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IntroductionFuzzy 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.
Fuzzy C-Mean Algorithm1. Select an initial fuzzy pseudo-partition, i.e. ,assign values to all uij.2. repeat3. 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.
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
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.
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
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