Dynamic clustering algorithm using fuzzy c means
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This document proposes a dynamic clustering algorithm using fuzzy c-means clustering. It begins with an introduction to fuzzy c-means clustering and its limitations when the chosen number of clusters is incorrect. It then proposes a dynamic clustering algorithm that starts with a fixed number of clusters but can automatically increase the number of clusters during iterations based on the data, improving purity. The algorithm is described and examples are provided to illustrate its effectiveness at forming clear clusters after iterations and determining when clustering has terminated.
![applicable everywhere if one wants to
classify some objects into several
categories, we commonly encounter. [4]
Classification of data is done in general
using Fuzzy C-mean clustering, though this
classification needs a number of clusters
as input and the optimal convergence
depends on the initial cluster centers
selection. Therefore, Fuzzy C-means
method is not suitable for classifying large
data set. [5]
When similar characteristic data is
grouped then it is known as data
classification. It is done to enhance human
understanding of data structure and to
describe the data behaviour by building
models. K-means clustering algorithm,
Fuzzy C-mean clustering algorithm,
neural-net etc. were developed, just to be
applied in this field of study. [8]
Fuzzy C-means clustering classifies the
unclassified data after performing pre-
classification. Y.Lim applied Fuzzy C-means
clustering to colour image segmentation.
He obtained the appropriate threshold
value to get the number of cluster using
scale space filtering and 1st and 2nd
differentiation. Then image data was pre-
classified on the basis of appropriate
threshold value. Finally, fine-classification
was performed using Fuzzy C-means
clustering.
Though Fuzzy C-means clustering is not
generally used for large data classification
Y.G.Jin used a method, subtractive and
gravity Fuzzy C-means clustering to
overcome this problem. [7] Subtractive
clustering is used to get the number of
cluster and the cluster centers used for
pre-classifications. Then during pre-
classification, for the unclassified data
gravity Fuzzy C-means clustering is used
which actually overcomes the deficiency
of Fuzzy C-means clustering.
This algorithm is a process for showing
that dataset can be differentiated and
formulated into groups but it can be seen
that every data has some specifications
such as difference between each nodes of
data, difference of distance, different
weights for data nodes that makes it
worse to simplify how to group each node
points in such a way that will show better
classification and use for data nodes. [6]
This algorithm is also used to separate the
data in different magnitude of cluster by
using the logic of the fuzzy theory. This
division depends on various criteria, such
as, distance between two data nodes,
choosing centroid and membership
function that mean we do not have
accurate data cluster size. [1]
4. Implementation
4.1. Fuzzy C-means: [3]
The main concept of FCM is to find a Fuzzy
pseudo-partition to minimize the cost
function.
Cost Function:
s.t.
In the above formula,
yj= featured data to be clustered;
nl= center of each cluster;
vjl= fuzzy partition corresponding to
feature data;
m= number of feature data;
L= number of cluster;
δ= exponent to adjust fuzzy degree.
The updating steps are as follows:
E-Step:
M-Step:](https://image.slidesharecdn.com/dynamicclusteringalgorithmusingfuzzyc-means-130611225524-phpapp02/85/Dynamic-clustering-algorithm-using-fuzzy-c-means-2-320.jpg)
![E-step is used to get new center of each
cluster and M-step is used to update the
fuzzy partition. When E-step and M-step
are repeated, cluster center m and fuzzy
partition u are updated, until the cost
function reaches the minimal value, or it
cannot be reduced anymore, we get the
final cluster information.
4.2. The Fuzzy C-means Algorithm: [2]
1. Initialize V=[vjx] matrix, V(0)
2. At k-step: calculate the centers
vectors centroid(k)=[centroidz] with
V(l)
3. Update V(l) , V(l+1)
4. If || V(l+1) - V(l)||< then STOP;
otherwise return to step 2.
5. Results and Discussion
After we implemented this algorithm we
got the following outputs.
Fig5.1: Input Feature Vector
In the above figure we can see the input
feature Vectors. Here we can see the
clusters formed after implementation of
the algorithm. We can also see the
belonging of certain data points to a
particular cluster.
Fig5.2: Input Feature Vector after
Iterations
Here, in the above figure we can see the
input feature vectors after iterations.](https://image.slidesharecdn.com/dynamicclusteringalgorithmusingfuzzyc-means-130611225524-phpapp02/85/Dynamic-clustering-algorithm-using-fuzzy-c-means-3-320.jpg)
![After several iterations we can see the
clearly formed clusters. The points which
were distantly connected to a particular
cluster have moved in closer to it after
several iterations.
Fig5.3: Termination Measure
In the above figure we have shown the
termination measure after several
iterations.
6. Proposed Algorithm
6.1. The Dynamic Fuzzy C-mean
Algorithm:
1. Initialize V=[vjx] matrix, V(0)
2. At k-step: calculate the centers
vectors centroid(k)=[centroidz] with
V(l)
3. Update V(l) , V(l+1)
4. Find the maximum number of
objects closer to yj where j=1 to n.
5. Calculate the distance between all
objects yj to that of cluster centroid.
[d(yj, centroidz)]
6. If d(yj,centroidz) where j=1,2,...., c,
is higher, then yj is new centroid
and centroid=centroid+1.
7. If || V(l+1) - V(l)||< then STOP;
otherwise return to step 2.
8. Conclusion
There are many algorithms present which
can be applied to implement clustering.
Fuzzy C-mean clustering algorithm is easy
and efficient to implement. Our research
proposes the dynamic clustering of the
data set and also belonging of points
between clusters. The proposed algorithm
if implemented, then, at each and every
iteration, based on the distance between
the objects, new clusters will be formed
and new centroid will be generated. The
main benefit of this algorithm is to
develop a high performance algorithm
which will reduce the number of iterations
and also will converge faster than the
existing Fuzzy C-mean algorithm.
9. Reference
1. Abu-Zanona M.A., El-Zaghmouri B.M,
Fuzzy C-Means Clustering Algorithm
Modification and Adaptation for
Application, World of Computer
Science and Information Technology
Journal (WCSIT), ISSN: 2221-0741, Vol.
2, No. 1, 42-45, 2012.
2. Bezdek.J.C(1981): “Pattern
Recognition with Fuzzy Objective](https://image.slidesharecdn.com/dynamicclusteringalgorithmusingfuzzyc-means-130611225524-phpapp02/85/Dynamic-clustering-algorithm-using-fuzzy-c-means-4-320.jpg)
Dynamic clustering algorithm using fuzzy c means
- 1. DYNAMIC CLUSTERING ALGORITHM USING FUZZY C- MEANS J Anuradha, Wrishin Bhattacharya, Tanuja Senapaty School of Computing Science and Engineering VIT University, Vellore – 14. 1. ABSTRACT Here, in this paper we are introducing a dynamic clustering algorithm using fuzzy c-mean clustering algorithm. We will try to process several sets patterns together to find a common structure. The structure is finalized by interchanging prototypes of the given data and by moving the prototypes of the subsequent clusters toward each other. In regular FCM clustering algorithm, fixed numbers of clusters are chosen and those are pre- defined. If, in case, the number of chosen clusters is wrong, then the final result will degrade the purity of the cluster. In our proposed algorithm this drawback will be overcome by using dynamic clustering architecture. Here we will take fixed number of clusters in the beginning but on iterations the algorithm will increase the number of clusters automatically depending on the nature and type of data, which will increase the purity of the result at the end. A detailed clustering algorithm is developed on a basis of the standard FCM method and will be illustrated by means of numeric examples. Keywords: Cluster, dynamic clustering, objective function, membership function, fuzzy membership. 2. Introduction A cluster is defined as a group of similar type of objects. The objects belonging to same cluster are of same type, and objects belonging to different clusters are of different types. When we have to group N number of patterns in C number of clusters which has high rate of similarity in its own class and low rate of similarity with respect to other classes then it becomes a problem. The main goal of “Objective-Function” supported clustering algorithm is to determine a partition for a cluster. The fuzzy C-means algorithm represents each cluster by its centre of gravity. Let us consider an example. Suppose, there is large amount of data about client information which is distributed in different databases and if we now try to mine such data, an intelligent approach would be to analyze each database locally and then combine the results at globally abstract level, which also satisfy some of the security concerns of the client as well. In this situation, one can cluster each subpopulation locally as a module which will enable faster convergence of clustering. Finally, after the union between the different modules of data we are converged to a stable clustering. Fuzzy C-means is derived by incorporating fuzzy sets, rough sets and c-means framework together. Overlapped partitions are efficiently handled by fuzzy c-means membership. 3. Literature Review From the early 1950’s, pattern recognition became a field of study. From the 1960’s, Fuzzy Theory was started to be used in the field of patter recognition and clustering analysis. Pattern Recognition is not the only application of cluster analysis. Based on many criteria, cluster analysis can be used in social groupings, retrieval of information etc. So, we can say that it is
- 2. applicable everywhere if one wants to classify some objects into several categories, we commonly encounter. [4] Classification of data is done in general using Fuzzy C-mean clustering, though this classification needs a number of clusters as input and the optimal convergence depends on the initial cluster centers selection. Therefore, Fuzzy C-means method is not suitable for classifying large data set. [5] When similar characteristic data is grouped then it is known as data classification. It is done to enhance human understanding of data structure and to describe the data behaviour by building models. K-means clustering algorithm, Fuzzy C-mean clustering algorithm, neural-net etc. were developed, just to be applied in this field of study. [8] Fuzzy C-means clustering classifies the unclassified data after performing pre- classification. Y.Lim applied Fuzzy C-means clustering to colour image segmentation. He obtained the appropriate threshold value to get the number of cluster using scale space filtering and 1st and 2nd differentiation. Then image data was pre- classified on the basis of appropriate threshold value. Finally, fine-classification was performed using Fuzzy C-means clustering. Though Fuzzy C-means clustering is not generally used for large data classification Y.G.Jin used a method, subtractive and gravity Fuzzy C-means clustering to overcome this problem. [7] Subtractive clustering is used to get the number of cluster and the cluster centers used for pre-classifications. Then during pre- classification, for the unclassified data gravity Fuzzy C-means clustering is used which actually overcomes the deficiency of Fuzzy C-means clustering. This algorithm is a process for showing that dataset can be differentiated and formulated into groups but it can be seen that every data has some specifications such as difference between each nodes of data, difference of distance, different weights for data nodes that makes it worse to simplify how to group each node points in such a way that will show better classification and use for data nodes. [6] This algorithm is also used to separate the data in different magnitude of cluster by using the logic of the fuzzy theory. This division depends on various criteria, such as, distance between two data nodes, choosing centroid and membership function that mean we do not have accurate data cluster size. [1] 4. Implementation 4.1. Fuzzy C-means: [3] The main concept of FCM is to find a Fuzzy pseudo-partition to minimize the cost function. Cost Function: s.t. In the above formula, yj= featured data to be clustered; nl= center of each cluster; vjl= fuzzy partition corresponding to feature data; m= number of feature data; L= number of cluster; δ= exponent to adjust fuzzy degree. The updating steps are as follows: E-Step: M-Step:
- 3. E-step is used to get new center of each cluster and M-step is used to update the fuzzy partition. When E-step and M-step are repeated, cluster center m and fuzzy partition u are updated, until the cost function reaches the minimal value, or it cannot be reduced anymore, we get the final cluster information. 4.2. The Fuzzy C-means Algorithm: [2] 1. Initialize V=[vjx] matrix, V(0) 2. At k-step: calculate the centers vectors centroid(k)=[centroidz] with V(l) 3. Update V(l) , V(l+1) 4. If || V(l+1) - V(l)||< then STOP; otherwise return to step 2. 5. Results and Discussion After we implemented this algorithm we got the following outputs. Fig5.1: Input Feature Vector In the above figure we can see the input feature Vectors. Here we can see the clusters formed after implementation of the algorithm. We can also see the belonging of certain data points to a particular cluster. Fig5.2: Input Feature Vector after Iterations Here, in the above figure we can see the input feature vectors after iterations.
- 4. After several iterations we can see the clearly formed clusters. The points which were distantly connected to a particular cluster have moved in closer to it after several iterations. Fig5.3: Termination Measure In the above figure we have shown the termination measure after several iterations. 6. Proposed Algorithm 6.1. The Dynamic Fuzzy C-mean Algorithm: 1. Initialize V=[vjx] matrix, V(0) 2. At k-step: calculate the centers vectors centroid(k)=[centroidz] with V(l) 3. Update V(l) , V(l+1) 4. Find the maximum number of objects closer to yj where j=1 to n. 5. Calculate the distance between all objects yj to that of cluster centroid. [d(yj, centroidz)] 6. If d(yj,centroidz) where j=1,2,...., c, is higher, then yj is new centroid and centroid=centroid+1. 7. If || V(l+1) - V(l)||< then STOP; otherwise return to step 2. 8. Conclusion There are many algorithms present which can be applied to implement clustering. Fuzzy C-mean clustering algorithm is easy and efficient to implement. Our research proposes the dynamic clustering of the data set and also belonging of points between clusters. The proposed algorithm if implemented, then, at each and every iteration, based on the distance between the objects, new clusters will be formed and new centroid will be generated. The main benefit of this algorithm is to develop a high performance algorithm which will reduce the number of iterations and also will converge faster than the existing Fuzzy C-mean algorithm. 9. Reference 1. Abu-Zanona M.A., El-Zaghmouri B.M, Fuzzy C-Means Clustering Algorithm Modification and Adaptation for Application, World of Computer Science and Information Technology Journal (WCSIT), ISSN: 2221-0741, Vol. 2, No. 1, 42-45, 2012. 2. Bezdek.J.C(1981): “Pattern Recognition with Fuzzy Objective
- 5. Function Algorithms”, Plenum Press, New York. 3. Gath.I, Unsupervised Optimal Fuzzy Clustering, IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE. VOL. I I . NO. 7. JULY 1989 4. Hall L.O, A Comparison of Neural Network and Fuzzy Clustering Techniques in Segmenting Resonance Images of the Brain, IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 3, NO. 5, SEPTEMBER 1992. 5. Jin Y.G, Kwon.O.S, Kim .T.K “DATA CLASSIFICATION BASED ON SUBTRACTIVE AND GRAVITY FUZZY C- MEANS CLUSTERING”, Fuzzy Logic and Intelligent Technologies for Nuclear Science and Industry, Proceedings of the 3rd International Films Workshop Antwerp, Belgium, September 14_ 16- 1998. 6. Jiang.H, Generalized Fuzzy Clustering Model using Fuzzy C-Means. 7. Lim Y.W, Lee S.U.K, “ON THE COLOR IMAGE SEGMENTATION ALGORITHM BASED ON THE THRESHOLDING AND THE FUZZY C-MEANS TECHNIQUE”, Pattern Recognition, Vol.23,no.9,pp.935-952,1990. 8. Xie X.L, A validity measure for Fuzzy Clustering.