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 …..