Xian Jin, “A Fuzzy C-Means Clustering Algorithm Based on Improved Quantum Genetic Algorithm”, International Journal of Database Theory and Application Vol.9, No.1 (2016), pp.227-236

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

[3]Dinesh Komarasamy and Vijayalakshmi Muthuswamy, “A Novel Approach for Dynamic Load Balancing with
Efective Bin Packing and VM Reconiguration in Cloud “, Indian Journal of Science and Technology, Vol 9(11), DOI: 10.17485/ijst/2016/v9i11/89290, March 2016

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