International Journal of Engineering Trends and Technology (IJETT) – Volume 8 Number 4- Feb 2014
Bonding By Space Method for Effective
Distance Calculation
Mr.V.Harikrishnan#1, Ms.D.Dhanabakyam#2
1
Assistant Professor, Department of BCA, K.S.Rangasamy College of Arts & Science, Tiruchengode,
Namakkal DT, Tamilnadu, India-637215.
2
Assistant Professor, Department of Computer Applications, Sri Jayendra Saraswathy Maha Vidhyalaya
College of Arts & Science, Coimbatore, Tamilnadu, India.
Abstract
The Bonding by Space (BBS) model is
used to estimate distance between transactions
with local consistency and global connectivity
information. The ant colony optimization (ACO)
techniques are used for the data clustering
process. Visited path management behavior of
ants is used in the ant colony optimization
schemes. The ant colony-clustering algorithm is
integrated with the Bonding by Space based
distance measure. The fuzzy logic techniques are
used to analyze complex relationships between
the objects.
Data partitioning using machine learning
techniques are performed with the distance
measures. Similarity between the transactions is
estimated using the distance measurement
algorithms such as Euclidian distance measure
and cosine distance measure algorithm.
Transaction assignments for the clusters are
carried out with respect to the distance measures.
Keywords
Fuzzy logic, aco, distance, partitioning, cluster,
weights
1. INTRODUCTION
Cluster analysis is an important research
branch of data mining. The process of grouping a
set of physical or abstract objects into classes of
similar objects is called clustering. A cluster is a
collection of data objects that are similar to one
another within the same cluster and are dissimilar
to the objects in other clusters [1].
Ants and other gregarious insects are
with decentralized, self-organization, pheromones
communication, cooperation and other characters.
Ant colony algorithm imitates such intelligent
behavior and applys it to the solution of hard
computational problems. The inspiration of ant
colony cluster comes of the accumulation of ant
bodies and classification of ant larvae. The
earliest work in this area was initiated by
Deneubourg et al [2]. According to the similarity
between data object and its surrounding objects,
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The Euclidian distance measure and
cosine distance measure algorithms consider the
local consistency factor only. Theses distance
measures did not consider the global connectivity
information.
The distance with connection distance
measure model is enhanced with fuzzy logic. The
transaction
weights
are
updated
using
fuzzification process. All the attribute weight
values are updated with a fuzzy set weight value.
The Bonding by Space model is tuned to estimate
distance between the transactions using the fuzzy
set values. The distance measure model efficiently
handles the uneven transaction distributions. The
ant colony-clustering algorithm is also improved
with fuzzy logic. The similarity computations are
carried out with fuzzy distance measurement
models. Un-even data distribution handling,
accurate distance measure and cluster accuracy
are the features of the proposed clustering
algorithm.
the algorithm decides ants randomly moving,
picking up or dropping data objects in order to
achieve the purpose of clustering data. This
algorithm is actually a density-based clustering,
which is difficult to solve the data sets with
uneven density distribution. Many researchers put
forward some improvement or new ant colony
clustering algorithm (ACCA) [3], [4], [5], [6].
But, the core idea of such algorithms is based on
the comparison of the Euclidean distances
between the objects, or objects and cluster’s
centroid. Each object is distributed to a cluster
based on the cluster center to which it is the
nearest. These pure Euclidean distance-based ant
colony clustering algorithms are suitable for
discovering spherical-shaped clusters and
encounter difficulty at discovering clusters of
arbitrary shapes. However, facing the real
problems, the data sets have a wide variety of
complex structures, which requires ant colony
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International Journal of Engineering Trends and Technology (IJETT) – Volume 8 Number 4- Feb 2014
clustering algorithm, which has the capability of
finding arbitrary shape clusters.
This paper proposes an improved
formula for calculating the distance. The basic
idea is that based on the traditional Euclidean
distance, we introduce a measure of connectivity
between the data. Such improved distance
between data objects, as criteria of data clustering,
can reflect not only the local consistency which
refers that data points close in location will have a
high affinity, but also the global connectivity
which refers that data points locating in the same
manifold structure will have a high affinity [10].
The method proposed in this paper is better able
to describe the inherent clustering characters of
the data sets and suitable for data sets with uneven
density distribution, comparing with the densitybased methods. The experiments demonstrate that
the ant colony-clustering algorithm based on the
improved distance calculation formula can
discover clusters of arbitrary shapes.
2. FUZZY LOGIC CONCEPTS
This section gives a view on fuzzy logic
and fuzzy sets.
2.1. Fuzzy Logic
Fuzzy logic starts with and builds on a
set of user supplied human language rules. The
fuzzy systems convert these rules to their
mathematical equivalents. This simplifies the job
of the system designer and the computer, and
results in much more accurate representation of
the way systems behave in the real world.
Additional benefits of fuzzy logic include its
simplicity and its flexibility. Fuzzy logic can
handle problems with imprecise and incomplete
data, and it can model nonlinear functions of
arbitrary complexity. It not a good plant model, or
if the system is changing, the fuzzy will produce a
better solution than conventional control
techniques, “ says Bob Varely, a Senior Systems
Engineer at Harris Corp., an aerospace company
in Palm Bay, Florida.
A fuzzy system can create to match any
set of input data. The Fuzzy Logic Toolbox makes
this particularly easy by supplying adaptive
techniques such as adaptive techniques such as
adaptive neuro-fuzzy inference systems (ANFIS)
and fuzzy subtractive clustering. Fuzzy logic
models, called fuzzy inference systems, consist of
a number of confidential “if then” rules. For the
designer who understands the systems, these rules
are easy to write, and as many rules as necessary
can be supplied to describe the system adequately.
In fuzzy logic, unlike standard
conditional logic, the truth of any statement is a
matter of degree. The inference rule is the form of
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p - > q. Fuzzy logic is possible to say (.5*p) - >
(.5*q). For example the rule if then, both
variables, cold and on, amp to ranges of values.
Fuzzy inference systems rely on membership
functions to explain to the computer how to
calculate the correct value between 0 and 1. The
degree to which any fuzzy statement is true is
denoted by a value between 0 and 1. Not only do
the rule-based approach and flexible membership
function
scheme
make
fuzzy
systems
straightforward to create, but they also simplify
the design of systems and ensure that it can easily
update and maintain the system over time.
2.2. Fuzzy Set
Fuzzy Set Theory formalized by
Professor Lofti Zadeh at the University of
California in 1965. What Zadeh proposed is very
much a paradigm shift that first gained acceptance
in the Far East and its successful application has
ensured its adoption around the world. A
paradigm is a set of rules and a regulation, which
defines boundaries and tells what to do be
successful application, has ensured its adoption
around the world. Bivalent Set Theory can be
somewhat it describes a humanistic problem
mathematically.
The most obvious limiting features of
bivalent sets that can be seen clearly from the
diagram is that they are mutually exclusive it is
not possible to have membership of more than one
set. Clearly it is not accurate to define a transition
from a quantity such as warm to hot by the
application of one degree Fahrenheit of heat. In
the real world a smooth unnoticeable drift from
warm to hot would occur. This natural
phenomenon can describe more accurately by
Fuzzy Set Theory.
3.
DATA
CONNECTIVITY
BASED
DISTANCE ESTIMATION
As shown in Fig.1 (a), data object a and
object b belong to the same cluster, object c and
object a, b belong to different clusters. The
definition of clustering requests that data objects
that are similar to one another within the same
cluster and are dissimilar to the objects in other
clusters. Then we would expect the similarity
between object a and object b is higher than
object a and object c. According to the Euclidean
distance, d(a, c) is less than d(a ,b ), which
determines that the similarity between object a
and object c is higher than the similarity between
object a and object b. Thus Euclidean distance
does not reflect the global consistency of data
sets, and can not find clusters of arbitrary shape.
As shown in Fig.1 (c) and Fig.1 (d), the
number of data points between object a and object
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b of (c) is less than that of (d). It is clear that
object a and object b of (d) are more likely to
belong to the same cluster, comparing with object
a and object b of (c) (The Euclidean distance
between object a and object b in (c) and (d) is
equal). Suppose we link each data point of (c) and
(d) to its nearest three data points (In the figure,
does not draw all the connections), you will
discover that the number of reachability paths
between object a and object b in (c) is less than
that in (d). From this idea, we can use the number
of reachability paths to measure the connectivity
between data points, and further measure the
similarity between data points.
Figure 1.(a) An illustration of that the Euclidian
distance metric can not reflect the global
consistency; (b) An illustration of that the weight
of n+1- steps reachability paths between data
points is less than that of n-steps reachability
paths; (c) and (d).An illustration of the number of
reachability paths can measure the connectivity
between data points
Based on the same way, in Fig.1 (b), we
can get the number of reachability paths between
object a and object d is much larger than that of
object a and object b. If we simply use the number
of reachability paths to measure the similarity
between data points, the similarity of object a and
object d is higher than that of object a and object
b, which is obviously unreasonable. So, data
processing is necessary, in order to make that the
weight of n+1-steps reachability paths between
data points is less than that of n-steps reachability
paths.
An improved distance calculation
method proposed based on the above data
characteristics can reflect the local compactness
of the data, simultaneously reflect the global
connectivity between the data. We put forth a
novel formula for distance calculation, named
BBS (distance with connection).
Establish the adjacency matrix of the
data
set.
First,
calculate
the
set
i  {ij   , j  1,...L} of the data object
xi (i  1,...N ) L (L> 0) nearest-neighbor [8],
using the Euclidean distance formula. Then, link
xi and (i= 1,…,N) L (L> 0) , and the link is
undirected. In this way, we construct an
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undirected graph G = (X, V) and the adjacency
matrix R  [ R y ]N  N of the data set, where:
R ij 

1 , x j   i orx i   i
0 , otherwise
V is the set of all links between data.
Rs = [Rij s]N×N , Rij s is the number of s-steps
reachability paths between xi and xj .
Definition 1: The connectivity between data
object xi and xj is defined as:
step
Conn( xi , x j )   conn s ( xi , x j )
s 1
Where step(1  step  N ) is a parameter
representing the maximum step.
The higher the connectivity between data
point xi and xj is, the more reachability paths
between xi and xj will have, which reflects the
higher similarity between xi and xj , and the more
close distance of xi and xj . Hence, BBS (xi ,xj) 
1/Conn(xi,xj) . Furthermore, the defintion of BBS
(xi,xj) should reflect the local consistency at the
same time, then we difined the BBS distance of
data objects as follows.
Definition 2: The BBS distance between data
object xi and xj is difined as:
(3)
Where Dis ( x , x ) 
i
j
m
x
iv
 x jv
2
v 1
is the Euclidean distance between xi and xj, m
denotes the number of the data object attributes,
Max is a very large positive constant, M is a
positive constant.
If Dis(xi,xj) is short and Conn(xi,xj) is
high, BBS(xi ,xj) will be small. Then, the data
object xi and xj will be clustered into the same
cluster with a high probability. If Dis(xi ,xj) is
short, but Conn(xi,xj) is low, BBS(xi,xj) will still be
large. Then, object xi and xj will not be grouped in
the same cluster. So, formula (3) can effectively
avoid data objects with little connectivity or large
distance clustering into the same category.
Simultaneously, due to the definition of
connectivity is not limited to the neighborhood
within a radius r of a given object, but the L
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International Journal of Engineering Trends and Technology (IJETT) – Volume 8 Number 4- Feb 2014
nearest data points is directly accessible, so, in a
dense region the definition of neighborhood is
relatively narrow; in a sparse region, its definition
is relatively wide. Comparing with density based
methods, such as DBSCAN, BBS can produce
more natural clustering results, and is suitable for
data sets with uneven density distributed.
Obviously, BBS satisfies the following basic
properties:
 BBS(x,x) ≥ 0 , if and only if x = y,
equality holds;
 BBS(x, x) = BBS(x, x).
BBS does not always satisfy the triangle
inequality, so the definition of BBS is a
generalized distance.
4. ANT COLONY OPTIMIZATION
The ant colony optimization algorithm
(ACO), is a probabilistic technique for solving
computational problems which can be reduced to
finding good paths through graphs.
This algorithm is a member of ant
colony algorithms family, in swarm intelligence
methods, and it constitutes some metaheuristic
optimizations. Initially proposed by Marco Dorigo
in 1992 in his PhD thesis , the first algorithm was
aiming to search for an optimal path in a graph;
based on the behavior of ants seeking a path
between their colony and a source of food. The
original idea has since diversified to solve a wider
class of Numerical problems, and as a result,
several problems have emerged, drawing on
various aspects of the behavior of ants.
5.
ANT
COLONY
CLUSTERING
ALGORITHM BASED ON BBS
This article improves the ant colonyclustering algorithm proposed in the literature [3].
Given {x1,x2,….xN} a data set of N objects, and
K(0<K<N), the number of clusters to form,
clustering analysis organizes the N objects into K
clusters, in order to minimize the clustering
objective function F, where each object
x1(i=1,….N) has m attributes, expressed as
{xi1,xi2,….xim}.
The objective function is computed as follows:
Min F(w,C)=
(4)
Subject to:
K
w
ij
 1, i  1,2,...., N -
(5)
j 1
N
w
ij
 1, j  1,2,...., K - (6)
i 1
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Here, w is an N-by-K weighting matrix, its
elements:
w ij 

1, ifx i  cluster
0 , ifx i  cluster
j
j
(7)
Cj denotes the centroid of clusterj
(j=1,….K). Instead of taking the mean value of
the objects in a cluster as a reference point, we
pick actual objects to represent the clusters, using
one representative object per cluster. Here, each
representative object is the medoid, or most
centrally located object, of its cluster, in order to
diminish sensitivity of the algorithm to outliers.
In the algorithm, we use R(R≥5) ants to
build solutions sq={cq1, cq2,----., cqN}(q=1,….,R),
string of length N, where cqi(i=1,2,….,N) is the
class identifier of data object xi and cqj 
{cluster1, .. clusterk}. cqi= cqj means that the object
xi and xj belong to the same cluster in the solution
builded by the qth ant. On the contrary, cqi ≠ cqj
denotes that the object xi and xj belong to different
clusters [3]. For example, given N=5,
{x1,x2,….x5}, and K=3, suppose a feasible solution
s1{cluster1,cluster3,cluster2,cluster2,,cluster3},
which means xi is assigned to cluster1 , x2 is
assigned to cluster3 , and the rest may be deduced
by analogy.
At the beginning of the algorithm,
initialize the N-by-N BBS distance matrix
according to formula (3), where we recommend to
set parameters, L and step, within the range 3% N
≤ L 4% N ≤ and N /(K  L) ≤ step≤ N..
Subsequently, the calculation, involving BBS
distance, can be efficiently completed in linear
time. Here, need to note that if A and B both are
N-by-N matrix, the computational complexity of
A×B is O(nlog7) n ο , using of Strassen matrix
multiplication. So the computational complexity
of initializing BBS distance matrix is between
(nlog7) and O (n2). Then, initialize the N-by- K
pheromone matrix [τij ]N × N and each element with
an initial pheromone value τ0 , where the matrix
elements τij denotes the concentration of
pheromone of object xi relative to clusterj . In each
loop, each artificial ant constructs a pheromonebased solution, thereafter updates the pheromone
matrix based on the quality of the solutions
achieved. Under the guidance of pheromone
matrix, ants improve the quality of solution step
by step until a stopping criterion.
5.1. Solution construction
In ant colony algorithm, the ants
construct solution (S) by means of formula (8).
Ant, located at object x1(i=1,….N), selects
clusterj (j=1,….K) in probability Pij .
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International Journal of Engineering Trends and Technology (IJETT) – Volume 8 Number 4- Feb 2014
 ij  ij   pathif 

Pij 
TABLE I. AVERAGE F-MEASURE OF
CLUSTERING [9]

, j  1,...., K
K


    path 
ik
ik
ik
k 1
(8)
Where Pij is the probability distribution
of object xi belonging to
clusterj.nij= 1/BBS(xi,Cj)
represents heuristic information value
BBS(xi,Cj) is the BBS distance between object xi
and the center of clusterj).  is the heuristic
factor, indicating the relative importance of
heuristic information. ij path denotes the number
of excellent ants , which construct good solutions
and group xi into the clusterj . If pathij is very
large, we can speculate that building good
solution must group xi into the clusterj. Formula
(8) reflects that if pathij is very large, then xi is
grouped into the clusterj with a high probability.
In this way, we can build a good solution rapidly,
that is, the ant-colony clustering algorithm can
fast convergence.
5.2. Pheromone Update Rule
After each loop of the algorithm, i.e.,
when R(R≥5) ants have completed a solution, we
sort the solutions according to the clustering
object function value in ascending order. Then,
we
get
'
q
'
q1

'
'
'

S_sorted= s1 , s2 ,...., sR ,
'
q2
Data
ant
ant
ant Combinatio Algorithm
set colony 1 colony 2 colony 3 n of the 3 suggested
(SACA) (SACA) (SACA)
ant
in this
colonies
paper
(MACCA)
Iris
0.918
0.910
0.915
0.927
0.925
Comparing with Fig.2 (a) and Fig.2 (c),
which are the simulation results on Line, Circlesquare synthetic data sets obtained from ant
colony clustering algorithm(ACCA) in the
literature [3], Fig.2 (b) and Fig.2 (d), the
simulation results on Line, Circle-square synthetic
and the center ( Cj ) of clusterj in solution sq′ .
Selecting 20% better solutions as the basis for
pheromone updating, not only records relatively
better solutions through updating pheromone trail,
but also expands the algorithm's search space,
avoiding stagnation deriving from an excessive
trail level on the moves of one solution.
where
'
qN
s  {c , c ,....c } , (q=1,….R)of which use
the
top
20%
better
solutions
(S_best=
to
s  S _ sorted ,1  q  20% R  Z })
"
q
update the pheromone matrix.
updating formula is as follows:
Pheromone
 ij (t  1)  (1   ) ij (t )    ij (t )
∆
( ) = 0,
( ,
ℎ
,
=
"
(9)
path
ij


path
path
'
ij  1 , is q
ij
 S _ best
, otherwise
'
 c qi
 cluster
j
Where _ ρ,0≤ρ≤1, is a user-defined parameter
called evaporation coefficient, Q is a positive
constant. BBS(x i C sq' j ) represents the BBS
distance between xi.
Based on the data in the table, we can see
that this algorithm is better than the average
performance of SACA, somewhat less effective
than MACCA.
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Figure 2. Clusters obtained by the algorithms in
literature [3] and this paper for synthetic data sets
6.
FUZZY
ENABLED
CLUSTERING
SCHEME
The data-clustering scheme is designed
with dynamic distance measures. Ant colony
clustering algorithm is integrated with the
dynamic distance measure. The distance with
connection measure is used to maintain the local
consistency and global connectivity factors. The
distance with connection measure model is
enhanced using the fuzzy logic technique. The
fuzzy enhancement is done in two areas. Distance
estimation function is enhanced with fuzzy
models to handle uneven data distributions. The
ant colony-clustering algorithm is enhanced with
fuzzy relationship analysis model.
The fuzzy logic based ant colony
clustering algorithm is designed with dynamic
distance measure and fuzzy enabled ant colony
clustering models. Fuzzification, fuzzy based
distance estimation, clustering process and cluster
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International Journal of Engineering Trends and Technology (IJETT) – Volume 8 Number 4- Feb 2014
analysis phases. The fuzzification phase is used to
convert the attribute weight for each transaction
into fuzzy sets. The distance and global
connectivity are analyzed using the fuzzy weight
values. The clustering process is done with the
dynamic distance based ant colony clustering
algorithm and fuzzy enabled ant colony clustering
algorithm. Cluster accuracy is analyzed in the
cluster analysis.
The attribute weight values for each
transaction are passed into the fuzzification
process. All the attribute weight values are
converted into fuzzy weight values. The fuzzy
weight conversion process updates the weight
values with in a range of 0 to 1. The weight value
distribution is not even in some data sets. The
fuzzification process removes the overhead to
calculate distance in uncertain data distributions.
The distance measure estimates the distance with
global connectivity factors. Fuzzy sets are applied
to the distance estimation process. The dynamic
distance based ant colony clustering algorithm is
enhanced with fuzzy comparison for dynamic
similarity analysis. All the transaction analysis is
carried out with fuzzy enabled weight values. The
cluster results are updated using the actual
attribute weights. The comparison process is
performed with fuzzy weights. The cluster
analysis is done with a set of parameters. The
precision/recall and fitness measures are used in
the cluster analysis process.
The distance with connection distance
measure model is enhanced with fuzzy logic. The
transaction
weights
are
updated
using
fuzzification process.
The distance measure
model efficiently handles the uneven transaction
distributions. The ant colony-clustering algorithm
is also improved with fuzzy logic. The similarity
computations are carried out with fuzzy distance
measurement models. The system enhances the
distance estimation process. Fuzzy logic
techniques are used to improve the distance
estimation process. Global relationship is used in
the system. Ant colony clustering is improved
with fuzzy scheme. The system divided into four
modules
 Distance Analysis
 Fuzzification Process
 Ant Colony Clustering
 Fuzzy Ant Colony Clustering
Distance analysis module is designed to
measure local global distance. Fuzzification
module is designed to estimate fuzzy weights for
transactions. Ant colony clustering module is
designed with distance with connection model.
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Fuzzy weights are used in fuzzy based ant colony
clustering module
6.1 Distance Analysis
Distance analysis is performed to
estimate transaction relevancy. Local and global
distance estimation schemes are used in the
system. Local distance is estimated with the
current transaction information only. Global
distance estimation uses the transaction details
and support information
6.2. Fuzzification Process
Fuzzy model is used to assign weights in
a range between 0 to 1. Transaction weights are
converted into fuzzy based weights. Fuzzy weight
is used for the distance estimation. Support value
is also calculated using fuzzy weights
6.3. Ant Colony Clustering
Ant colony optimization is used for the
clustering process. Transaction weights are used
in the clustering process. Transaction comparison
is done with the ant behavior. One pass analysis is
used in the system.
6.4. Fuzzy Ant Colony Clustering
Clustering process is performed using
fuzzy weights. Fuzzy weights based distance is
used for the relevancy estimation. Global distance
is used for the clustering process. Fuzzy relations
are integrated with the ant colony clustering
model
7. CONCLUSION
Based on the Euclidean distance between
objects, the system uses data connectivity and an
improved formula for calculating the distance
named BBS. BBS reflects not only the local
consistency but also the global connectivity
between objects. It also overcomes the
disadvantage of Euclidean distance in data
clustering. Then, we improve the ant colonyclustering algorithm by using BBS and fuzzy
logic concepts. Our experimental results on both
synthetic and real world data sets show that the
improved algorithm can discover clusters with
arbitrary shape and is better than the clustering
effect of earlier techniques.
8. REFERENCES
[1] Jiawei Han and Micheline Kamber.Data
Mining Concepts and Techniques, San Francisco:
Morgan kaufmann, 2006, pp.383.
[2] Deneubourg JL, Goss S, et a1. “The dynamics
of collective sorting: robot-like ant and ant-like
robot,”. In: M eyer JA, Wilson SW ed.
Proceedings first conference on simulation of
adaptive behavior: from animals to animats.
Cambridge, MA: MIT Press, 1991, pp.356–365.
http://www.ijettjournal.org
Page 196
International Journal of Engineering Trends and Technology (IJETT) – Volume 8 Number 4- Feb 2014
[3] Shiyong Li, Baojiang Zhao. “Ant Colony
Clustering Algorithm,” easurement & Control,
vol.
11,
NO.15,
2007,
pp.159–1592,
2007.15(11):1590–1592.
[4] Parag M. Kanade and Lawrence O. Hall,
“Fuzzy Ants and Clustering, IEEE Transactions
on Systems,” man, and Cybernetics-part a:
systems and humans, vol. 37, no. 5, September
2007, pp. 758–769.
[5] Xinbin Yang, Jinggao Sun and Dao Huang, “A
New Clustering Method Based on Ant Colony
Algorithm,” Proceeding of the 4th World Congress
on Intelligent Control and Automation June 10–
14, 2002, pp.2222-2226.
[6] Jian Gao. “Cluster Analysis Based on Parallel
Ant Colony Adaptive Algorithm,” Computer
Engineering and Application, vol. 25, 2003,
pp.78–79, 2003.25:78–79.
[7] Maoguo Gong and Liefeng Bo. “DensitySensitive Evolutionary Clustering,” The 11th
Pacific- Asia Conference on Knowledge
Discovery and Data Mining, Springer-Verlag
Berlin Heidelberg ,2007, pp.507–514.
[8] Julia Handl and Joshua Knowles. “An
Evolutionary Approach to Multiobjective
Clustering,” IEEE Transactions on Evolutionary
Computation, vol. 11, no. 1, Feb.2007, pp.60.
[9] Yan YANG and Fan Jin, “Mohamed
Kamel.Clustering Combination Based on Ant
Colony Algorithm,” Journal of the China Railway
Socity, vol. 4, No. 26, 2004, pp.6–69.
[10] Miguel A.Sanz-Bobi and Mario Castro
“IDSAI: A Distributed System for Intrusion
Detection Based on Intelligent Agent” 5th
International Conference on Internet Monitoring
and Protection, IEEE, 2010.
ISSN: 2231-5381
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