This document discusses fuzzy clustering, which allows data points to belong to multiple clusters with a degree of membership. It describes fuzzy c-means clustering, which was improved in 1981 by Bezdek. The algorithm involves initializing membership values, computing centroids, recomputing membership values iteratively until convergence. Pros include a more natural representation of overlapping clusters, while cons include sensitivity to initialization and needing to specify the number of clusters. Applications include image segmentation, enhancement, and change detection.

1. Fuzzy Clustering
By:- Akshay Chaudhari

2. Agenda
 Introduction
 Fuzzy C-Mean clustering
 Algorithm
 Complexity analysis
 Pros. and cons.
 References

3. Introduction
Fuzzy clustering is a method of clustering which
allows one piece of data to belong to two or
more clusters.
In other words, each data is a member of every
cluster but with a certain degree known as
membership value.
This method (developed by Dunn in 1973 and
improved by Bezdek in 1981) is frequently used
in pattern recognition.

4. Applications
Image segmentation
Medical imaging
X-ray Computer Tomography (CT)
Magnetic Resonance Imaging (MRI)
Position Emission Tomography (PET)
Image and speech enhancement
Edge detection
Video shot change detection

5. Fuzzy C-means Clustering

6. Fuzzy C-means Clustering

7. Fuzzy C-means Clustering

8. Fuzzy C-means Clustering

9. Fuzzy C-means Clustering

10. Fuzzy C-means Clustering

11. Fuzzy C-means Clustering

12. Fuzzy C-Mean Algorithm
1. Select an initial fuzzy pseudo-partition, i.e. ,assign values
to all uij.
2. repeat
3. Compute the centroid of each cluster using fuzzy
pseudo-partition.
4. Recompute fuzzy pseudo-partition, i.e., the uij.
5. until the centroids don’t change.

13. Algorithm

14. An example
 X=[3 7 10 17 18 20] and assume C=2
0.1 0.2 0.6 0.3 0.1 0.5
 Initially, set U randomly U=
0.9 0.8 0.4 0.7 0.9 0.5
N
∑u m
x
ij i
cj = i =1
N
∑u
i =1
m
ij
 Compute centroids, cj using , assume m=2
1
uij = 2
C
 || xi − c j ||  m −1
 c1=13.16; c2=11.81
∑  || x − c || 

k =1 

i k 
 Compute new membership values, uij using
0.43 0.38 0.24 0.65 0.62 0.59
U=
 New U: 0.57 0.62 0.76 0.35 0.38 0.41
 Repeat centroid and membership computation until changes in
membership values are smaller than say 0.01

15. Complexity analysis
 Time complexity of the fuzzy c mean algorithm is
O(ndc2i)
 Where
 i number FCM over entire dataset.
 n number of data points.
 c number of clusters
 d number of dimensions
where… i grows very slowly with n,c and d.

16. Pros. & Cons.
 Pros:
 Allows a data point to be in multiple clusters
 A more natural representation of the behavior of genes
genes usually are involved in multiple functions
 Cons:
 Need to define c, the number of clusters
 Need to determine membership cutoff value
 Clusters are sensitive to initial assignment of centroids
Fuzzy c-means is not a deterministic algorithm

17. References
 http://home.dei.polimi.it/matteucc/Clustering/tutorial_h
tml/cmeans.html
 http://en.wikipedia.org/wiki/Fuzzy_clustering
 Section 9.2 from Introduction to Data Mining by Tan,
Kumar, Steinbach

18. Thank You …..