Research Article Fuzzy Rules for Ant Based Clustering ...

Research ArticleFuzzy Rules for Ant Based Clustering Algorithm

Amira Hamdi12 Nicolas Monmarcheacute2 Mohamed Slimane2 and Adel M Alimi1

1REGIM-Lab Research Groups in Intelligent Machines University of Sfax ENIS BP 1173 3038 Sfax Tunisia2Polytech Tours University of Tours Tours France

Correspondence should be addressed to Amira Hamdi amirahamdi7ieeeorg

Received 8 July 2016 Accepted 8 September 2016

Academic Editor Katsuhiro Honda

Copyright copy 2016 Amira Hamdi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper provides a new intelligent technique for semisupervised data clustering problem that combines the Ant System (AS)algorithmwith the fuzzy 119888-means (FCM) clustering algorithm Our proposed approach called F-ASClass algorithm is a distributedalgorithm inspired by foraging behavior observed in ant colonyTThe ability of ants to find the shortest path forms the basis of ourproposed approach In the first step several colonies of cooperating entities called artificial ants are used to find shortest pathsin a complete graph that we called graph-data The number of colonies used in F-ASClass is equal to the number of clusters indataset Hence the partition matrix of dataset founded by artificial ants is given in the second step to the fuzzy 119888-means techniquein order to assign unclassified objects generated in the first stepThe proposed approach is tested on artificial and real datasets andits performance is compared with those of119870-means119870-medoid and FCM algorithms Experimental section shows that F-ASClassperforms better according to the error rate classification accuracy and separation index

1 Introduction

How do ants optimize food search How do social spidersbuild communal nest Why does a flock of birds fly in a v-shaped formation How do termites build collectively theirsophisticated nest structure How do honey bee swarmscooperatively select their new nesting site How does fireflyflash its light in a wonderful pattern How does a colonycoordinate its behavior How is it possible for social insectsand animals to coordinate their actions and create complexpatternsHowdo such agents perform complex tasks withoutany direction and coordination between themselves Howagents in colony perform a work locally for global goalwith sufficient flexibility as they are not controlled centrallyCollective behaviors in swarms of insects or animals haveattached the attention of researches They have proposedseveral intelligent models to solve a wide range of complexproblems This branch of artificial intelligence is addressedas swarm intelligence The key components of swarm intelli-gence are self-organization emergence and stigmergy

Self-organization is ldquoa processwhereby pattern at the globallevel of a system emerges solely from numerous interactions

among the lower-level components of the system Moreoverthe rules specifying interactions among the systemrsquos com-ponents are executed using only local information withoutreference to the global patternrdquo [1] In short it can be ldquoa setof dynamical mechanisms whereby structures appear at theglobal level of a system from interactions of its lower-levelcomponentsrdquo [2]

Emergence seems to be the explication of what self-organizing systems produce In this context the whole is notjust the sum of its parts it gets a surplus meaning that it is notcaptured by its part alone The idea of emergence was firstlydeveloped in [3] to explain indirect task coordination in thecontext of building behavior of termites Grasse [3] showedthat the coordination of building activities does not dependon the workers themselves but is mainly achieved by the neststructure

The underlying idea of this paper is to propose a newapproach to data clustering problem We will show thatthe use of fuzzy logic combined with swarm intelligencetechnique yields robust results

The remainder of the paper is organized as follows InSection 2 we present an overview of data clustering problem

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016 Article ID 8198915 16 pageshttpdxdoiorg10115520168198915

2 Advances in Fuzzy Systems

and swarm intelligence tools for data clustering A generaldescription of our proposed approach called F-ASClass isgiven in Section 3 Experimental results and comparisonwithhard and fuzzy algorithms are reported in Section 4 Finallyconcluding remarks are drawn in Section 5

2 Literature Review

21 Problem Definition Cluster analysis is a technique thatorganizes data by abstracting underlying structure either asa grouping of objects Each group consists of objects that aresimilar between themselves and dissimilar to objects of othergroups

Each object corresponds to a vector of 119872 numericalvalues which correspond to the119872 numerical attributes Therelationships between objects are generated into a dissim-ilarity matrix in which rows and columns correspond toobjects As objects can be represented by points in numericalspace the dissimilarity between two objects can be defined asdistance between the two corresponding points Any distancecan be used as dissimilarity measures The most commonlyused dissimilarity matrix is theMinkowskimetric

119889119903(119874119894 119874119895) =

119898

sum

119896=1

(119908119896

10038161003816100381610038161003816119909119894119896 119909119895119896

10038161003816100381610038161003816

119903

)

1119903

(1)

where 119908119896is a weighting factor that will be set to 1 thereafter

According to the value of 119903 (119903 ge 1) the following measuresare obtained Manhattan distance (119903 = 1) Euclidean distance(119903 = 2) and Chebyshev distance (119903 = infin) As mentioned in[4] Euclidean distance is themost common of theMinkowskimetric

22 Clustering Algorithms The grouping step can be per-formed in a number of ways In [5] different approaches toclustering data are described

(i) PartitioningHierarchical Classification Partitionalclustering technique identifies the partition thatoptimizes a clustering criterion defined on a subset ofobjects (locally) or over all of the objects (globally)Hierarchical clustering technique builds a sequenceof nested partitions that are visualized by exampleby a dendrogram

(ii) HardFuzzy Classification A hard clustering algo-rithm allocates each object to a single cluster duringits operations Hence the clusters are disjoint A fuzzyclustering algorithm associates each object with everycluster using a membership function The output ofsuch algorithm is a clustering but not a partition

(iii) DeterministicStochastic Optimization in partitionalapproach can be accomplished using traditional tech-nique or through a random search of the state spaceconsisting of all possible labeling

(iv) SupervisedUnsupervised Classification An unsuper-vised classification uses only the dissimilarity matrixNo information on the object class is provided to the

method (objects are said unlabeled) In supervisedclassification objects are labeled while knowing theirdissimilarities The problem is then to constructhyperplanes separating objects according to theirclass The unsupervised classification objective isdifferent from that of the supervised case in the firstcase the goal is to discover groups of objects while inthe second known groups are considered and the goalis to discover what makes them different or to classifynew objects whose class is unknown

Our proposed technique presented in this paper whichwe call F-ASClass belongs to fuzzy-semisupervised parti-tional clustering technique It uses fuzzy rules and stochas-tic behavior to partition dataset into specified number ofclusters For the present paper it suffices to note that thefollowing techniques (119870-means 119870-medoid and FCM) areused to improve F-ASClass algorithm A comparative studybetween us will be presented in Section 4

119870-means algorithm is a hard-unsupervised learningalgorithm that appears to partition dataset into a specifiednumber of clusters The technique presented in [6] consistsof starting with 119896 groups each of which consists of a singlerandomly selected object and thereafter adding each newobject to its closest cluster center After an object is added toa group the mean of that group is adjusted in order to takeaccount of the new added object The algorithm is deemed tohave converged when the assignments no longer change

The 119870-medoid algorithm described in [7] is based uponthe search of representative objects of each cluster (calledmedoid) which should represent the various aspects of thestructure of the data 119870-medoid algorithm is related to the119870-means algorithm The main difference between 119870-meansand 119870-medoid stands in calculating the cluster center Themedoid is a statistic which represents that data member of adataset whose average dissimilarity to all the other membersof the set is minimal Therefore a medoid unlike meanis always a member of the dataset It represents the mostcentrally located data item of the dataset

The fuzzy 119888-means (FCM) algorithm firstly presented in[8] and improved in [9 10] allows one samples to belongto two or more one clusters The first idea which aims tocharacterize an individual objectsrsquo similarity to all the clusterswas introduced in [11] In this context the similarity an objectshares with each cluster is represented with a membershipfunction whose values are between zero and one Object indatasetwill have amembership in every clustermembershipsclose to unity indicate a high degree of similarity between theobject and a cluster while memberships close to zero involvelittle similarity between the object and that cluster

FCMmethod differs from previously presented119870-meansand 119870-medoid algorithms by the fact that the centroid of acluster is the mean of all samples in the dataset weightedby their degree of belonging to the cluster The degree ofbelonging is presented by a function of the distance ofthe sample from the centroid which includes a parameter

Advances in Fuzzy Systems 3

controlling for the highest weight given to the closest sampleAll these techniques are sensitive to initial condition

Another fuzzy classification model is studied in [12]which constructs the membership function on the basis ofavailable statistical data by using an extension of the well-known contamination neighborhood Reference [13] presentsa new fuzzy technique using an adaptive network of fuzzylogic connectives to combine class boundaries generated bysets of discriminant functions in order to address the ldquocurseof dimensionalityrdquo in data analysis and pattern recognition

Reference [14] is intended to solve the problem ofdependence of clustering results on the use of simple andpredetermined geometrical models for clusters In this con-text the proposed algorithm computes a suited convex hullrepresenting the cluster It determines suitable membershipfunctions and hence represents fuzzy clusters based on theadopted geometrical model that it is used during the fuzzydata partitioning within an online sequential procedure inorder to calculate the membership function

23 Swarm Intelligence Tools for Data Clustering ProblemWe start with an illustration of swarm intelligence tools thathave been developed to solve clustering problems ParticleSwarm Optimization [15] Artificial Bee Colony [16] Fireflyalgorithm [17] Fish swarm algorithm [18] and Ant ColonyAlgorithm In [19] the basic data mining terminologies arelinked with some of the works using swarm intelligencetechniques A comprehensive review of the state-of-the-artant based clustering methods can be found in [20]

The first model of antsrsquo sorting behavior has been doneby Deneubourg et al [21] where a population of ants arerandomly moving in a 2-dimensional grid and are allowedto drop or load objects using simple local decision rulesand without any central control The general idea is thatisolated items should be picked up and dropped at someother locations where more items of that type are presentBased on this existing work Lumer and Faieta [22] haveextended it to clustering data problems The idea is todefine dissimilarity between objects in the space of objectattributes Each ant remembers a small number of locationswhere it has successfully picked up an object And so whendeposing a new item this memory is used in order to biasthe direction in which the ant will move ant tends to movetowards the location where it last dropped a similar itemFrom these basic models in [23] Monmarche has proposedan ant based clustering algorithm namely AntClass whichintroduces clustering in a population of artificial ants capableto carry heaps of objects Furthermore this ant algorithm ishybridized with the119870-means algorithm In [24] a number ofmodifications have been introduced on both LF andAntClassalgorithm and authors have proposed AntClust which is anant based clustering algorithm for image segmentation InAntClust a rectangular grid was replaced by a discrete arrayof cells Each pixel is placed in a cell and all cells of the arrayare connected to the nest of antsrsquo colony Each ant performs anumber of moves between its nest and the array and decides

with a probabilistic rule whether or not to drop its pixel Ifthe ant becomes free it searches for a new pixel to pick up[24]

According to [25] another important real antrsquos collectivebehavior namely the chemical recognition system of antswas used to resolve an unsupervised clustering problem In[26] Azzag et al considers another biologically observedbehavior in which ants are able to build mechanical structurethanks to a self-assembling behavior This can be observedthrough the formation of drops constituted of ants or thebuilding of chains by ants with their bodies in order to linkleaves togetherThemain idea here is to consider that each antrepresents a data and is initially placed on a fixed point calledthe support which corresponds to the root of the tree Thebehavior of an ant consists of moving on already fixed ants tofix itself to a convenient location in the tree This behavior isdirected by the local structure of the tree and by the similaritybetween data represented by ants When all ants are fixed inthe tree this hierarchical can be interpreted as a partitioningof the data [26]

Bird flocks and schools clearly display structural orderand appear to move as single coherent entity [27] In [28 29]it has been demonstrated that flying animal can be used tosolve data clustering problem The main idea is to considerthat individuals represent data to cluster and that they movefollowing local behavior rule in a way after few movementshomogeneous individual clusters appear and move togetherIn [30] Abraham et al propose a novel fuzzy clusteringalgorithm called MPSO which is based on a deviant varietyof the PSO algorithm

Social phenomena also exists in the case of spiders inthe Anelosimus eximius case individuals live together sharethe same web and cooperate in various activities such ascollective web building spiders are gathered in small clustersunder the vegetal leaves included in the web and distributedon the whole silky structure In [31] the environment modelsthe natural vegetation and is implemented as a square gridin which each position corresponds to a stake Stakes can beof different heights to model the environmental diversity ofthe vegetation Spiders are always situated on top of stakesand behave according to three several independent items (amovement item a silk fixing item and a return to web item)This model was transposed to region detection in image

In [32] Hamdi et al propose a new swarm-based algo-rithm for clustering based on the existing work of [21 23 28]which uses antsrsquo segregation behavior to group similar objectstogether birdsrsquo moving behavior to control next relativepositions for a moving ant and spidersrsquo homing behavior tomanage antsrsquo movements with conflicting situations

In [33] we proposed using the stochastic principles of antcolonies in conjunction with the geometric characteristics ofthe beersquos honeycomb This algorithm was called AntBee andit was improved in [34] In this context we used fractal rulesto improve the convergence of the algorithm

Another example of Ant Clustering algorithm calledAntBee algorithm is developed in [29] The proposedapproach uses the stochastic principles of ant colonies inconjunction with the geometric characteristics of the beersquos


honeycomb and the basic principles of stigmergy Animproved AntBee called FractAntBee was proposed [24] thatincorporated main characteristics of fractal theory

According to [35] a novel approach to image segmen-tation based on Ant Colony System (ACS) is proposed InACS algorithm an artificial ant colony is capable of solvingthe traveling salesman problem [36] As in ACO for the TSPin ACO-based algorithms for clustering each ant tries to finda cost-minimizing path where the nodes of the path are thedata points to be clustered Like in the TSP the cost ofmovingfrom data point 119909

119894to 119909119895is the distance 119889(119894 119895) between these

points measured by some appropriate dissimilarity metricThus the next point to be added to the path tends to be similarto the last point on the path An important way inwhich thesealgorithms deviate from ACO algorithms is that the ants donot necessarily visit all data points [37]

3 Proposed Methodology

In ASClass algorithm we have assumed that a graph 119866 =

1198741 119874

119899 of 119899 object has been collected by the domain

expert where each object is a vector of 119896 numerical valuesV1 V

119896 For measuring the similarity between objects we

will use in the following Euclidean distance between twovectors denoted by119863 which is used for edges in graph119866 [38]The complete set of parameters of ourmodel will be presentedin Section 33

Initially all the objects will be scattered randomly on thegraph119866 each node in the graph represents an object119874 in thedatasetsThe edge that connects two objects in the graph-datarepresents ameasure of dissimilarity between these objects inthe database A class is represented by a route connecting a setof objects In ASClass we chose to use more than one colonyof ants the number of colonies needed here is equal to thenumber of classes in the database Initially for each colony119898 artificial ants are placed on a selected object called ldquonest-objectrdquo For each cluster we randomly chose one and only oneldquonest-objectrdquo The simulation model is detailed in Figure 1

The objective is to find the shortest route between thegiven objects and return to nest-object while keeping inmind that each object can be connected to the path onlyonce The path traced by the ant represents a cluster in thedataset Figure 2 shows a possible result of ASClass algorithmexecution on the graph of Figure 1 It may be noted that at theend of algorithm each colony gives a collective traced-paththat represent a cluster in the partition

For each colony an artificial agent possesses three behav-ioral rules

(i) a movement item inspired by ant foraging behavior(ii) an object fixing item inspired by the collective weav-

ing in social spiders(iii) a return to web item

31 Movement Item An artificial ant 119896 is an agent whichmoves from an object 119894 to an object 119895 on a dataset graph It

Colony of ants

Artificial ant

Nest-object of cluster 1



Figure 1 Graph used in ASClass algorithm [38]

Figure 2 Results of ASClass algorithm on graph used in Figure 1tree clusters are drawn each of them is created by a colony of ants inthe ASClass algorithm


decides which object to reach among the objects it can accessfrom its current positionThe agent selects the object tomoveon according to a probability density 119875

119894119895depending on the

trail accumulated on edges and on the heuristic value whichwas chosen here to be a dissimilarity measures

119875119896

119894119895(119905) =

[120591119894119895]

120572

[120578119894119895]

120573

sum119897isin119873119896

119894

[120591119894119897]120572

[120578119894119897]120573

if 119895 isin 119873119896

119894 (2)

120572 and 120573 are two parameters which determine the relativeinfluence of the pheromone trail and the dissimilarity mea-sures and 119873

119894is the set of objects which ant 119896 has not yet

connected to its tour

32 Object Fixing Item When an ant reaches an object it canfix it on its path according to a contextual probability 119875fix if adecision is made ants draw a new edge between the currentobject and the last fixed object otherwise it returns to weband updates itsmemory (it decides to delete the object amongthe objects it can access from its current position)

The probability to fix the object to the path is defined as

119875fix119896

119894(119905) =

2119891 (119874119894) if 119891 (119874

119894) lt 119896fix

1 if otherwise(3)

where 119896fix is constant119891(119874119894) is measure of the average similarity of the object119874

119894

with the objects119874119895forming the path created by the ant 119896 and

it is calculated as follows

119891 (119874119894) = max

1

119873119879119870

sum

119874119895isin119879119896

1 minus

119889 (119874119894 119874119895)

120579

0

(4)

where 120579 is a scaling factor determining the extent towhich thedissimilarity between two objects is taken into account and119873119879119896

is the number of objects forming the tour constructed bythe ant 119896

At each time step if an ant decides to fix new objects itupdates the pheromone trail on the arcs it has crossed in itstour This is achieved by adding a quantity Δ120591

119894119895to its arc and

it is defined as follows

Δ120591119896

119894119895(119905) =

1

119862119896

if edge (119894 119895) belongs to 119879119896

0 otherwise

120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) forall (119894 119895) isin 119879

119896

(5)

where 119862119896is the length of the tour 119879

119896built by the ant 119896 and 120591

0

is constant

After all ants have constructed their tours the pheromonetrails are lowered This is done by the following rules

120591119894119895= (1 minus 120588) 120591

119894119895 (6)

where 0 lt 120588 lt 1 and it is the pheromone trail evaporationEquation (6) becomes

120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) + (1 minus 120588) 120591

119894119895forall (119894 119895) isin 119879

119896 (7)

33 Return toWeb Item If the decision ismade ant returns tothe last fixed object and selects the objet tomove on accordingto its updated memory

This process is achieved for each colony At the end ofprocess we obtain 119862 routes which represent the 119862 clusters inthe datasets

All notations used in ASClass algorithm are shown asfollows

120591119894119895(119905) the amount of pheromone on edge which

connects city 119894 to city 119895 at time 119905120572 parameter that control the relative importance ofthe trail intensity 120591

119894119895

120573 parameter that control the visibility 120578119894119895

120578119894119895 the visibility

120588 the coefficient of decay119873119879119896 the number of objects forming the tour con-

structed by the ant 119896

119869119896 set of objects which ant 119896 has not yet connected toits tour119875119894119895(119905) transition probability from object 119894 to object 119895

120579 scaling factor119876 constant119871119896 the length of the tour built by the 119896th ant

119896fix constant119898 number of ants in colony

Its general principle is defined as shown in Algorithm 1

34 Improvements of ASClass Algorithm In studying theasymptotic behavior of ASClass algorithm we make theconvenient assumption there are still some objects which arenot assigned to any cluster when theASClass algorithm stopsWe called these objects Outliers This phenomenon is causedby the fact that an object can belong to two or more clusters(the same object can be added by several colonies to theirpaths) Solutions that we propose in this paper are presentedin the following

341 First Solution ASClassi Algorithms In order to findpartition for unclassifiable objects (Outliers) we proposeapplying respectively 119870-means algorithm 119870-medoid algo-rithm and FCM algorithm on dataset of Outliers samples


Initialize randomly the 119899 objects in a 2D environmentChoose randomly the nest-object for each cluster we assume 119874init this objectFor each nest-object determine the cluster119862init larr 119862(119874selected)

End ForFor each cluster determine the number of objects119873119887(119862init)

End ForFor all colony doFor 119879 = 1 to 119879max do119874current larr 119874selectedFor all ants doRepeat119894 larr 119874currentChoose in the list 119869119896

119894(list of objects not visited) an object 119895 according to the below formula

119875119896

119894119895(119905) =

[120591119894119895]

120572

[120578119894119895]

120573

sum119897isin119869119896

119894

[120591119894119897]120572

[120578119894119897]120573

if 119895 isin 119869119896

119894

0 if 119895 notin 119869119896

119894

119874current larr 119895

Until (119873 minus 119869119896

119894) = 119873119887(119862init)

Place a quantity of pheromone on the route according to the following equation

Δ120591119896

119894119895(119905) =

119876

119871119896

(119905) if edge (119894 119895) belongs to 119879119896

0 otherwiseEnd forEvaporate the tracks using the following equation

120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) + (1 minus 120588) 120591

119894119895forall (119894 119895) isin 119879

119896

End forEnd for

Algorithm 1

Table 1 Hybrid ASClass119894 algorithms

Index Algorithm 119860 (second step) Proposed models1 119870-means ASClass1

2 119870-medoid ASClass2

3 FCM ASClass3

Our proposed method ASClass presented in the previoussection can also be used as an initialization step for thesealgorithms As can be seen in Table 1 we called respectivelyASClass1 ASClass2 and ASClass3 our proposed versions ofASClass

The ASClassi procedure consists of simply starting with119896 groups The initialization part is identical to ASClassWe are able to show in Figure 3 that the output of firststep consists of classified and unclassified objects Classifiedsamples are represented by different symbol (red green andblue) Unclassified ones are represented by black point

As can be seen in Figure 4 unclassified objects resultingat the first step (black point) are given as input parameter toASClassi in the second step

Cluster 3

Outliers

Cluster 1

Cluster 2

ASClass algorithm

Figure 3 Step 1 of both ASClassi and F-ASClass algorithms

Cluster 3

Outliers

Cluster 1

Cluster 2

Algorithm A Fusion process

Figure 4 Step 2 of ASClassi algorithms


Cluster 1

Cluster 2Cluster 3

FCM algorithm

Figure 5 Step 2 of F-ASClass algorithm

Algorithm119860 (different values of 119860 are given in Table 1) isapplied on dataset of unclassified objects to reassigning themto the appropriate cluster 119860 algorithm create new clusterswith the same specified number 119870 These clusters will bemerged with existing clusters created at the first step

342 Second Solution F-ASClass Algorithm The result of thepartition founded by ASClass in the first step is given as inputparameter to FCM algorithm in the second step An element119906119894119895of partition matrix represents the grade of membership of

object 119909119894in cluster 119888

119895 Here 119906

119894119895is a value that described the

membership of object 119894 to classWe will initialize the partition matrix (membership

matrix) given to FCM function by 1 according to classifiedobject and 119891119911 according to unclassified objects

For the classified objects if object 119894 is in class 119895 119906119894119895= 1

otherwise 119906119894119895= 0 For the unclassified objects 119906

119894119895= 119891119911 for

all class 119895 (119895 = 1 2 3 119870) meaning that an unclassifiedobject 119894 has the same membership to all class 119895 The sum ofmembership values across classes must equal one

119891119911 =

1

119870

(8)

Figure 5 shows that the output of the first step (see Figure 3)will be given as input parameter in the second step

4 Experimental Results

41 Artificial and Real Data To evaluate the contributionof our method we use several numerical datasets includ-ing artificial and real databases from the Machine Learn-ing Repository [39] Concerning the artificial datasets thedatabases art1 art2 art3 art5 and art6 are generated withGaussian laws and with various difficulties (classes overlaynonrelevant attributes etc) and art4 data is generated withuniform law The general information about the databases issummarized in Table 2 For each data file the following fieldsare given the number of objects (119873) the number of attributes(119873Att) and the number of clusters expected to be found in thedatasets (119873

119862)

Art1 dataset is the type of data most frequently usedwithin previous work on ant based clustering algorithm [2223]

Table 2 Main characteristics of artificial and real databases used inour tests

Datasets 119873Att 119873 119870 Class distributionArt1[4 400] 2 400 4 (100 100 100 100)

Art2[2 1000] 2 1000 2 (500 500)

Art3[4 400] 2 1100 4 (500 50 500 50)

Art4[2 200] 2 200 2 (100 100)

Art5[9 900] 2 900 9 (100 100 100)

Art6[4 400] 8 400 4 (100 100 100 100)

Iris[3 150] 4 150 3 (50 50 50)

Thyroid[3 215] 5 215 3 (150 35 30)

Pima[2 768] 8 768 2 (500 268)

The data are normalized in [0 1] the measure of sim-ilarity is based on Euclidian distance and the algorithmsparameters used in our tests were always the same for alldatabases and all algorithms

As can be seen in Table 2 Fisherrsquos Iris dataset contains 3classes of 50 instances where each class refers to a type of Irisplant An example of class discovery ofASClass on Iris datasetis shown in Figure 7

Five versions of the ASClass algorithm were coded andtested with test data in order to determine accuracy or resultsOne implementation presented basic features of ASClasswithout improvements ASClass1 ASClass2 and ASClass3are improvements versions of the basic ASClass in which wepropose applying respectively the first solution described inSection 341 F-ASClass is a fuzzy-ant clustering solutionsdescribed in Section 342

42 Evaluation Functions The quality of the clusteringresults of the different algorithms on the test sets are com-pared using the following performances measures The firstperformance measure is the error classification (Ec) index Itis defined as

Ec = 1 minus

sum119870

119896=1Recall

119896

119870

Recall119896=

119899119896119896

119899119896

(9)

The second measure used in this paper is the accuracycoefficients It determines the ratio of correctly assignedobjects

Accuracy =sum119870

119896=1119899119896119896

119873

(10)

where 119899119896119896

is the number of objects of the 119896th class correctlyclassified 119899

119896is the number of objects in cluster 119862

119896 and119873 is

the total number of objects in dataset Not ass represents thepercentage of not assigned objects in the predicted partition


Table 3 Error classification and accuracy coefficients found on Irisdataset Average on 20 trials 100 iterations per trial

Datasets 119898 Ec Accuracy Not ass

Iris[3 150]

5 0310 0663 287010 0320 0651 281320 0309 0662 304330 0275 0696 283740 0292 0667 294350 0249 0724 2680100 0251 0729 2870

Moreover we will use separation index It is defined in[40] as

119878 =

sum119870

119894=1sum119873

119895=1(120583119894119895)

2 10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2

119873 lowastmin119894119895

10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2 (11)

In [5] 119906119894119895is defined as follows 119906

119894119895(119894 = 1 119870 119895 =

1 119873) is the membership of any fuzzy partition Thecorresponding hard partition of 119906

119894119895is defined as 119908

119894119895fl 1

if argmax119894119906119894119895 119908119894119895

fl 0 otherwise Cluster should be wellseparated thereby a smaller 119878 indicates the partition in whichall the clusters are overall compact and separate to each other

The centroid of 119894 is represented by the parameter 120592119894

43 Results This section is divided into two parts First westudy the behavior of ASClass with respect to the number ofants in each colony using Iris dataset Second we compare F-ASClass with the 119870-means 119870-medoid FCM and ASClassi(119894 = 1 2 3) using test datasets presented in Table 2 Thiscomparison can be done on the basis of average classificationerror and average value of accuracy

431 Parameters Setting of ASClass Algorithm We proposestudying the speed to find the optimal solution defined as thenumber of iterations as a function of the number of ants inASClass So we evaluate the performance of ASClass varyingthe number of ants from 5 to 100 given a fixed number ofiterations (100 iterations per trial)

Results presented in Figures 6(a)ndash6(g) show that thelength of the best tour made by each colony of ants isimproved very fast in the initial phase of the algorithm Oncethe second phase has been reckoned new good solutions startto appear but a phenomenon of local optima is discovered inthe last phase In this stage the system ceased to explore newsolutions and therefore will not be improved any more Thisprocess called in the literature of ACO algorithms unipathbehavior and it would indicate the situation in which the antsfollow the same path and so create the same cluster Thiswas due to a much higher quantities of artificial pheromonesdeposed by ants following that path than on all the others

On one hand we can observe from Table 3 that the bestresults with ASClass (corresponds to the smallest value oferror of classification Ec = 0249 and the largest value ofaccuracy Ac = 0729) are obtained when the number of antsis equal to 50 (number of samples in each cluster = 50) It isclear here that the number of ants improve the efficiency ofsolution quality a run with 100 ants is more search effectivethan with 5 ants This can be explained by the importance ofcommunication in colony through trail in using many ants(in case of 100 ants) This propriety called synergistic effectmakes ASClass attractiveness on its ability of finding quicklyan optimal solution

As a semisupervised clustering algorithm we can useconfusion matrix as a visualization tool to evaluate theperformance of ASClass It contains information about thenumber per class of well classifies and mislabeled samplesFurthermore it contains information about the capability ofsystem to not mislabel one cluster as another An example ofconfusion matrix on Iris dataset is shown in Figures 8(a)ndash8(g)

It is important to note that Iris dataset presents thefollowing characteristic one cluster is linearly separable fromthe other two the latter are not linearly separable from eachother We have presented this characteristic in Figure 7Results presented in Figures 8(a)ndash8(g) prove that ASClassmay be capable of identifying this characteristic

It is clear to see that for all examples in Figures 8(a)ndash8(g) the illustrated results show that all predicted samples ofcluster 1 are correctly assigned as cluster 1 Besides no instanceof cluster 1 is misclassified as class 2 and class 3 Howevera large number of predicted samples of cluster 2 may bemisclassified into cluster 3 and a few number aremisclassifiedinto cluster 1 In case of cluster 3 there are some samplesmisclassified into cluster 2 and no one is misclassified intocluster 1

As can be seen in Figures 8(a)ndash8(g) which representconfusion matrix for only clustered data the confusionmatrix did not contain the total 150 instances distribution ofIris data in the case of ASClass there are still some objectswhich are not assigned to any cluster when the algorithmstops

432 Comparative Analysis In order to compare the effec-tiveness of our proposed improvements of ASClass we applythem to datasets presented inTable 2We also use tree indexesof cluster validity error classification (Ec) accuracy andseparation (119878) coefficient ldquoThe index should make goodintuitive sense should have a basis in theory and should bereadily computablerdquo [4]

Note that underline bold face indicates the best and boldface indicates the second best In 119870-means 119870-medoid andFCM algorithms the predefined input number of clusters 119870is set to the known number of classes in each dataset

Tables 4 and 5 contain respectively the mean standarddeviations mode minimal and maximal value of errorrate of classification and the accuracy obtained by the four


272829

33132333435

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(a)

26

27

28

29

3

31

32

33

34

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(b)

10 20 30 40 7060 80 9050 1000Iterations

26

27

28

29

3

31

32

33

Aver

age l

engt

h of

bes

t tou

r

(c)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(d)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(e)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(f)

Aver

age l

engt

h of

bes

t tou

r

33

32

31

3

29

28

27

2610 20 30 40 50 60 70 80 90 1000

Iterations(g)

Figure 6 Results obtained by ASClass algorithm obtained on Iris dataset (a) 119898 = 5 (b) 119898 = 10 (c) 119898 = 20 (d) 119898 = 30 (e) 119898 = 40 (f)119898 = 50 (g)119898 = 100


Table 4 Mean standard deviation mode min and max values of error rate achieved by each clustering algorithm (119870-means 119870-medoidFCM and ASClass) over 20 trials on dataset presented in Table 2

Datasets Algorithm Mean Std Mode Min Max

Art1[4 400]

119870-means 0806 0180 0947 05225 09950119870-medoid 0773 0158 0760 05150 09750

FCM 0555 1139 times 10minus16 0555 05550 05550ASClass 0471 0 0471 04713 04713

Art2[2 1000]

119870-means 0548 0490 0980 0020 0980119870-medoid 0404 0482 0020 0020 0980

FCM 0980 2278 times 10minus16 0980 0980 0980

ASClass 0039 0 0039 0039 0039

Art3[4 100]

119870-means 0749 0227 0585 0222 0989119870-medoid 0810 0141 0549 0549 0995

FCM 0644 0 0644 0644 0644ASClass 0507 8183 times 10minus16 0508 0506 0508

Art4[2 200]

119870-means 0450 0510 0 0 1119870-medoid 0572 0494 1 0 1


Art5[9 900]

119870-means 0908 0065 0873 0802 0990119870-medoid 0883 0133 0985 0455 0996


Art6[4 400]

119870-means 0641 0273 0750 0 1119870-medoid 0827 0176 1 0427 1

FCM 0750 0 0750 0750 0750ASClass 0089 0002 0088 0085 0095

Iris[3 150]

119870-means 0576 0336 0480 0113 1119870-medoid 0710 0229 0126 0126 1

FCM 0746 2278 times 10minus16 0746 0746 0746ASClass 0257 5695 times 10minus17 0257 0257 0257

Thyroid[3 215]

119870-means 0774 0166 0922 0239 0922119870-medoid 0594 0261 0666 0128 0952


Pima[2 768]

119870-means 0461 0125 0371 0371 0628119870-medoid 0504 0116 0616 0325 0641


Unlabeled Iris dataset

0

01

02

03

04

05

06

07

08

09

1

1080701 040 0503 090602

1

2

3

Figure 7 Two-dimensional projection of Iris dataset onto first twoprincipal components

automatic clustering algorithms (119870-means119870-medoid FCMand ASClass)

In Tables 6 and 7 we report respectively the meanstandard deviations mode minimal and maximal values ofthe value of error rate of classification and accuracy obtainedby the four automatic clustering algorithms (ASClass1ASClass2 ASClass3 and F-ASClass)

Table 8 contains respectively the mean standard devi-ations mode minimal and maximal value of separationobtained by all fuzzy approaches FCM ASClass3 and F-ASClass

In general the comparative results presented in Table 4indicate that our proposed algorithm generates more com-pact clusters as well as lower error rates than the otherclustering algorithms The only exception is on the Pimadataset where ASClass does not achieve the minimum errorrates


(40)

789(3)

2105(8)

7105(27)

0 1481(4)

8519(23)

1 2

1

2

3

3

0 010000

(a)

10000

(41)

769

(3)

(3)

1538

(6)

7692

(30)

0

0

0

1304 8696

(20)

1

1

2

2

3

3

(b)

10000

(42)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(c)

10000

(29)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(d)

10000

(38)

10000

(26)

4286(18)

5714(24)

0

0

0

0

01

1

2

2

3

3

(e)

10000

(35)

9333

(14)

8571

(36)

0

0

01

1

2

2

3

3

714(3)

714(3)

667(1)

(f)

10000

(38)

9286

(26)

5714

(24)

0

0

01

1

2

2

3

3

238(1)

4048(17)

714(2)

(g)

Figure 8 Confusion matrix obtained with ASClass on Iris dataset for 100 iterations (a) 119898 = 5 (b) 119898 = 10 (c) 119898 = 20 (d) 119898 = 30 (e)119898 = 40 (f)119898 = 50 (g)119898 = 100

Table 5 also conforms to the fact that ASClass algorithmremains clearly and consistently superior to the other threetechniques in terms of the clustering accuracy The onlyexception is also seen on Pima dataset where the best valueis generated by the 119870-means algorithm

In Tables 6 and 7 we report respectively the meanstandard deviations mode minimal and maximal values of

error rate classification and accuracy obtained by ASClass1ASClass2 ASClass3 and F-ASClass algorithms It is clearfrom these tables that F-ASClass algorithm can minimizethe value error classification and maximize accuracy forall datasets The only exception is for Art3 and Pimadatasets The best values on error rate classification for theseboth datasets are given by ASClass1 The best value for


Table 5 Mean standard deviation mode min and max values of accuracy classification achieved by each clustering algorithm (119870-means119870-medoid FCM and ASClass) over 20 trials on dataset presented in Table 2


Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]

119870-means 0279 0236 0049 0002 0702119870-medoid 0201 0195 0344 9090 times 10minus4 0674


Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686


accuracy is given by ASClass2 for Art3 and by ASClass1 forPima

Moreover F-ASClass appears to be the most stable algo-rithm by having the lowest value of standard deviation for allindexes validity on all datasets

Both ASClass1 and ASClass2 are unable to find newpartition for Art4 dataset where the number of unclassifiedobjects is too low (unclassified objects = 3) However bothASClass3 and F-ASClass can be executed on small datasets

This feature proves that fuzzy-hybrid techniques are moreefficient than hard-hybrid technique

Table 8 shows that the most partition is made by F-ASClass and the second best value is given by ASClass3 thusa smallest 119878 indeed indicates a valid optimal partition Ascan be seen in (11) the more separate the clusters the larger(119874119895minus1198741198942) and the smaller 119878 Thus the fuzzy clustering rule

used in F-ASClass is to modify the compactness measure byminimizing 119878


Table 6 Average on 20 trials Mean standard deviation mode min and max values of error rate achieved by each clustering (ASClass1ASClass2 ASClass3 and F-ASClass) algorithm over 20 trials on dataset presented in Table 2


Art1[4 400]

ASClass1 0678 0096 07625 0434 0782ASClass2 0686 0095 0755 0462 0790ASClass3 0585 2278 times 10minus16 0585 0585 0585F-ASClass 0545 0 0545 0545 0545

Art2[2 1000]


Art3[4 100]

ASClass1 0609 0071 0616 0409 0703ASClass2 0633 0070 0689 0473 0705ASClass3 0619 1139 times 10minus16 0619 0619 0619F-ASClass 0676 1139 times 10minus16 0675 0675 0675

Art4[2 200]

ASClass1 mdash mdash mdash mdash mdashASClass2 mdash mdash mdash mdash mdashASClass3 0020 0 0020 0020 0020F-ASClass 0015 0 0015 0015 0015

Art5[9 900]


Art6[4 400]


Iris[3 150]

ASClass1 0335 0050 0293 0273 0420ASClass2 0345 0060 0313 0240 0426ASClass3 0306 0 0306 0306 0306F-ASClass 0106 4271 times 10minus17 0 0 0004

Thyroid[3 215]


Pima[2 768]


5 Conclusion

We have presented in this paper a new fuzzy-swarm-algorithm called F-ASClass for data clustering in a knowledgediscovery context F-ASClass introduces new fuzzy-heuristicfor the ant colony inspired from the spider web constructionWe have also proposed using several colonies of ants the

number of colonies in F-ASClass depended on the numberof clusters in databases to be classified Fuzzy rules are intro-duced to find a partition for unclassified objects generatedby the work of artificial ants The experimental results showthat the proposed algorithm is able to achieve an interestingresult with respect to the well-known clustering algorithm


Table 7 Mean standard deviation mode min and max values of accuracy classification achieved by each clustering algorithm (ASClass1ASClass2 ASClass3 and F-ASClass) over 20 trials on dataset presented in Table 2


Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]

ASClass1 mdash mdash mdash mdash mdashASClass2 mdash mdash mdash mdash mdashASClass3 0980 2278 times 10minus16 0980 0980 0980F-ASClass 0985 2278 times 10minus16 0985 0985 0985

Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


119870-means 119870-medoid and FCM and our proposed hybridalgorithms ASClass ASClass1 ASClass2 and ASClass3

In the future we will propose a new approach called F-CIC or ldquoFuzzy-Collective Intelligence of Coloniesrdquo to masterthe complexity In this context we will propose new fuzzyrules to manage communication between colonies However

the remarkable success of our fuzzy-metaheuristic proposedin this paper can serve as a starting point for new approachin engineering and computer science In our approach basicprinciples of swarm intelligence are still present but againuntimely we will propose using a good improvement withfuzzy rules and fractal theory


Table 8 Mean standard deviation mode min and max values of separation index achieved by each clustering algorithm (FCM ASClass3and F-ASClass) over 20 trials on dataset presented in Table 2


Art1[4 400]FCM 0003 4449 times 10minus19 0003 0003 0003

ASClass3 3013 times 10minus5 6026 times 10minus4 0 0 0012F-ASClass 6212 times 10minus6 1242 times 10minus4 0 0 0002







Art5[9 900]FCM 0001 0 0001 0001 0001




Iris[3 150]FCM 0006 1779 times 10minus18 0006 0006 0006

ASClass3 9837 times 10minus5 0001 0 0 0014F-ASClass 3171 times 10minus5 3884 times 10minus4 0 0 0004

Thyroid[3 215]FCM 00089 3559 times 10minus18 0008 0008 0008


Pima[2 768]FCM 0012 5339 times 10minus18 0012 0012 0012


Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to acknowledge the financial supportof this work by grants from General Direction of ScientificResearch (DGRST) Tunisia under the ARUB program

References

[1] S Camazine J L Deneubourg N R Franks J Sneyd GTheraulaz and E Bonabeau Self-Organization in BiologicalSystems Princeton University Press 2001

[2] E Bonabeau M Dorigo and G Theraulaz Swarm IntelligenceFrom Natural to Artificial Systems Oxford University PressNew York NY USA 1999

[3] P P Grasse ldquoLa reconstruction du nid et les coordinations inter-individuelles chez Bellicositermes natalensis et Cubitermes spLa theorie de la stigmergie Essai drsquointerpretation du comporte-ment des termites constructeursrdquo Insectes Sociaux vol 6 pp41ndash81 1959

[4] AK Jain andRCDubesAlgorithms for ClusteringData Pren-tice Hall Advanced Reference Series Prentice Hall EnglewoodCliffs NJ USA 1988

[5] A K Jain M N Murty and P J Flynn ldquoData clustering areviewrdquo ACM Computing Surveys vol 31 no 3 pp 264ndash3231999

[6] J MacQueen ldquoSome methods for classification and analysis ofmultivariate observationsrdquo in Proceedings of the 5th BerkeleySymposium on Mathematical Statistics and Probability pp 281ndash297 University of California Press 1967

[7] L Kaufman and P J Rousseeuw ldquoClustering by means ofMedoidsrdquo in Statistical Data Analysis Based on the L1-Norm andRelated Methods Y Dodge Ed pp 405ndash416 North-HollandNew York NY USA 1987

[8] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[9] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Plenum Press New York NY USA 1981

[10] J C Bezdek R Ehrlich and W Full ldquoFCM the fuzzy c-meansclustering algorithmrdquo Computers amp Geosciences vol 10 no 2-3pp 191ndash203 1984


[11] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

[12] L VUtkin ldquoFuzzy one-class classificationmodel using contam-ination neighborhoodsrdquo Advances in Fuzzy Systems vol 2012Article ID 984325 10 pages 2012

[13] N J Pizzi and W Pedrycz ldquoClassifying high-dimensionalpatterns using a fuzzy logic discriminant networkrdquo Advances inFuzzy Systems vol 2012 Article ID 920920 7 pages 2012

[14] L Liparulo A Proietti and M Panella ldquoFuzzy clustering usingthe convex hull as geometrical modelrdquo Advances in FuzzySystems vol 2015 Article ID 265135 13 pages 2015

[15] MOmranA Salman andA Engelbrecht ldquoImage classificationusing particle swarm optimizationrdquo in Proceedings of the Con-ference on Simulated Evolution and Learning vol 1 pp 370ndash374Singapore November 2002

[16] D Karaboga and C Ozturk ldquoA novel clustering approach Arti-ficial Bee Colony (ABC) algorithmrdquo Applied Soft Computingvol 11 no 1 pp 652ndash657 2011

[17] J Senthilnath S N Omkar and V Mani ldquoClustering usingfirefly algorithm performance studyrdquo Swarm and EvolutionaryComputation vol 1 no 3 pp 164ndash171 2011

[18] L Xiao ldquoA clustering algorithm based on artificial fish schoolrdquoin Proceedings of the 2nd International Conference on ComputerEngineering andTechnology (ICCET rsquo10) pp 766ndash769 ChengduChina 2010

[19] C Grosan A Abraham and M Chis Swarm Intelligence inData Mining Studies in Computational Intelligence (SCI) vol34 Springer Berlin Germany 2006

[20] A Hamdi N Monmarche A M Alimi and M Slimane ldquoArti-ficial ants for automatic classificationrdquo in Artificial Ants FromCollective Intelligence to Real-Life Optimization and Beyond NMonmarche F Guinand and P Siarry Eds pp 265ndash290 ISTE-WILEY Hoboken NJ USA 2010

[21] J Deneubourg J S Goss S Pasteels J Fresneau and DLachaud ldquoSelf-organization mechanisms in ant societies (II)learning in foraging and division of laborrdquo in From Individualto Collective Behavior in Social Insects Experientia Supplemen-tum pp 177ndash196 Birkhauser 1987

[22] E Lumer and B Faieta ldquoDiversity and adaptation in popula-tions of clustering antsrdquo in Proceedings of the 3rd InternationalConference on Simulation of Adaptive Behavior FromAnimals toAnimats (SAB rsquo94) pp 501ndash508 MIT Press Cambridge MassUSA 1994

[23] N Monmarche ldquoOn data clustering with artficial antsrdquo inProceedings of the Workshop on Data Mining with EvolutionaryAlgorithms Research Directions (AAAI rsquo99 amp GECCO rsquo99 ) AA Freitas Ed pp 23ndash26 Orlando Fla USA 1999

[24] S Ouadfel and M Batouche ldquoAn efficient ant algorithm forswarm-based image clusteringrdquo Journal of Computer Sciencevol 3 no 3 pp 162ndash167 2007

[25] N Labroche N Monmarche and G Venturini ldquoAntClust antclustering and web usage miningrdquo in Proceedings of the Geneticand Evolutionary Computation Conference (GECCO rsquo03) pp25ndash36 Chicago Ill USA July 2003

[26] H Azzag N Monmarche M Slimane G Venturini and CGuinot ldquoAntTree a new model for clustering with artificialantsrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation pp 2642ndash2647 December 2003

[27] G W Reynolds ldquoFlocks herds and schools a distributedbehavioral modelrdquo in Proceedings of the 14th Annual Conferenceon Computer Graphics and Interactive Techniques (SIGGRAPHrsquo87) pp 25ndash34 1987

[28] G Proctor and C Winter ldquoInformation flocking data visu-alisation in virtual worlds using emergent behavioursrdquo inProceedings of the 1st International Conference on VirtualWorlds(VW rsquo98) pp 168ndash176 Paris France 1998

[29] S Aupetit N Monmarche M Slimane C Guinot and GVenturini ldquoClustering and dynamic data visualizationwith arti-ficial flying insect rdquo inGenetic and Evolutionary ComputationmdashGECCO 2003 vol 2723 of Lecture Notes in Computer Sciencepp 140ndash141 Springer Berlin Germany 2003

[30] A Abraham S Das and S Roy ldquoSwarm intelligence algorithmsfor data clusteringrdquo in Soft Computing for Knowledge Discoveryand DataMining pp 279ndash313 Springer Berlin Germany 2008

[31] C Bourjot V Chevrier and V Thomas ldquoA new swarm mech-anism based on social spiders colonies from web weaving toregion detectionrdquoWeb Intelligence and Agent Systems vol 1 no1 pp 47ndash64 2003

[32] A Hamdi N Monmarche A M Alimi and M SlimaneldquoSwarmClass a novel data clustering approach by a hybridiza-tion of an ant colony with flying insectsrdquo in Ant ColonyOptimization and Swarm Intelligence vol 5217 of Lecture Notesin Computer Science pp 412ndash414 Springer Berlin Germany2010

[33] A Hamdi N Monmarche A M Alimi and M SlimaneldquoAntBee clustering with artificial ants in hexagonal gridrdquo inProceedings of the International Conference of Artificial Evolu-tion pp 462ndash472 Angers France October 2011

[34] A Hamdi N Monmarche M Slimane and A M AlimildquoFractAntBee algorithm for data clustering problemrdquo Journal ofComputer Science and Information Security (IJCSIS) vol 14 no7 2016

[35] S Ouadfel and M Batouche ldquoMRF-based image segmentationusing Ant Colony Systemrdquo Electronic Letters on ComputerVision and Image Analysis vol 2 no 1 pp 12ndash24 2003

[36] M Dorigo and L Maria Gambardella Ant Colonies for theTraveling Salesman Problem BioSystems Press 1997

[37] A Hamdi N Monmarche A M Alimi and M SlimaneldquoTspClass a novel data clustering approach by artificial antsrdquoin Proceedings of the International Conference on Metaheuristicsand Nature Inspired Computing (META rsquo10) pp 27ndash31 DjerbaTunisia 2010

[38] A Hamdi N Monmarche M Slimane and A M Alimi ldquoSpi-derrsquos behavior for ant based clustering algorithmrdquo inProceedingsof the International Conference on Intelligent Systems Designand Applications (ISDA rsquo15) pp 254ndash259 Marrakech MoroccoDecember 2015

[39] M Lichman UCI Machine Learning Repository Universityof California School of Information and Computer ScienceIrvine Calif USA 2013 httparchiveicsucieduml

[40] A M Bensaid L O Hall J C Bezdek et al ldquoValidity-guided(Re)clustering with applications to image segmentationrdquo IEEETransactions on Fuzzy Systems vol 4 no 2 pp 112ndash123 1996

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



and swarm intelligence tools for data clustering A generaldescription of our proposed approach called F-ASClass isgiven in Section 3 Experimental results and comparisonwithhard and fuzzy algorithms are reported in Section 4 Finallyconcluding remarks are drawn in Section 5

2 Literature Review

21 Problem Definition Cluster analysis is a technique thatorganizes data by abstracting underlying structure either asa grouping of objects Each group consists of objects that aresimilar between themselves and dissimilar to objects of othergroups

Each object corresponds to a vector of 119872 numericalvalues which correspond to the119872 numerical attributes Therelationships between objects are generated into a dissim-ilarity matrix in which rows and columns correspond toobjects As objects can be represented by points in numericalspace the dissimilarity between two objects can be defined asdistance between the two corresponding points Any distancecan be used as dissimilarity measures The most commonlyused dissimilarity matrix is theMinkowskimetric

119889119903(119874119894 119874119895) =

119898

sum

119896=1

(119908119896

10038161003816100381610038161003816119909119894119896 119909119895119896

10038161003816100381610038161003816

119903

)

1119903

(1)

where 119908119896is a weighting factor that will be set to 1 thereafter

According to the value of 119903 (119903 ge 1) the following measuresare obtained Manhattan distance (119903 = 1) Euclidean distance(119903 = 2) and Chebyshev distance (119903 = infin) As mentioned in[4] Euclidean distance is themost common of theMinkowskimetric

22 Clustering Algorithms The grouping step can be per-formed in a number of ways In [5] different approaches toclustering data are described

(i) PartitioningHierarchical Classification Partitionalclustering technique identifies the partition thatoptimizes a clustering criterion defined on a subset ofobjects (locally) or over all of the objects (globally)Hierarchical clustering technique builds a sequenceof nested partitions that are visualized by exampleby a dendrogram

(ii) HardFuzzy Classification A hard clustering algo-rithm allocates each object to a single cluster duringits operations Hence the clusters are disjoint A fuzzyclustering algorithm associates each object with everycluster using a membership function The output ofsuch algorithm is a clustering but not a partition

(iii) DeterministicStochastic Optimization in partitionalapproach can be accomplished using traditional tech-nique or through a random search of the state spaceconsisting of all possible labeling

(iv) SupervisedUnsupervised Classification An unsuper-vised classification uses only the dissimilarity matrixNo information on the object class is provided to the

method (objects are said unlabeled) In supervisedclassification objects are labeled while knowing theirdissimilarities The problem is then to constructhyperplanes separating objects according to theirclass The unsupervised classification objective isdifferent from that of the supervised case in the firstcase the goal is to discover groups of objects while inthe second known groups are considered and the goalis to discover what makes them different or to classifynew objects whose class is unknown

Our proposed technique presented in this paper whichwe call F-ASClass belongs to fuzzy-semisupervised parti-tional clustering technique It uses fuzzy rules and stochas-tic behavior to partition dataset into specified number ofclusters For the present paper it suffices to note that thefollowing techniques (119870-means 119870-medoid and FCM) areused to improve F-ASClass algorithm A comparative studybetween us will be presented in Section 4

119870-means algorithm is a hard-unsupervised learningalgorithm that appears to partition dataset into a specifiednumber of clusters The technique presented in [6] consistsof starting with 119896 groups each of which consists of a singlerandomly selected object and thereafter adding each newobject to its closest cluster center After an object is added toa group the mean of that group is adjusted in order to takeaccount of the new added object The algorithm is deemed tohave converged when the assignments no longer change

The 119870-medoid algorithm described in [7] is based uponthe search of representative objects of each cluster (calledmedoid) which should represent the various aspects of thestructure of the data 119870-medoid algorithm is related to the119870-means algorithm The main difference between 119870-meansand 119870-medoid stands in calculating the cluster center Themedoid is a statistic which represents that data member of adataset whose average dissimilarity to all the other membersof the set is minimal Therefore a medoid unlike meanis always a member of the dataset It represents the mostcentrally located data item of the dataset

The fuzzy 119888-means (FCM) algorithm firstly presented in[8] and improved in [9 10] allows one samples to belongto two or more one clusters The first idea which aims tocharacterize an individual objectsrsquo similarity to all the clusterswas introduced in [11] In this context the similarity an objectshares with each cluster is represented with a membershipfunction whose values are between zero and one Object indatasetwill have amembership in every clustermembershipsclose to unity indicate a high degree of similarity between theobject and a cluster while memberships close to zero involvelittle similarity between the object and that cluster

FCMmethod differs from previously presented119870-meansand 119870-medoid algorithms by the fact that the centroid of acluster is the mean of all samples in the dataset weightedby their degree of belonging to the cluster The degree ofbelonging is presented by a function of the distance ofthe sample from the centroid which includes a parameter





















1198741 119874











Colony of ants

Artificial ant










119875119896

119894119895(119905) =

[120591119894119895]

120572

[120578119894119895]

120573

sum119897isin119873119896

119894

[120591119894119897]120572

[120578119894119897]120573

if 119895 isin 119873119896

119894 (2)






119875fix119896

119894(119905) =

2119891 (119874119894) if 119891 (119874

119894) lt 119896fix

1 if otherwise(3)


119894



119891 (119874119894) = max

1

119873119879119870

sum

119874119895isin119879119896

1 minus

119889 (119874119894 119874119895)

120579

0

(4)






Δ120591119896

119894119895(119905) =

1

119862119896


0 otherwise

120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) forall (119894 119895) isin 119879

119896

(5)



0

is constant


120591119894119895= (1 minus 120588) 120591

119894119895 (6)


120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) + (1 minus 120588) 120591

119894119895forall (119894 119895) isin 119879

119896 (7)






119894119895


120578119894119895 the visibility














119875119896

119894119895(119905) =

[120591119894119895]

120572

[120578119894119895]

120573

sum119897isin119869119896

119894

[120591119894119897]120572

[120578119894119897]120573

if 119895 isin 119869119896

119894

0 if 119895 notin 119869119896

119894


Until (119873 minus 119869119896

119894) = 119873119887(119862init)


Δ120591119896

119894119895(119905) =

119876

119871119896



120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) + (1 minus 120588) 120591

119894119895forall (119894 119895) isin 119879

119896

End forEnd for

Algorithm 1




3 FCM ASClass3




Cluster 3

Outliers

Cluster 1

Cluster 2

ASClass algorithm


Cluster 3

Outliers

Cluster 1

Cluster 2




Cluster 1

Cluster 2Cluster 3

FCM algorithm





119895 Here 119906






119894119895= 119891119911 for


119891119911 =

1

119870

(8)




119862)




Art2[2 1000] 2 1000 2 (500 500)

Art3[4 400] 2 1100 4 (500 50 500 50)

Art4[2 200] 2 200 2 (100 100)

Art5[9 900] 2 900 9 (100 100 100)

Art6[4 400] 8 400 4 (100 100 100 100)

Iris[3 150] 4 150 3 (50 50 50)

Thyroid[3 215] 5 215 3 (150 35 30)

Pima[2 768] 8 768 2 (500 268)





Ec = 1 minus

sum119870

119896=1Recall

119896

119870

Recall119896=

119899119896119896

119899119896

(9)


Accuracy =sum119870

119896=1119899119896119896

119873

(10)

where 119899119896119896



119896 and119873 is





Iris[3 150]

5 0310 0663 287010 0320 0651 281320 0309 0662 304330 0275 0696 283740 0292 0667 294350 0249 0724 2680100 0251 0729 2870


119878 =

sum119870

119894=1sum119873

119895=1(120583119894119895)

2 10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2

119873 lowastmin119894119895

10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2 (11)


119894119895(119894 = 1 119870 119895 =


119894119895is defined as 119908

119894119895fl 1

if argmax119894119906119894119895 119908119894119895















272829

33132333435

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(a)

26

27

28

29

3

31

32

33

34

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(b)

10 20 30 40 7060 80 9050 1000Iterations

26

27

28

29

3

31

32

33

Aver

age l

engt

h of

bes

t tou

r

(c)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(d)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(e)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(f)

Aver

age l

engt

h of

bes

t tou

r

33

32

31

3

29

28

27

2610 20 30 40 50 60 70 80 90 1000

Iterations(g)





Art1[4 400]

119870-means 0806 0180 0947 05225 09950119870-medoid 0773 0158 0760 05150 09750


Art2[2 1000]

119870-means 0548 0490 0980 0020 0980119870-medoid 0404 0482 0020 0020 0980

FCM 0980 2278 times 10minus16 0980 0980 0980

ASClass 0039 0 0039 0039 0039

Art3[4 100]

119870-means 0749 0227 0585 0222 0989119870-medoid 0810 0141 0549 0549 0995


Art4[2 200]

119870-means 0450 0510 0 0 1119870-medoid 0572 0494 1 0 1


Art5[9 900]

119870-means 0908 0065 0873 0802 0990119870-medoid 0883 0133 0985 0455 0996


Art6[4 400]

119870-means 0641 0273 0750 0 1119870-medoid 0827 0176 1 0427 1

FCM 0750 0 0750 0750 0750ASClass 0089 0002 0088 0085 0095

Iris[3 150]

119870-means 0576 0336 0480 0113 1119870-medoid 0710 0229 0126 0126 1


Thyroid[3 215]

119870-means 0774 0166 0922 0239 0922119870-medoid 0594 0261 0666 0128 0952


Pima[2 768]

119870-means 0461 0125 0371 0371 0628119870-medoid 0504 0116 0616 0325 0641



0

01

02

03

04

05

06

07

08

09

1

1080701 040 0503 090602

1

2

3







(40)

789(3)

2105(8)

7105(27)

0 1481(4)

8519(23)

1 2

1

2

3

3

0 010000

(a)

10000

(41)

769

(3)

(3)

1538

(6)

7692

(30)

0

0

0

1304 8696

(20)

1

1

2

2

3

3

(b)

10000

(42)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(c)

10000

(29)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(d)

10000

(38)

10000

(26)

4286(18)

5714(24)

0

0

0

0

01

1

2

2

3

3

(e)

10000

(35)

9333

(14)

8571

(36)

0

0

01

1

2

2

3

3

714(3)

714(3)

667(1)

(f)

10000

(38)

9286

(26)

5714

(24)

0

0

01

1

2

2

3

3

238(1)

4048(17)

714(2)

(g)








Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]



Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686











Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in























1198741 119874











Colony of ants

Artificial ant










119875119896

119894119895(119905) =

[120591119894119895]

120572

[120578119894119895]

120573

sum119897isin119873119896

119894

[120591119894119897]120572

[120578119894119897]120573

if 119895 isin 119873119896

119894 (2)






119875fix119896

119894(119905) =

2119891 (119874119894) if 119891 (119874

119894) lt 119896fix

1 if otherwise(3)


119894



119891 (119874119894) = max

1

119873119879119870

sum

119874119895isin119879119896

1 minus

119889 (119874119894 119874119895)

120579

0

(4)






Δ120591119896

119894119895(119905) =

1

119862119896


0 otherwise

120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) forall (119894 119895) isin 119879

119896

(5)



0

is constant


120591119894119895= (1 minus 120588) 120591

119894119895 (6)


120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) + (1 minus 120588) 120591

119894119895forall (119894 119895) isin 119879

119896 (7)






119894119895


120578119894119895 the visibility














119875119896

119894119895(119905) =

[120591119894119895]

120572

[120578119894119895]

120573

sum119897isin119869119896

119894

[120591119894119897]120572

[120578119894119897]120573

if 119895 isin 119869119896

119894

0 if 119895 notin 119869119896

119894


Until (119873 minus 119869119896

119894) = 119873119887(119862init)


Δ120591119896

119894119895(119905) =

119876

119871119896



120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) + (1 minus 120588) 120591

119894119895forall (119894 119895) isin 119879

119896

End forEnd for

Algorithm 1




3 FCM ASClass3




Cluster 3

Outliers

Cluster 1

Cluster 2

ASClass algorithm


Cluster 3

Outliers

Cluster 1

Cluster 2




Cluster 1

Cluster 2Cluster 3

FCM algorithm





119895 Here 119906






119894119895= 119891119911 for


119891119911 =

1

119870

(8)




119862)




Art2[2 1000] 2 1000 2 (500 500)

Art3[4 400] 2 1100 4 (500 50 500 50)

Art4[2 200] 2 200 2 (100 100)

Art5[9 900] 2 900 9 (100 100 100)

Art6[4 400] 8 400 4 (100 100 100 100)

Iris[3 150] 4 150 3 (50 50 50)

Thyroid[3 215] 5 215 3 (150 35 30)

Pima[2 768] 8 768 2 (500 268)





Ec = 1 minus

sum119870

119896=1Recall

119896

119870

Recall119896=

119899119896119896

119899119896

(9)


Accuracy =sum119870

119896=1119899119896119896

119873

(10)

where 119899119896119896



119896 and119873 is





Iris[3 150]

5 0310 0663 287010 0320 0651 281320 0309 0662 304330 0275 0696 283740 0292 0667 294350 0249 0724 2680100 0251 0729 2870


119878 =

sum119870

119894=1sum119873

119895=1(120583119894119895)

2 10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2

119873 lowastmin119894119895

10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2 (11)


119894119895(119894 = 1 119870 119895 =


119894119895is defined as 119908

119894119895fl 1

if argmax119894119906119894119895 119908119894119895















272829

33132333435

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(a)

26

27

28

29

3

31

32

33

34

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(b)

10 20 30 40 7060 80 9050 1000Iterations

26

27

28

29

3

31

32

33

Aver

age l

engt

h of

bes

t tou

r

(c)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(d)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(e)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(f)

Aver

age l

engt

h of

bes

t tou

r

33

32

31

3

29

28

27

2610 20 30 40 50 60 70 80 90 1000

Iterations(g)





Art1[4 400]

119870-means 0806 0180 0947 05225 09950119870-medoid 0773 0158 0760 05150 09750


Art2[2 1000]

119870-means 0548 0490 0980 0020 0980119870-medoid 0404 0482 0020 0020 0980

FCM 0980 2278 times 10minus16 0980 0980 0980

ASClass 0039 0 0039 0039 0039

Art3[4 100]

119870-means 0749 0227 0585 0222 0989119870-medoid 0810 0141 0549 0549 0995


Art4[2 200]

119870-means 0450 0510 0 0 1119870-medoid 0572 0494 1 0 1


Art5[9 900]

119870-means 0908 0065 0873 0802 0990119870-medoid 0883 0133 0985 0455 0996


Art6[4 400]

119870-means 0641 0273 0750 0 1119870-medoid 0827 0176 1 0427 1

FCM 0750 0 0750 0750 0750ASClass 0089 0002 0088 0085 0095

Iris[3 150]

119870-means 0576 0336 0480 0113 1119870-medoid 0710 0229 0126 0126 1


Thyroid[3 215]

119870-means 0774 0166 0922 0239 0922119870-medoid 0594 0261 0666 0128 0952


Pima[2 768]

119870-means 0461 0125 0371 0371 0628119870-medoid 0504 0116 0616 0325 0641



0

01

02

03

04

05

06

07

08

09

1

1080701 040 0503 090602

1

2

3







(40)

789(3)

2105(8)

7105(27)

0 1481(4)

8519(23)

1 2

1

2

3

3

0 010000

(a)

10000

(41)

769

(3)

(3)

1538

(6)

7692

(30)

0

0

0

1304 8696

(20)

1

1

2

2

3

3

(b)

10000

(42)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(c)

10000

(29)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(d)

10000

(38)

10000

(26)

4286(18)

5714(24)

0

0

0

0

01

1

2

2

3

3

(e)

10000

(35)

9333

(14)

8571

(36)

0

0

01

1

2

2

3

3

714(3)

714(3)

667(1)

(f)

10000

(38)

9286

(26)

5714

(24)

0

0

01

1

2

2

3

3

238(1)

4048(17)

714(2)

(g)








Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]



Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686











Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







119875119896

119894119895(119905) =

[120591119894119895]

120572

[120578119894119895]

120573

sum119897isin119873119896

119894

[120591119894119897]120572

[120578119894119897]120573

if 119895 isin 119873119896

119894 (2)






119875fix119896

119894(119905) =

2119891 (119874119894) if 119891 (119874

119894) lt 119896fix

1 if otherwise(3)


119894



119891 (119874119894) = max

1

119873119879119870

sum

119874119895isin119879119896

1 minus

119889 (119874119894 119874119895)

120579

0

(4)






Δ120591119896

119894119895(119905) =

1

119862119896


0 otherwise

120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) forall (119894 119895) isin 119879

119896

(5)



0

is constant


120591119894119895= (1 minus 120588) 120591

119894119895 (6)


120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) + (1 minus 120588) 120591

119894119895forall (119894 119895) isin 119879

119896 (7)






119894119895


120578119894119895 the visibility














119875119896

119894119895(119905) =

[120591119894119895]

120572

[120578119894119895]

120573

sum119897isin119869119896

119894

[120591119894119897]120572

[120578119894119897]120573

if 119895 isin 119869119896

119894

0 if 119895 notin 119869119896

119894


Until (119873 minus 119869119896

119894) = 119873119887(119862init)


Δ120591119896

119894119895(119905) =

119876

119871119896



120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) + (1 minus 120588) 120591

119894119895forall (119894 119895) isin 119879

119896

End forEnd for

Algorithm 1




3 FCM ASClass3




Cluster 3

Outliers

Cluster 1

Cluster 2

ASClass algorithm


Cluster 3

Outliers

Cluster 1

Cluster 2




Cluster 1

Cluster 2Cluster 3

FCM algorithm





119895 Here 119906






119894119895= 119891119911 for


119891119911 =

1

119870

(8)




119862)




Art2[2 1000] 2 1000 2 (500 500)

Art3[4 400] 2 1100 4 (500 50 500 50)

Art4[2 200] 2 200 2 (100 100)

Art5[9 900] 2 900 9 (100 100 100)

Art6[4 400] 8 400 4 (100 100 100 100)

Iris[3 150] 4 150 3 (50 50 50)

Thyroid[3 215] 5 215 3 (150 35 30)

Pima[2 768] 8 768 2 (500 268)





Ec = 1 minus

sum119870

119896=1Recall

119896

119870

Recall119896=

119899119896119896

119899119896

(9)


Accuracy =sum119870

119896=1119899119896119896

119873

(10)

where 119899119896119896



119896 and119873 is





Iris[3 150]

5 0310 0663 287010 0320 0651 281320 0309 0662 304330 0275 0696 283740 0292 0667 294350 0249 0724 2680100 0251 0729 2870


119878 =

sum119870

119894=1sum119873

119895=1(120583119894119895)

2 10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2

119873 lowastmin119894119895

10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2 (11)


119894119895(119894 = 1 119870 119895 =


119894119895is defined as 119908

119894119895fl 1

if argmax119894119906119894119895 119908119894119895















272829

33132333435

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(a)

26

27

28

29

3

31

32

33

34

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(b)

10 20 30 40 7060 80 9050 1000Iterations

26

27

28

29

3

31

32

33

Aver

age l

engt

h of

bes

t tou

r

(c)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(d)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(e)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(f)

Aver

age l

engt

h of

bes

t tou

r

33

32

31

3

29

28

27

2610 20 30 40 50 60 70 80 90 1000

Iterations(g)





Art1[4 400]

119870-means 0806 0180 0947 05225 09950119870-medoid 0773 0158 0760 05150 09750


Art2[2 1000]

119870-means 0548 0490 0980 0020 0980119870-medoid 0404 0482 0020 0020 0980

FCM 0980 2278 times 10minus16 0980 0980 0980

ASClass 0039 0 0039 0039 0039

Art3[4 100]

119870-means 0749 0227 0585 0222 0989119870-medoid 0810 0141 0549 0549 0995


Art4[2 200]

119870-means 0450 0510 0 0 1119870-medoid 0572 0494 1 0 1


Art5[9 900]

119870-means 0908 0065 0873 0802 0990119870-medoid 0883 0133 0985 0455 0996


Art6[4 400]

119870-means 0641 0273 0750 0 1119870-medoid 0827 0176 1 0427 1

FCM 0750 0 0750 0750 0750ASClass 0089 0002 0088 0085 0095

Iris[3 150]

119870-means 0576 0336 0480 0113 1119870-medoid 0710 0229 0126 0126 1


Thyroid[3 215]

119870-means 0774 0166 0922 0239 0922119870-medoid 0594 0261 0666 0128 0952


Pima[2 768]

119870-means 0461 0125 0371 0371 0628119870-medoid 0504 0116 0616 0325 0641



0

01

02

03

04

05

06

07

08

09

1

1080701 040 0503 090602

1

2

3







(40)

789(3)

2105(8)

7105(27)

0 1481(4)

8519(23)

1 2

1

2

3

3

0 010000

(a)

10000

(41)

769

(3)

(3)

1538

(6)

7692

(30)

0

0

0

1304 8696

(20)

1

1

2

2

3

3

(b)

10000

(42)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(c)

10000

(29)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(d)

10000

(38)

10000

(26)

4286(18)

5714(24)

0

0

0

0

01

1

2

2

3

3

(e)

10000

(35)

9333

(14)

8571

(36)

0

0

01

1

2

2

3

3

714(3)

714(3)

667(1)

(f)

10000

(38)

9286

(26)

5714

(24)

0

0

01

1

2

2

3

3

238(1)

4048(17)

714(2)

(g)








Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]



Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686











Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in








119875119896

119894119895(119905) =

[120591119894119895]

120572

[120578119894119895]

120573

sum119897isin119869119896

119894

[120591119894119897]120572

[120578119894119897]120573

if 119895 isin 119869119896

119894

0 if 119895 notin 119869119896

119894


Until (119873 minus 119869119896

119894) = 119873119887(119862init)


Δ120591119896

119894119895(119905) =

119876

119871119896



120591119894119895= 1205910+

119898

sum

119896=1

Δ120591119896

119894119895(119905) + (1 minus 120588) 120591

119894119895forall (119894 119895) isin 119879

119896

End forEnd for

Algorithm 1




3 FCM ASClass3




Cluster 3

Outliers

Cluster 1

Cluster 2

ASClass algorithm


Cluster 3

Outliers

Cluster 1

Cluster 2




Cluster 1

Cluster 2Cluster 3

FCM algorithm





119895 Here 119906






119894119895= 119891119911 for


119891119911 =

1

119870

(8)




119862)




Art2[2 1000] 2 1000 2 (500 500)

Art3[4 400] 2 1100 4 (500 50 500 50)

Art4[2 200] 2 200 2 (100 100)

Art5[9 900] 2 900 9 (100 100 100)

Art6[4 400] 8 400 4 (100 100 100 100)

Iris[3 150] 4 150 3 (50 50 50)

Thyroid[3 215] 5 215 3 (150 35 30)

Pima[2 768] 8 768 2 (500 268)





Ec = 1 minus

sum119870

119896=1Recall

119896

119870

Recall119896=

119899119896119896

119899119896

(9)


Accuracy =sum119870

119896=1119899119896119896

119873

(10)

where 119899119896119896



119896 and119873 is





Iris[3 150]

5 0310 0663 287010 0320 0651 281320 0309 0662 304330 0275 0696 283740 0292 0667 294350 0249 0724 2680100 0251 0729 2870


119878 =

sum119870

119894=1sum119873

119895=1(120583119894119895)

2 10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2

119873 lowastmin119894119895

10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2 (11)


119894119895(119894 = 1 119870 119895 =


119894119895is defined as 119908

119894119895fl 1

if argmax119894119906119894119895 119908119894119895















272829

33132333435

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(a)

26

27

28

29

3

31

32

33

34

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(b)

10 20 30 40 7060 80 9050 1000Iterations

26

27

28

29

3

31

32

33

Aver

age l

engt

h of

bes

t tou

r

(c)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(d)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(e)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(f)

Aver

age l

engt

h of

bes

t tou

r

33

32

31

3

29

28

27

2610 20 30 40 50 60 70 80 90 1000

Iterations(g)





Art1[4 400]

119870-means 0806 0180 0947 05225 09950119870-medoid 0773 0158 0760 05150 09750


Art2[2 1000]

119870-means 0548 0490 0980 0020 0980119870-medoid 0404 0482 0020 0020 0980

FCM 0980 2278 times 10minus16 0980 0980 0980

ASClass 0039 0 0039 0039 0039

Art3[4 100]

119870-means 0749 0227 0585 0222 0989119870-medoid 0810 0141 0549 0549 0995


Art4[2 200]

119870-means 0450 0510 0 0 1119870-medoid 0572 0494 1 0 1


Art5[9 900]

119870-means 0908 0065 0873 0802 0990119870-medoid 0883 0133 0985 0455 0996


Art6[4 400]

119870-means 0641 0273 0750 0 1119870-medoid 0827 0176 1 0427 1

FCM 0750 0 0750 0750 0750ASClass 0089 0002 0088 0085 0095

Iris[3 150]

119870-means 0576 0336 0480 0113 1119870-medoid 0710 0229 0126 0126 1


Thyroid[3 215]

119870-means 0774 0166 0922 0239 0922119870-medoid 0594 0261 0666 0128 0952


Pima[2 768]

119870-means 0461 0125 0371 0371 0628119870-medoid 0504 0116 0616 0325 0641



0

01

02

03

04

05

06

07

08

09

1

1080701 040 0503 090602

1

2

3







(40)

789(3)

2105(8)

7105(27)

0 1481(4)

8519(23)

1 2

1

2

3

3

0 010000

(a)

10000

(41)

769

(3)

(3)

1538

(6)

7692

(30)

0

0

0

1304 8696

(20)

1

1

2

2

3

3

(b)

10000

(42)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(c)

10000

(29)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(d)

10000

(38)

10000

(26)

4286(18)

5714(24)

0

0

0

0

01

1

2

2

3

3

(e)

10000

(35)

9333

(14)

8571

(36)

0

0

01

1

2

2

3

3

714(3)

714(3)

667(1)

(f)

10000

(38)

9286

(26)

5714

(24)

0

0

01

1

2

2

3

3

238(1)

4048(17)

714(2)

(g)








Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]



Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686











Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Cluster 1

Cluster 2Cluster 3

FCM algorithm





119895 Here 119906






119894119895= 119891119911 for


119891119911 =

1

119870

(8)




119862)




Art2[2 1000] 2 1000 2 (500 500)

Art3[4 400] 2 1100 4 (500 50 500 50)

Art4[2 200] 2 200 2 (100 100)

Art5[9 900] 2 900 9 (100 100 100)

Art6[4 400] 8 400 4 (100 100 100 100)

Iris[3 150] 4 150 3 (50 50 50)

Thyroid[3 215] 5 215 3 (150 35 30)

Pima[2 768] 8 768 2 (500 268)





Ec = 1 minus

sum119870

119896=1Recall

119896

119870

Recall119896=

119899119896119896

119899119896

(9)


Accuracy =sum119870

119896=1119899119896119896

119873

(10)

where 119899119896119896



119896 and119873 is





Iris[3 150]

5 0310 0663 287010 0320 0651 281320 0309 0662 304330 0275 0696 283740 0292 0667 294350 0249 0724 2680100 0251 0729 2870


119878 =

sum119870

119894=1sum119873

119895=1(120583119894119895)

2 10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2

119873 lowastmin119894119895

10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2 (11)


119894119895(119894 = 1 119870 119895 =


119894119895is defined as 119908

119894119895fl 1

if argmax119894119906119894119895 119908119894119895















272829

33132333435

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(a)

26

27

28

29

3

31

32

33

34

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(b)

10 20 30 40 7060 80 9050 1000Iterations

26

27

28

29

3

31

32

33

Aver

age l

engt

h of

bes

t tou

r

(c)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(d)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(e)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(f)

Aver

age l

engt

h of

bes

t tou

r

33

32

31

3

29

28

27

2610 20 30 40 50 60 70 80 90 1000

Iterations(g)





Art1[4 400]

119870-means 0806 0180 0947 05225 09950119870-medoid 0773 0158 0760 05150 09750


Art2[2 1000]

119870-means 0548 0490 0980 0020 0980119870-medoid 0404 0482 0020 0020 0980

FCM 0980 2278 times 10minus16 0980 0980 0980

ASClass 0039 0 0039 0039 0039

Art3[4 100]

119870-means 0749 0227 0585 0222 0989119870-medoid 0810 0141 0549 0549 0995


Art4[2 200]

119870-means 0450 0510 0 0 1119870-medoid 0572 0494 1 0 1


Art5[9 900]

119870-means 0908 0065 0873 0802 0990119870-medoid 0883 0133 0985 0455 0996


Art6[4 400]

119870-means 0641 0273 0750 0 1119870-medoid 0827 0176 1 0427 1

FCM 0750 0 0750 0750 0750ASClass 0089 0002 0088 0085 0095

Iris[3 150]

119870-means 0576 0336 0480 0113 1119870-medoid 0710 0229 0126 0126 1


Thyroid[3 215]

119870-means 0774 0166 0922 0239 0922119870-medoid 0594 0261 0666 0128 0952


Pima[2 768]

119870-means 0461 0125 0371 0371 0628119870-medoid 0504 0116 0616 0325 0641



0

01

02

03

04

05

06

07

08

09

1

1080701 040 0503 090602

1

2

3







(40)

789(3)

2105(8)

7105(27)

0 1481(4)

8519(23)

1 2

1

2

3

3

0 010000

(a)

10000

(41)

769

(3)

(3)

1538

(6)

7692

(30)

0

0

0

1304 8696

(20)

1

1

2

2

3

3

(b)

10000

(42)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(c)

10000

(29)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(d)

10000

(38)

10000

(26)

4286(18)

5714(24)

0

0

0

0

01

1

2

2

3

3

(e)

10000

(35)

9333

(14)

8571

(36)

0

0

01

1

2

2

3

3

714(3)

714(3)

667(1)

(f)

10000

(38)

9286

(26)

5714

(24)

0

0

01

1

2

2

3

3

238(1)

4048(17)

714(2)

(g)








Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]



Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686











Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Iris[3 150]

5 0310 0663 287010 0320 0651 281320 0309 0662 304330 0275 0696 283740 0292 0667 294350 0249 0724 2680100 0251 0729 2870


119878 =

sum119870

119894=1sum119873

119895=1(120583119894119895)

2 10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2

119873 lowastmin119894119895

10038171003817100381710038171003817119874119895minus 120592119894

10038171003817100381710038171003817

2 (11)


119894119895(119894 = 1 119870 119895 =


119894119895is defined as 119908

119894119895fl 1

if argmax119894119906119894119895 119908119894119895















272829

33132333435

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(a)

26

27

28

29

3

31

32

33

34

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(b)

10 20 30 40 7060 80 9050 1000Iterations

26

27

28

29

3

31

32

33

Aver

age l

engt

h of

bes

t tou

r

(c)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(d)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(e)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(f)

Aver

age l

engt

h of

bes

t tou

r

33

32

31

3

29

28

27

2610 20 30 40 50 60 70 80 90 1000

Iterations(g)





Art1[4 400]

119870-means 0806 0180 0947 05225 09950119870-medoid 0773 0158 0760 05150 09750


Art2[2 1000]

119870-means 0548 0490 0980 0020 0980119870-medoid 0404 0482 0020 0020 0980

FCM 0980 2278 times 10minus16 0980 0980 0980

ASClass 0039 0 0039 0039 0039

Art3[4 100]

119870-means 0749 0227 0585 0222 0989119870-medoid 0810 0141 0549 0549 0995


Art4[2 200]

119870-means 0450 0510 0 0 1119870-medoid 0572 0494 1 0 1


Art5[9 900]

119870-means 0908 0065 0873 0802 0990119870-medoid 0883 0133 0985 0455 0996


Art6[4 400]

119870-means 0641 0273 0750 0 1119870-medoid 0827 0176 1 0427 1

FCM 0750 0 0750 0750 0750ASClass 0089 0002 0088 0085 0095

Iris[3 150]

119870-means 0576 0336 0480 0113 1119870-medoid 0710 0229 0126 0126 1


Thyroid[3 215]

119870-means 0774 0166 0922 0239 0922119870-medoid 0594 0261 0666 0128 0952


Pima[2 768]

119870-means 0461 0125 0371 0371 0628119870-medoid 0504 0116 0616 0325 0641



0

01

02

03

04

05

06

07

08

09

1

1080701 040 0503 090602

1

2

3







(40)

789(3)

2105(8)

7105(27)

0 1481(4)

8519(23)

1 2

1

2

3

3

0 010000

(a)

10000

(41)

769

(3)

(3)

1538

(6)

7692

(30)

0

0

0

1304 8696

(20)

1

1

2

2

3

3

(b)

10000

(42)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(c)

10000

(29)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(d)

10000

(38)

10000

(26)

4286(18)

5714(24)

0

0

0

0

01

1

2

2

3

3

(e)

10000

(35)

9333

(14)

8571

(36)

0

0

01

1

2

2

3

3

714(3)

714(3)

667(1)

(f)

10000

(38)

9286

(26)

5714

(24)

0

0

01

1

2

2

3

3

238(1)

4048(17)

714(2)

(g)








Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]



Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686











Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




272829

33132333435

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(a)

26

27

28

29

3

31

32

33

34

Aver

age l

engt

h of

bes

t tou

r

10 20 30 40 50 60 70 80 90 1000Iterations(b)

10 20 30 40 7060 80 9050 1000Iterations

26

27

28

29

3

31

32

33

Aver

age l

engt

h of

bes

t tou

r

(c)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(d)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(e)

10 20 30 40 50 60 70 80 90 1000Iterations

27

275

28

285

29

295

3

305

31

315

Aver

age l

engt

h of

bes

t tou

r

(f)

Aver

age l

engt

h of

bes

t tou

r

33

32

31

3

29

28

27

2610 20 30 40 50 60 70 80 90 1000

Iterations(g)





Art1[4 400]

119870-means 0806 0180 0947 05225 09950119870-medoid 0773 0158 0760 05150 09750


Art2[2 1000]

119870-means 0548 0490 0980 0020 0980119870-medoid 0404 0482 0020 0020 0980

FCM 0980 2278 times 10minus16 0980 0980 0980

ASClass 0039 0 0039 0039 0039

Art3[4 100]

119870-means 0749 0227 0585 0222 0989119870-medoid 0810 0141 0549 0549 0995


Art4[2 200]

119870-means 0450 0510 0 0 1119870-medoid 0572 0494 1 0 1


Art5[9 900]

119870-means 0908 0065 0873 0802 0990119870-medoid 0883 0133 0985 0455 0996


Art6[4 400]

119870-means 0641 0273 0750 0 1119870-medoid 0827 0176 1 0427 1

FCM 0750 0 0750 0750 0750ASClass 0089 0002 0088 0085 0095

Iris[3 150]

119870-means 0576 0336 0480 0113 1119870-medoid 0710 0229 0126 0126 1


Thyroid[3 215]

119870-means 0774 0166 0922 0239 0922119870-medoid 0594 0261 0666 0128 0952


Pima[2 768]

119870-means 0461 0125 0371 0371 0628119870-medoid 0504 0116 0616 0325 0641



0

01

02

03

04

05

06

07

08

09

1

1080701 040 0503 090602

1

2

3







(40)

789(3)

2105(8)

7105(27)

0 1481(4)

8519(23)

1 2

1

2

3

3

0 010000

(a)

10000

(41)

769

(3)

(3)

1538

(6)

7692

(30)

0

0

0

1304 8696

(20)

1

1

2

2

3

3

(b)

10000

(42)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(c)

10000

(29)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(d)

10000

(38)

10000

(26)

4286(18)

5714(24)

0

0

0

0

01

1

2

2

3

3

(e)

10000

(35)

9333

(14)

8571

(36)

0

0

01

1

2

2

3

3

714(3)

714(3)

667(1)

(f)

10000

(38)

9286

(26)

5714

(24)

0

0

01

1

2

2

3

3

238(1)

4048(17)

714(2)

(g)








Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]



Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686











Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Art1[4 400]

119870-means 0806 0180 0947 05225 09950119870-medoid 0773 0158 0760 05150 09750


Art2[2 1000]

119870-means 0548 0490 0980 0020 0980119870-medoid 0404 0482 0020 0020 0980

FCM 0980 2278 times 10minus16 0980 0980 0980

ASClass 0039 0 0039 0039 0039

Art3[4 100]

119870-means 0749 0227 0585 0222 0989119870-medoid 0810 0141 0549 0549 0995


Art4[2 200]

119870-means 0450 0510 0 0 1119870-medoid 0572 0494 1 0 1


Art5[9 900]

119870-means 0908 0065 0873 0802 0990119870-medoid 0883 0133 0985 0455 0996


Art6[4 400]

119870-means 0641 0273 0750 0 1119870-medoid 0827 0176 1 0427 1

FCM 0750 0 0750 0750 0750ASClass 0089 0002 0088 0085 0095

Iris[3 150]

119870-means 0576 0336 0480 0113 1119870-medoid 0710 0229 0126 0126 1


Thyroid[3 215]

119870-means 0774 0166 0922 0239 0922119870-medoid 0594 0261 0666 0128 0952


Pima[2 768]

119870-means 0461 0125 0371 0371 0628119870-medoid 0504 0116 0616 0325 0641



0

01

02

03

04

05

06

07

08

09

1

1080701 040 0503 090602

1

2

3







(40)

789(3)

2105(8)

7105(27)

0 1481(4)

8519(23)

1 2

1

2

3

3

0 010000

(a)

10000

(41)

769

(3)

(3)

1538

(6)

7692

(30)

0

0

0

1304 8696

(20)

1

1

2

2

3

3

(b)

10000

(42)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(c)

10000

(29)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(d)

10000

(38)

10000

(26)

4286(18)

5714(24)

0

0

0

0

01

1

2

2

3

3

(e)

10000

(35)

9333

(14)

8571

(36)

0

0

01

1

2

2

3

3

714(3)

714(3)

667(1)

(f)

10000

(38)

9286

(26)

5714

(24)

0

0

01

1

2

2

3

3

238(1)

4048(17)

714(2)

(g)








Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]



Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686











Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




(40)

789(3)

2105(8)

7105(27)

0 1481(4)

8519(23)

1 2

1

2

3

3

0 010000

(a)

10000

(41)

769

(3)

(3)

1538

(6)

7692

(30)

0

0

0

1304 8696

(20)

1

1

2

2

3

3

(b)

10000

(42)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(c)

10000

(29)

10000

(26)

789(3)

2895(11)

6316(24)

0

0 0

01

1

2

2

3

3

(d)

10000

(38)

10000

(26)

4286(18)

5714(24)

0

0

0

0

01

1

2

2

3

3

(e)

10000

(35)

9333

(14)

8571

(36)

0

0

01

1

2

2

3

3

714(3)

714(3)

667(1)

(f)

10000

(38)

9286

(26)

5714

(24)

0

0

01

1

2

2

3

3

238(1)

4048(17)

714(2)

(g)








Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]



Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686











Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Art1[4 400]

119870-means 0193 0180 0052 0052 0477119870-medoid 0226 0158 0050 0025 0485


Art2[2 1000]

119870-means 0452 0490 0020 0020 0980119870-medoid 0596 0482 0980 0020 0980


Art3[4 100]



Art4[2 200]

119870-means 0550 0510 1 0 1119870-medoid 0427 0494 0 0 1


Art5[9 900]

119870-means 0091 0065 0126 0010 0197119870-medoid 0116 0133 0014 0003 0544


Art6[4 400]

119870-means 0358 0273 0250 0 1119870-medoid 0172 0176 0 0 0572

FCM 0250 0 0250 0250 0250ASClass 0918 0002 0918 0912 0921

Iris[3 150]

119870-means 0424 0336 0520 0 0886119870-medoid 0289 0229 0 0 0873


Thyroid[3 215]

119870-means 0212 0281 0032 0032 0888119870-medoid 0448 0341 0697 0023 0944


Pima[2 768]

119870-means 0550 0164 0668 0332 0668119870-medoid 0505 0158 0334 0320 0686











Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]


5 Conclusion






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Art1[4 400]


Art2[2 1000]


Art3[4 100]


Art4[2 200]


Art5[9 900]


Art6[4 400]


Iris[3 150]


Thyroid[3 215]


Pima[2 768]
















Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in














Art5[9 900]FCM 0001 0 0001 0001 0001










Competing Interests


Acknowledgments


References

















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Research Article Fuzzy Rules for Ant Based Clustering ...

Documents

Transcript of Research Article Fuzzy Rules for Ant Based Clustering ...