Research Article Intuitionistic Fuzzy Possibilistic C Means Clustering Algorithms

Transcription

1 Hindawi Publishing Cororation Advanes in Fuzzy Systems Volume 2015, Artile ID 2827, 17 ages htt://dx.doi.org/ /2015/2827 Researh Artile Intuitionisti Fuzzy Possibilisti C Means Clustering Algorithms Arindam Chaudhuri Samsung Researh & Develoment Institute, Noida 20104, India Corresondene should be addressed to Arindam Chaudhuri; arindamhdthesis@gmail.om Reeived 24 August 2014; Aeted 4 Otober 2014 Aademi Editor: Ferdinando Di Martino Coyright 2015 Arindam Chaudhuri. This is an oen aess artile distributed under the Creative Commons Attribution Liense, whih ermits unrestrited use, distribution, and rerodution in any medium, rovided the original work is roerly ited. Intuitionisti fuzzy sets (IFSs) rovide mathematial framework based on fuzzy sets to desribe vagueness in data. It finds interesting and romising aliations in different domains. Here, we develo an intuitionisti fuzzy ossibilisti C means (IFPCM) algorithm to luster IFSs by hybridizing onets of FPCM, IFSs, and distane measures. IFPCM resolves inherent roblems enountered with information regarding membershi values of objets to eah luster by generalizing membershi and nonmembershi with hesitany degree. The algorithm is extended for lustering interval valued intuitionisti fuzzy sets (IVIFSs) leading to interval valued intuitionisti fuzzy ossibilisti C means (IVIFPCM). The lustering algorithm has membershi and nonmembershi degrees as intervals. Information regarding membershi and tyiality degrees of samles to all lusters is given by algorithm. The exeriments are erformed on both real and simulated datasets. It generates valuable information and rodues overlaed lusters with different membershi degrees. It takes into aount inherent unertainty in information atured by IFSs. Some advantages of algorithms are simliity, flexibility, and low omutational omlexity. The algorithm is evaluated through luster validity measures. The lustering auray of algorithm is investigated by lassifiation datasets with labeled atterns. The algorithm maintains areiable erformane omared to other methods in terms of ureness ratio. 1. Introdution Clustering algorithms [1, 2]formanintegralartofomutational intelligene and attern reognition researh. Clusteringanalysisisommonlyusedasanimortanttooltolassify olletion of objets into homogeneous grous, suh that objetswithinagivengrouaresimilartoeahotherwhereas objets within different grous are dissimilar to eah other. The onet is based on the notion of similarity, whih is a basi omonent of intelligene and ubiquitous to sientifi endeavor. Clustering finds numerous aliations [] aross a variety of disilines suh as taxonomy, image roessing, information retrieval, data mining, attern reognition, mirobiology, arhaeology and geograhial analysis, and so forth. It is an exloratory tool for deduing the nature of data by roviding labels to individual objets that desribe how the data searate into grous. It has imroved the erformane of other systems by searating the roblem domain into manageable subgrous [4]. Often researhers are onfronted with the hallenging datasets that are large and unlabeled. There are many methods available in exloratory data analysis [5, 6] by whih researhers an eluidate these data. Clustering an unlabeled dataset X={x 1,...,x n } R is artitioning of X into 1 < < n subgrous suh that eah subgrou reresents natural substruture in X. This is done by assigning labels to vetors in X and hene to objets generating X.A artition of is a set of n values that an be onveniently reresented as nmatrix =[u ik ].Thereare generally three sets of artition matries [7, 8]: M n ={U R n :0 u ik 1 i,k; k i u ik >0} (1) M fn ={U M n : u ik =1 k; u ik >0 i} (2) M hn ={U M fn :u ik =0 1 i k}. () The matrix M n in (1) has the roerty that for any k there exists at least an index i suh that u ik is greater than 0. The matrix M fn in (2) states that if u ik is equal to 1foranyk, itisobviousthat u ik is greater than 0.

2 2 Advanes in Fuzzy Systems The matrix M hn in () is formed by boolean matries as a subset of matrix M fn in (2). Theequations(1), (2), and() thus define the sets of ossibilisti, fuzzy, or robabilisti and ris artitionsof X, resetively. Hene, there are thus four kinds of label vetors, but fuzzy and robabilisti label vetors are mathematially idential having entries between 0 and 1 that sum to 1 over eah olumn. The reason these matries are alled artitions follows from the interretation of their entries. If U is ris or fuzzy, u ik is taken as a membershi of x k in ith artitioning fuzzy subset or luster of X. If U in M fn is robabilisti, u ik is the osterior robability Prob(i/x k ).IfUin M n is ossibilisti, it has entries between 0and1thatdonotneessarilysumto1overanyolumn. In this ase, u ik is taken as the ossibility that x k belongs to lass i. An alternate interretation of ossibility u ik is that it measures the tyiality of x k to luster i. Itisobserved that M hn M fn M n. A lustering algorithm C finds U M hn (M fn,m n ) whih best exlains and reresents an unknown struture in X withresettothemodelthat defines C.ForU in M fn,=1is reresented uniquely by the hard 1 artition, 1 n =[1 1] whih unequivoally assigns all n times n objets to a single luster, and = nis reresented uniquely by U = I n,then nidentity matrix u to a ermutation of olumns. In this ase, eah objet is in its own singleton luster. Choosing =1or =nrejets thehyothesisthatx ontains lusters. In the last few deades, variety of lustering tehniques [, 5, 6, 9, 10] has been develoed to lassify data. Clustering tehniques are broadly divided into hierarhial and artition methods. Hierarhial lustering [5] generates hierarhial tree of lusters alled dendrogram whih an be either divisive or agglomerative []. The former is a to-down slitting tehnique whih starts with all objets in one luster and forms hierarhy by dividing objets into smaller lusters in an iterative roedure until the desired number of lusters is ahieved or onsidered objets whih is onstituted as unique luster. The latter starts by onsidering eah objet as luster, followed by omaring them amongst themselves using distane measure. The lusters with smaller distane are onsidered as onstituting unique grou and then merged. The merging roedure is reeated until the desirable number of lusters is ahieved or only one luster is left with all onsidered objets. Partition lustering method gives single artition of objets, with being the redefined number of lusters [11]. One of the most widely used artition lustering algorithms is fuzzy C means (FCM). FCM is a ombination of k means lustering algorithm and fuzzy logi [1, 7]. It works iteratively in whih the desired number of lusters and initial seeds are redefined. FCM algorithm assigns membershis to x k whih are inversely roortional to relative distane of x k to oint rototyes {k i } that are luster enters. For = 2,ifx k isequidistantfromtwo rototyes, the membershi of x k in eah luster will be the same regardless of absolute value of the distane of x k from two entroids as well as from other oints. The roblem this reates is noise oints whih are far but equidistant from entral struture of two lusters that an never be given equal membershi, when it seems far more natural that suh oints are given very low or no membershi in either luster. This roblem was overome by Krishnauram and Keller [8], who roosed ossibilisti C means (PCM) whih relaxes olumn sum onstraint in (2) so that sum of eah olumn satisfies the onstraint 0 < u ik. In other words, eah element of kth olumn an be between 0 and 1, as long as at least oneofthemisositive.theysuggestedthatvalueshould be interreted as tyiality of x k relative to luster i. They interreted eah row of U as ossibility distribution over X. The objetive funtion of PCM algorithm sometimes hels to identifyoutliers,thatis,noiseoints.however,barnietal. [12] ointed that PCM ays rie for its freedom to ignore noiseointssuhthatitisverysensitivetoinitializationsand sometimes generates oinident lusters. Moreover, tyiality an be very sensitive to the hoie of additional arameters needed by PCM algorithm. The oinident luster roblem of PCM algorithm was avoided by two ossibilisti fuzzy lustering algorithms roosed by Timm et al. [1 15]. They modified PCM objetive funtion by adding an inverse funtion of distanes between the luster enters. This term ats in reulsive nature and avoids oinident lusters. In [1, 14], Timm et al. used the same onet to modify objetive funtion as used by Gustafson and Kessel [16] lustering algorithm. These algorithms exloit benefits of both fuzzy and ossibilisti lustering. Pal et al. [17] justified the need for both ossibility, that is, tyiality and membershi values, and roosed a model and orresonding algorithm alled fuzzy ossibilisti C means (FPCM). This algorithm normalizes ossibility values, so that the sum of ossibilities of all data oints in a luster is 1. Although FPCM is muh less rone to errors enountered by FCM and PCM, ossibility values are very small when size of dataset inreases. The notion of intuitionisti fuzzy set (IFS) oined by Atanassov [22] for fuzzy set generalizations has interesting and useful aliations in different domains suh as logi rogramming, deision making roblems, and medial diagnostis [2 26]. This generalization resents degrees of membershi and nonmembershi with a degree of hesitany. Thus, knowledge and semanti reresentation beome more meaningful and aliable [27, 28]. Sometimes it is not aroriate to assume that membershi and nonmembershi degree of an objet are exatly defined [29], but value ranges or value intervals an be assigned. In suh ases, IFS an be generalized and interval valued intuitionisti fuzzy set (IVIFS) [29] an be defined whose omonents are intervals rather than exat numbers. IFSs and IVIFSs have been found to be very useful to desribe and deal with vague and unertain data [28, 0]. With this motivation, it is desirable to develo some ratial aroahes to lustering IFSs and IVIFSs. Intuitionisti fuzzy similarity matrix was defined by [1] and thereby intuitionisti fuzzy equivalene matrix was develoed. The work in [1] gaveanaroahtotransform intuitionisti fuzzy similarity matries into intuitionisti fuzzy equivalene matries, based on whih a roedure for lustering intuitionisti fuzzy sets was roosed. Some methods for alulating assoiation oeffiients of IFSs or IVIFSs and orresonding lustering algorithm were introdued by [2]. The algorithm used derived assoiation oeffiients of IFSs or IVIFSs to onstrut an assoiation matrix and utilized

3 Advanes in Fuzzy Systems the roedure to transform it into an equivalent assoiation matrix. Referene [] introdued an intuitionisti fuzzy hierarhial algorithm for lustering IFSs whih is based on traditional hierarhial lustering roedure and intuitionisti fuzzy aggregation oerator. These algorithms annot rovide information about membershi degrees of objets to eah luster. In this work, an intuitionisti fuzzy ossibilisti C means (IFPCM) algorithm to luster IFSs is develoed. IFPCM is obtained by alying IFSs to FPCM whih is a known lustering method based on basi distane measures between IFSs [4, 5]. At eah stage of the algorithm seeds are modified and for eah IFS membershi and tyiality degrees to eah of the lusters are estimated. The algorithm ends when all given IFSs are lustered aording to estimated membershi and tyiality degrees. It overomes the inherent roblems enountered with information regarding membershi values of objets to eah luster by generalizing membershi and nonmembershi with hesitany degree. The algorithm is then extended to interval valued intuitionisti fuzzy ossibilisti C means (IVIFPCM) for lustering IVIFSs. The algorithms are illustrated through onduting exeriments on different datasets. The evaluation of the algorithm is erformed through luster validity measures. The lustering auray of the algorithm is determined by lassifiation datasets with labeled atterns. IFPCM algorithm is simle and flexible in nature and rovides information about membershi and tyiality degrees of samles to all lusters with low omutational omlexity. This aer is organized as follows. In the next setion, the onets of IFSs and IVIFSs are defined. FPCM lustering algorithm is given in Setion. The next setion resents IFPCM lustering algorithms for IFSs and IVIFSs, resetively. The exerimental results on both real world and simulated datasets are illustrated in Setion 5.Finally,in Setion 6 onlusions are given. 2. Intuitionisti Fuzzy Sets and Interval Valued Intuitionisti Fuzzy Sets In this setion, we resent some basi definitions assoiated with IFSs and IVIFSs. Definition 1. Considering X as universe of disourse [22], then IFS is defined as V={ x, μ V (x), V V (x) x X}. (4) In (4) μ V (x) and V V (x) are the membershi and nonmembershi degrees, resetively, satisfying the following onstraints: μ V :X [0, 1], x X μ V (x) [0, 1], V V :X [0, 1], x X V V (x) [0, 1]. Equation (5) issubjettotheonditionthatμ V (x)+v V (x) 1, x X. (5) Definition 2. For eah IFS V in X, ifπ V (x) = 1 μ V (x) V V (x), thenπ V (x) is alled hesitation degree (or intuitionisti index) [6] ofx to V. Obviouslyπ V (x) is seified in the range 0 π V (x) 1; eseiallyifπ V (x) = 0 x X, then IFS V is redued to fuzzy set. If π V (x) and V V (x) have 0 values suh that μ V (x) = V V (x) = 0 x X, thenifsv is omletely intuitionisti. Considering the fat that the elements x i ; i = 1,...,n in universe X have different imortane, let us assume w= (w 1,...,w n ) should be the weight vetor of x i ; i = 1,...,n with w i 0, n w i =1. (6) Xu [7] defined the following weighted Eulidean distane between IFSs V and W: D α (V, W) =( 1 2 n w i ((μ V (x i ) μ W (x i )) 2 +(V V (x i ) V W (x i )) 2 +(π V (x i ) π W (x i )) 2 )) In artiular, if w = (1/n,...,1/n), then(7) is redued to normalized Eulidean distane [4] whih is defined as follows: D α (V, W) n =( 1 ((μ 2n V (x i ) μ W (x i )) 2 +(V V (x i ) V W (x i )) 2 +(π V (x i ) π W (x i )) 2 )) Atanassov and Gargov [29] ointed out that sometimes it is not aroriate to assume that membershi and nonmembershi degrees of the element are exatly defined but value ranges or value intervals an be given. In this ontext, Atanassov and Gargov [29] extended IFS and introdued the onetofivifs,whihisharaterizedbyamembershi degree and a nonmembershi degree, whose values are intervals rather than exat numbers. Definition. An IVIFS V over X is an objet having the following form [29]: 1/2. 1/2. (7) (8) V ={ x, μ V (x), Ṽ V (x) x X}. (9) Here, μ V (x) = (x), (x)] [0, 1] and [ μl V μu V Ṽ V (x) = (x), [ṼL V Ṽ (x)] [0, 1] are intervals (x) = inf U V μl V μ V (x), (x) = μu V su μ V (x), (x) = inf ṼL V Ṽ V (x), (x) = su ṼU V Ṽ V (x)+ (x),and μu V Ṽ (x) 1, x X and U V π V (x) = (x), (x)], where [ πl V πu V

4 4 Advanes in Fuzzy Systems π (x) = (x), (x) = (x), x X. L V 1 μu V (x) VU V πu V 1 μl V (x) VL V In artiular, if μ V = (x) = (x) and μl V μu V V V (x) = (x) = ṼL V Ṽ U V (x),then V isreduedtoanifs. Now we extend the weighted Eulidean distane measure givenin (7) to IVIFS theory: D γ ( V, W) =( 1 4 n l=1 w l (( μ L V (x l) μ L W (x l)) 2 +( μ U V (x l) μ U W (x l)) 2 +(Ṽ L V (x l) Ṽ L W (x l)) 2 +(Ṽ U V (x l) Ṽ U W (x l)) 2 +( π L V (x l) π L W (x l)) 2 +( π U V (x l) π U W (x l)) 2 )) 1/2. (10) Partiularly, if w = (1/n,...,1/n), then(10) is redued to normalized Eulidean distane whih is given as follows: D γ ( V, W) n =( 1 w 4n l (( μ L V (x l) μ L W (x l)) 2 l=1 +( μ U V (x l) μ U W (x l)) 2 +(Ṽ L V (x l) Ṽ L W (x l)) 2 +(Ṽ U V (x l) Ṽ U W (x l)) 2 +( π L V (x l) π L W (x l)) 2 +( π U V (x l) π U W (x l)) 2 )). Fuzzy Possibilisti C Means Clustering Algorithm 1/2. (11) This setion illustrates FPCM lustering algorithm roosed by Pal et al. [17] in 1997 to exloit the benefits of fuzzy and ossibilisti modeling while irumventing their weaknesses. To orretly interret the data substruture, FPCM lustering uses both membershis (relative tyiality) and ossibilities (absolute tyiality). When we want to risly label a data oint, membershi is a lausible hoie as it is natural to assign a oint to luster whose rototye is losest to the oint. On the other hand, while estimating the entroids, tyiality is an imortant means for alleviating the undesirable effets of outliers. Here, the number of lusters is fixed a riori to a default value onsidering the dataset used in the aliation suh that it is omletely data driven. Generally it is advisable to avoid trivial lusters whih may be either too large or small. FPCM extends FCM lustering algorithm [17] by normalizing ossibility values so that sum of ossibilities of all data oints in a luster is 1. Although FPCM is muh less rone to the roblems of both FCM and PCM, the ossibility values are very small when size of the dataset inreases. Analogous to FCM lustering algorithm, the membershi term in FPCM is a funtion of data oint and all entroids. The tyiality term in FPCM is a funtion of data oint and luster rototye alone. That is, the membershi term is influened by the ositions of all luster enters whereas tyiality term is affeted by only one. Inororating the abovementioned faets the FPCM model is defined by the following otimization roblem [17]: min {J m,η (U, T, V; X) = (u m ik +tη ik )D2 ik } (U,T,V) k=1 subjet to m > 1, η > 1, 0 < u ik, t ik <1 k=1 (12) u ik =1 ki.e., U M fn (1) t ik =1 ii.e., T t M fn. (14) The transose of admissible T s is member of set M fn. T is viewed as a tyiality assignment of n objets to lusters. The ossibilisti term k=1 tη ik D2 ik distributes {t ik} with reset to all n data oints, but not with reset to all lusters. Under the usual onditions laed on -means otimization roblems, the first order neessary onditions for extrema of J m,η arestatedintermsofthefollowing theorem. Theorem FPCM (see [17]). If D ik = x k v i >0 iand k, m, η > 1 and X ontains at least distint data oints, then (U, T t, V) M fn M fn R may minimize J m,η only if u ik =( t ik =( n ( D 2/(m 1) 1 ik ) ) ; 1 i, 1 k n, (15) D jk ( D 2/(η 1) 1 ik ) ) ; 1 i, 1 k n, (16) D ij k i = n k=1 (um ik +tη ik ) x k n ; 1 i. (17) k=1 (um ik +tη ik ) The roof of the above theorem follows from [8]. FPCM has the same tye of singularity as FCM. FPCM does not suffer from the sensitivity roblem that PCM seems to exhibit. Unfortunately, when the number of data oints is large, the tyiality values will be very small. Thus, after FPCM-AO algorithm [8] for aroximating solutions to (12) based on iteration through (17) terminates, the tyiality values may need to be saled u. Conetually, this is not different than saling tyiality as is done in PCM. While saling seems to solve the small value roblem whih is aused by row

5 Advanes in Fuzzy Systems 5 sum onstraint on T, the saled values do not ossess any additional information about oints in the data. Thus saling {t ik } is an artifiial fix for a mathematial drawbak of FPCM. 4. Intuitionisti Fuzzy Possibilisti C Means Clustering Algorithms In this setion, we disuss intuitionisti fuzzy ossibilisti C means lustering algorithms for IFSs and IVIFSs, resetively Intuitionisti Fuzzy Possibilisti C Means Algorithm for IFSs. We develo the intuitionisti fuzzy ossibilisti C means (IFPCM) model and orresonding algorithm for IFSs. We take the basi distane measure D j in (7) as roximity funtion of IFPCM; the objetive funtion of IFPCM model an then be defined as follows: min (U,T,V) {J m,η (U, T, V; X) = (u m ik +tη ik )D2 α (Z k,v i )} k=1 subjet to m > 1, η > 1, 0 < u ik, t ik <1 k=1 u ik =1 ki.e., U M fn t ik =1 ii.e., T t M fn. (18) Here Z={Z 1,...,Z } are IFSs eah with n elements, is thenumberoflusters(1 ), andv = {V 1,...,V } are the rototyial IFSs, that is, entroid of the lusters. The arameter m is the fuzzy fator, u ik is the membershi degree of jth samle Z j to the ith luster, U = (u ik ) is matrix of order, arameter η is the tyiality fator, t ik is the tyiality of jth samle Z j to the ith luster, and T=(t ik ) is tyiality matrix. To solve the otimization roblem stated in (18),wemake use of Lagrange multilier method [9], whih is disussed below. Considering L= (u m ik +tη ik )D2 α (Z k,v i ) where, k=1 k=1 ξ k ( D 2 α (Z k,v i )= 1 2 t ik 1), n l=1 k=1 λ k ( u ik 1) w l ((μ Zk (x l ) μ Vi (x l )) 2 ) +(V Zk (x l ) V Vi (x l )) 2 +(π Zk (x l ) π Vi (x l )) 2. (19) (20) Furthermore, 1 i, 1 k ; let =0, u ik =0, t ik =0, λ k =0. ξ k (21) From the above system of equations, we have the following exressions: u ik = t ik = 1 r=1 (D α(z k,v i )/D α (Z k,v r )) 2/(m 1) ; 1 i, 1 k 1 r=1 (D α (Z k,v i )/D α (Z k,v r )) 2/(η 1) ; 1 i, 1 k. (22) (2) Now we roeed to omute V i ; i = 1,...,, the rototyial IFSs. Let us assume that μ Vi (x l ) = V Vi (x l ) = Fromtheaboveexressionwehave π Vi (x l ) =0. (24) μ Vi (x l )= k=1 (um ik +tη ik )μ Z k (x l ) k=1 (um ik +tη ik ) ; 1 i, 1 l n, (25) V Vi (x l ) = k=1 (um ik +tη ik ) V Z k (x l ) k=1 (um ik +tη ik ) ; 1 i, 1 l n, (26) π Vi (x l )= k=1 (um ik +tη ik )π Z k (x l ) k=1 (um ik +tη ik ) ; 1 i, 1 l n. (27) For simliity, we define weighted average oerator for IFSs as follows. Let A = {A 1,...,A } be a set of IFSs eah with n elements; let ω = {ω 1,...,ω } be a set of weights for IFSs, resetively, with ω j =1; and then the weighted oerator f is defined as f (A, ω) = { { { x l, ω j μ Aj (x l ), ω j V Aj (x l ) 1 l n } }. } (28)

6 6 Advanes in Fuzzy Systems Aording to (25) to (28), if we assume ω (i) ={ (u i1 +t i1 ) k=1 (u ik +t ik ),..., (u i +t i) k=1 (u ik +t ik ) }; 1 i, (29) the rototyial IFSs V={V 1,...,V } of the IFPCM model anbeomutedasfollows: V i =f(z,ω (i) ) = { { { x l, ω (i) ω (i) j μ Z j (x l ), j V Z j (x l ) 1 l n } } } 1 i. (0) Sinetheaboveequations(22), (2), and(0) are omutationally interdeendent, we exloit an iterative roedure similar to the FPCM algorithm to solve these equations. The stes of algorithm are as follows. IFPCM Algorithm Ste 1. Initializethe seed valuesv(0);let x=0and set ε>0. Ste 2(i).CalulateU(x) = (u ik (x)),where (a) if k,r,d α (Z k,v r (x)) > 0, then u ik (x) = 1/( r=1 (D α(z k,v i (x))/d α (Z k,v r (x))) 2/(m 1) ); 1 i, 1 k, (b) if k, r suh that D α (Z k,v r (x)) = 0,thenletu rk (x) = 1 and u ik (x) = 0 i =r. Ste 2(ii).CalulateT(x) = (t ik (x)),where (a) if k, r, D α (Z k,v r (x)) > 0, then t ik (x) = 1/( r=1 (D α(z k,v i (x))/d α (Z k,v r (x))) 2/(η 1) ); 1 i, 1 k, (b) if k, r suh that D α (Z k,v r (x)) = 0,thenlett rk (x) = 1 and t ik (x) = 0 i =r. Ste.CalulateV(x + 1) = {V 1 (x+1),...,v (x + 1)},where ω (i) (x+1) V i (x+1) =f(z, ω (i) (x+1)), 1 i (1) ={ (u i1 (x) +t i1 (x)) k=1 (u ik (x) +t ik (x)),..., (u i (x) +t i (x)) k=1 (u ik (x) +t ik (x)) }; 1 i. (2) Ste 4.If (D α(v i (x), V i (x + 1))/) < ε,thengotoste5; otherwise, let x=x+1, andreturntoste2. Ste 5.End, The seudoode of the IFPCM algorithm is given in Algorithm Interval Valued Intuitionisti Fuzzy Possibilisti C Means Algorithm for IVIFSs. If the olleted data are exressed as IVIFSs, then we extend IFPCM to interval valued intuitionisti fuzzy ossibilisti C means (IVIFPCM) model. We take the basi distane measure D γ in (10) as the roximity funtion of the IVIFCM. The objetive funtion of IVIFPCM model an be defined as follows: min {J m,η (U, T, V; X) = (U,T, V) (u m ik +tη ik )D2 γ ( Z k, V i )} k=1 subjet to m > 1, η > 1, 0 < u ik, t ik <1 k=1 () u ik =1 ki.e., U M fn (4) t ik =1 ii.e., T t M fn. (5) Here Z ={ Z 1,..., Z } are IVIFSs eah with n elements, is thenumberoflusters(1<<),and V ={ V 1,..., V } are the rototyial IVIFSs, that is, entroids of the lusters. The arameter m is the fuzzy fator, u ik is the membershi degree of jth samle Z j to the ith luster, U = (u ik ) is matrix of order, arameter η is the tyiality fator, t ik is the tyiality of jth samle Z j to the ith luster, and T=(t ik ) is tyiality matrix. To solve the otimization roblem stated in (0) to (5), we make use of Lagrange multilier method [9], whih is disussed below. Considering L= (u m ik +tη ik )D2 γ ( Z k, V i ) where, k=1 k=1 ξ k ( D 2 γ ( Z k, V i )= 1 4 t ik 1), n l=1 k=1 λ k ( w l (( μ L Z k (x l ) μ L V i (x l )) 2 +( μ U Z k (x l ) μ U V i (x l )) 2 +(Ṽ L Z k (x l ) Ṽ L V i (x l )) 2 +(Ṽ U Z k (x l ) Ṽ U V i (x l )) 2 u ik 1) +( π L Z k (x l ) π L V i (x l )) 2 +( π U Z k (x l ) π U V i (x l )) 2 ). (6) (7)

7 Advanes in Fuzzy Systems 7 Given an unlabeled dataset X={x 1,...,x n }, artition X into 1<<nlusters suh that objetive funtion J m,η (U,T,V;X)is minimized (1) Inut: Consider the seed values V (0) and assume x=0and set ε>0 (2) Outut: Generate lusters using the IFPCM lustering algorithm for IFSs () begin roedure (4) reeat (5) alulate U(x) = (u ik (x)) (6) begin (7) if ( k, r, D α (Z k,v r (x)) >0)then 1 (8) u ik (x) = r=1 (D α (Z k,v i (x))/d α (Z k,v r (x))) ; 2/(m 1) 1 i,1 k (9) end if (10) if ( k, r, D α (Z k,v r (x)) =0)then (11) u rk (x) =1and u ik (x) =0 i=r (12) end if (1) end (14) alulate T(x) = (t ik (x)) (15) begin (16) if ( k, r, D α (Z k,v r (x)) >0)then 1 (17) t ik (x) = r=1 (D α (Z k,v i (x))/d α (Z k,v r (x))) ; 2/(η 1) 1 i,1 k (18) end if (19) if ( k, r, D α (Z k,v r (x)) =0)then (20) t rk (x) =1and t ik (x) =0 i=r (21) end if (22) end (2) alulate V (x+1) ={V 1 (x+1),...,v (x+1)};v i (x+1) =f(z,ω (i) (x+1)),1 i ;ω (i) (x+1) = {(u i1 (x) +t i1 (x))/( k=1 (u ik (x) +t ik (x))),...,(u i (x) +t i (x))/( k=1 (u ik (x) +t ik (x)))}, 1 i (24) x=x+1 (25) until ( (D α (V i (x),v i (x+1))/) < ε) (26) end roedure Algorithm 1 Similar to IFPCM model, we establish the system of artial differentialfuntionsofl as follows: =0; u ik =0; t ik =0, λ k μ L V i (x l ) = μ U V i (x l ) = 1 i, 1 k, 1 i, 1 k, ξ k =0; V L V i (x l ) = V U V i (x l ) = 1 k, π L V i (x l ) =0, π U V i (x l ) =0 1 i, 1 l. The solution for the above system of equations is u ik = 1 r=1 (D γ( Z k, V i )/D γ ( Z k, V r )) 2/(m 1) ; 1 i, 1 k (8) (9) t ik = V i = f( Z, ω (i) ) = { { x l, [ { [ 1 r=1 (D γ( Z k, V i )/D γ ( Z k, V r )) 2/(η 1) ; ω (i) ω (i) 1 i, ω (i) 1 k, j μl Z (x j l ), j μu Z (x j l )], ] ω (i) (40) [ j ṼL Z (x j l ), j ṼU Z (x j l )] 1 l n }, [ ] } 1 i, (41) where ω (i) ={ (u i1 +t i1 ) k=1 (u ik +t ik ),..., (u i +t i) k=1 (u ik +t ik ) }; 1 i. (42) Beause (41) and (42) are omutationally interdeendent, we exloit a similar iteration roedure as follows.

8 8 Advanes in Fuzzy Systems IVIFPCM Algorithm Ste 1. Initializethe seed values V(0);let x=0and set ε>0. Ste 2(i).CalulateU(x) = (u ik (x)),where () if k, r, D γ ( Z k, V r (x)) > 0, then u ik (x) = 1/( r=1 (D γ( Z k, V i (x))/d γ ( Z k, V r (x))) 2/(m 1) ); 1 i, 1 k. (d) if k, r suh that D γ ( Z k, V r (x)) = 0,thenletu rk (x) = 1 and u ik (x) = 0 i =r. Ste 2(ii).CalulateT(x) = (t ik (x)),where () if k, r, D γ ( Z k, V r (x)) > 0, then t ik (x) = 1/( r=1 (D γ( Z k, V i (x))/d γ ( Z k, V r (x))) 2/(η 1) ); 1 i, 1 k, (d) if k, r suh that D γ ( Z k, V r (x)) = 0,thenlett rk (x) = 1 and t ik (x) = 0 i =r. Ste.Calulate V(x + 1) = { V 1 (x + 1),..., V (x + 1)},where V i (x+1) = f( Z, ω (i) (x+1)), 1 i ω (i) (x+1) ={ (u i1 (x) +t i1 (x)) k=1 (u ik (x) +t ik (x)),..., (u i (x) +t i (x)) k=1 (u ik (x) +t ik (x)) }; 1 i. (4) Ste 4.If (D γ( V i (x), V i (x + 1))/) < ε,thengotoste5; otherwise let x=x+1,and return to Ste 2. Ste 5.End The seudoode of the IVIFPCM algorithm is given in Algorithm Exerimental Results In this setion, we enumerate the results of exeriments erformed on both real world and simulated datasets [2]in order to demonstrate the effetiveness of IFPCM lustering algorithm. IFPCM algorithm is imlemented through MAT- LAB. We first exlain the stes of the algorithm by the use of some exerimental data whih is evaluated through luster validity measures. Next, the algorithm is alied to some lassifiation datasets, that is, data with labeled atterns in order to examine its lustering auray Aliation of IFPCM Algorithm on Exerimental Data. The arameters set in IFPCM algorithm are shown in Table 1. It is to be noted that if π(x) = 0 x X, then IFPCM is redued to FPCM algorithm. Hene, we resent a omarative erformane of both the algorithms. The exerimental data used here is investment ortfolio dataset whih ontains information regarding ten investments at the disosal of the investor to invest some money to be lassified in ICICI rudential finanial servies, India. Let Table 1: IFPCM arameters. Parameter Desrition f Name of inut file Number of lusters (default value = ) m Fuzzy fator (default value = 2) η Tyiality fator (default value = 2) w Tye of samle weights (default value = 0 (equal); user seified value = 1) s Tye of initial entroids (default value = 0 (random); user seified value = 1) i Maximumnumberofiterationstill onvergene (default value = 100) t Thresholdforiterationsstoage (default value = 0.001) I i ;,...,10betheinvestments desribed by six attributes, namely a 1 : investment rie; a 2 : advane mobilization; a : time eriod; a 4 : return on investment; a 5 :riskfator;a 6 : seurity fator. The weight vetor of these attributes is w= (0.20, 0.10, 0.0, 0.15, 0.10, 0.15). Theharateristisoften investments under six attributes are reresented by IFSs in Table 2. Simulated datasets are used for omaring with the exerimental data. We assume that there are three lasses in the simulated dataset, C i ; i = 1, 2,. ThenumberofIFSs in eah lass is onsidered as 00. The different lasses have different IFSs whih are haraterized as follows: (a) IFSs in C 1 have relatively high and ositive sores, (b) IFSs in C 2 have relatively high and unertain sores, () IFSs in C have relatively high and negative sores. Considering this, we generate simulated dataset as follows: (a) μ(x) U(0.7, 1) V(x)+π(x) U(0, 1 μ(x)) x C 1, (b) V(x) U(0.7, 1) μ(x)+π(x) U(0, 1 V(x)) x C 2, () π(x) U(0.7, 1) μ(x)+v(x) U(0, 1 π(x)) x C. Here, U(a, b) is the uniform distribution on the interval [a, b]. We thus generate a simulated dataset whih onsists of lasses omrising 900 IFSs Cluster Validity Measures. In IFPCM algorithm big hallenge lies in setting the arameter, that is,the number of lusters. To resolve this, we use two relative measures for fuzzy luster validity mentioned in [40], namely, artition oeffiient (PC) and lassifiation entroy (CE). The desritions of these two measures are given in Table.InPCand CE is number of samles in the dataset IFPCM Algorithm on Investment Portfolio Dataset. IFPCM algorithm is used to luster ten investments I i ; i = 1,...,10, involving the following stes. Ste 1. Let=and ε = Now randomly selet initial entroids V(0) fromthe dataset: V (0) = [ [ I 9 I 10 I 7 ] ]. (44)

9 Advanes in Fuzzy Systems 9 Given an unlabeled dataset X={x 1,...,x n }, artition X into 1<<nlusters suh that objetive funtion J m,η (U,T,V;X)is minimized (1) Inut: Consider the seed values V (0) and assume x=0and set ε>0 (2) Outut: Generate lusters using the IFPCM lustering algorithm for IVIFSs () begin roedure (4) reeat (5) alulate U(x) = (u ik (x)) (6) begin (7) if ( k, r, D γ ( Z k, V r (x)) >0)then (8) 1 u ik (x) = r=1 (D γ ( Z k, V i (x))/d γ ( Z k, V ; r (x))) 2/(m 1) 1 i,1 k (9) end if (10) if ( k, r, D γ ( Z k, V r (x)) =0)then (11) u rk (x) =1and u ik (x) =0 i=r (12) end if (1) end (14) alulate T(x) = (t ik (x)) (15) begin (16) if ( k, r, D γ ( Z k, V r (x)) >0)then (17) 1 t ik (x) = r=1 (D γ ( Z k, V i (x))/d γ ( Z k, V ; r (x))) 2/(η 1) 1 i,1 k (18) end if (19) if (D γ ( Z k, V r (x)) =0)then (20) t rk (x) =1and t ik (x) =0 i=r (21) end if (22) end (2) alulate V (x+1) ={ V 1 (x+1),..., V (x+1)}; V i (x+1) = f( Z, ω (i) (x+1)),1 i ;ω (i) (x+1) = {(u i1 (x) +t i1 (x))/( k=1 (u ik (x) +t ik (x))),...,(u i (x) +t i (x))/( k=1 (u ik (x) +t ik (x)))}, 1 i (24) x=x+1 (25) until ( (D γ ( V i (x), V i (x+1))/) < ε) (26) end roedure Algorithm 2 Ste 2(i). Calulate the membershi degrees and entroids iteratively. Aording to (15), we have U (0) = (45) [ ] [ ] Ste 2(ii). Calulate the tyiality degrees and entroids iteratively. Aording to (16), we have T (0) = (46) [ ] [ ] Ste. Aording to (17), we udate the entroids as follows: 0.66, , , , , , 0.79 V (1) = [ 0.760, , , , , , ]. (47) [ 0.679, , , , , , 0.19 ]

10 10 Advanes in Fuzzy Systems Ste 4. We made a hek whether to sto the iterations: D α (V i (0),V i (1)) = > (48) Sine this value exeeds the hosen threshold value, we ontinue with the next iteration. When x=1, U (1) = , [ ] [ ] T (1) = , [ ] [ ] 0.54, , , , , , V (2) = [ 0.755, , , , , , 0.99 ], [ 0.56, , , , , , ] D α (V i (1),V i (2)) = > (49) Sine this value exeeds the hosen threshold value, we ontinue with the next iteration. When x=2, U (2) = , [ ] [ ] T (2) = , [ ] [ ]

11 Advanes in Fuzzy Systems , , , , , , V () = [ 0.765, , , , , , 0.96 ], [ 0.528, , , , , , ] D α (V i (2),V i ()) = 0.00 < (50) Sine this value is less than the threshold value, we sto the iterations and alulate the values of U() and T() when x= : U () = , [ ] [ ] T () = [ ] [ ] (51) Aording to U(4) and T(4), luster validation measures V PC and V CE are alulated as 10 V PC = 1 (u 2 ik +t2 ik ) = 0.6, 10 k=1 V CE = 1 (u 10 ik log u ik +t ik log t ik ) = k=1 (52) If we further assume that u ik 0.75, t ik 0.65 I j C i (1 j 10, 1 i ),wherec i denotes luster i,thenwe have lusters as shown in Table Convergene of IFPCM Algorithm. Now, we roeed to investigate the onvergene of IFPCM algorithm on investment ortfolio dataset. The movements of objetive funtion values J m,η (U,T,V;X) are shown in Figure 1 along the iterations. As evident from Figure 1, the IFPCM algorithm dereases the objetive funtion value ontinuously by iterating two hases, namely, udating the membershi and tyiality degrees in (15) and (16) and udating rototyial IFSs in (17). The IFPCM algorithm has lower omutational omlexity as omared to other lustering algorithms [1, 5 7, 9]. The sae and time omlexities of IFPCM algorithm are O((n + ) + n) and O(In), where is the number of samles, n isthenumberofifssinsamle, is the number of lusters, and I is the maximum number of iterations reset for otimal value searh roess. Some advantages of IFPCM algorithm inlude simliity and flexibility, information about the membershi, and tyiality degrees of samles to all lusters and relatively low omutational omlexity Comarative Performane of IFPCM and FPCM on Investment Portfolio Dataset. In this subsetion, we resent a omarative erformane of IFPCM and FPCM algorithms. We first exeriment IFPCM algorithm on the simulated dataset.here,wesetaseriesof values in the range of 2 to 10 and omute V PC and V CE measures for eah lustering result. The results are given in Table 5,whereObj is the objetive funtion value after onvergene of IFPCM algorithm. The otimal values of luster validity measures are highlighted. As evident from Table 5,when = 4 V PC reahes its otimal value (maximum) and V CE also reahes its otimal value (minimum), this imlies that both V PC and V CE are aable of finding otimal number of lusters, that is,. However, this is not the ase for objetive funtion value. From Figure 2, as the number of lusters inreases, Obj dereases ontinuously and finally reahes when =4. Hene, the usage of V PC and V CE is justified in the evaluation of lustering results rodued by IFPCM algorithm. Next, we exeriment FPCM algorithm on the simulated dataset for omarison urose. The results are given in Table 6. The otimal values of luster validity measures are highlighted. As indiated by V PC and V CE values in Table 6, FPCM algorithm refers to luster the modified simulated datasets into three lusters whih are atually away from four true lusters in the data. In other words, FPCM algorithm annot identify all four lasses reisely. This further signifies the imortane of unertainty information in IFSs Examining Clustering Auray of IFPCM Algorithm. To assess the ability of IFPCM algorithm to exlore natural lusters in real world data, 11 lassifiation and two lustering datasets with numerial attributes are hosen from the University of California at the Irvine Mahine Learning Reository [41] and Knowledge Extration based on Evolutionary Learning Reository[42]. In Table 7, these datasets are summarized. The auray of IFPCM algorithm is adhered by removing the lass labels of data before alying the algorithm.

12 12 Advanes in Fuzzy Systems Table 2: Investment ortfolio dataset. a 1 a 2 a a 4 a 5 a 6 μ Ii (a 1 ) V Ii (a 1 ) μ Ii (a 2 ) V Ii (a 2 ) μ Ii (a ) V Ii (a ) μ Ii (a 4 ) V Ii (a 4 ) μ Ii (a 5 ) V Ii (a 5 ) μ Ii (a 6 ) V Ii (a 6 ) I I I I I I I I I I Table : Desrition of two luster validity riteria. Validity riteria Funtional desrition Otimal luster number PC CE V PC = 1 (u 2 ik +t2 ik ) k=1 V CE = 1 (u ik log u ik +t ik log t ik ) k=1 argmax (V PC,U,T,) argmin (V CE,U,T,) Objetive funtion Number of iterations Figure 1: The onvergene of IFPCM algorithm on investment ortfolio dataset. Objetive funtion/artition oeffiient Objetive funtion Partition oeffiient Number of lusters () Figure 2: Comarative erformane of Obj and V PC given different values. Table 4: Clustering result of the investment ortfolio dataset by IFPCM. Instane Cluster ID I 6,I 7 1 I 4,I 8 2 I 2,I 5,I 9 I 1,I,I 10 4 Eah attribute value of all datasets is resaled to a unit interval [0, 1] via linear transformation. The lustering results of the aliation of IFPCM algorithm on 11 lassifiation datasetsareshownintable 8,whereFPCMalgorithmresults as a benhmark fuzzy lustering method are also rovided. The threshold t for effetiveness measure is set to 0.1 for all the datasets, rovided that at least two lusters are exlored. For fairness of omarison between IFPCM and FPCM algorithms, the number of lusters that are needed by FPCM as a arameter for eah dataset is set to the number of lusters that are exlored by IFPCM. In this table, two suer ells for eah dataset are onfusion matries [4], whih reresent the lustering auray of IFPCM and FPCM algorithms on that data. In onfusion matrix, ell ij ontains the number of atterns with lass label i whih are groued by luster j. Aordingly, the ells in eah row of atual lass label aresummedutothenumberofatternsinthatlass.in addition, the summation of eah olumn s ells reresents the number of atterns in that luster. Ideally, otimal lusters are

13 Advanes in Fuzzy Systems 1 Table 5: IFPCM algorithm with different luster numbers on investment ortfolio dataset Obj V PC V CE Table 6: FPCM algorithm with different luster numbers on investment ortfolio dataset V PC V CE Table7:DatasetsusedforevaluatingIFPCMalgorithm. Dataset Number of attributes Number of lasses Number of samles Iris Thyroid Eoli Caner (Breast) Glass Vowel Wine Vehile WDBC Ionoshere 2 51 Sonar ahieved when atterns of eah lass are overed by only one luster and eah luster just ontains atterns of one lass. Suh a ase ourred for the first lass of the Iris dataset in both lustering methods. As evident from results in Table 8, the erformane of IFPCM is better as omared to FPCM algorithm in all datasets. Although onfusion matries for Vehile and WDBC datasets show almost idential overall erformane, the lustering auray of IFPCM algorithm for Iris, Thyroid, Caner, Glass, and Sonar datasets is omaratively better. On the other hand, FPCM algorithm obtains better erformane for Eoli, Vowel, Wine, and Ionoshere datasets. IFPCM algorithm exlores otential lusters that are embedded in datasets and needs only a distinguishing threshold t for effetiveness measure while the number of lusters in FPCM algorithm is rovided in advane. The lusters obtained by FPCM algorithm onvey no seifi ognitive interretation while those lusters exlored by IFPCM algorithm are identified by intuitionisti measure. This intuitionisti interretability whih reresents the lusters justifies the laim that IFPCM algorithm is more suitable for knowledge disovery in datasets. The IFPCM algorithm is more robust to outliers and noise in data. Moreover, the omutational ost of IFPCM algorithm is higher than that of FPCM algorithm as given in Table 9. Although some of the datasets whih are used in the exeriments are high dimensional, they are not too large. The aliation of IFPCM algorithm on large datasets onsumes greater CPU time. Sine, threshold t for effetiveness measure is set to 0.1 for all data given in Table 8, the number of lusters that are exlored for multilass datasets is less than their lasses. Consequently, IFPCM algorithm needs a lower threshold to exlore more lusters. Table 10 illustrates lustering results obtained for these datasets when threshold t is set to To omare the effetiveness of IFPCM algorithm with other fuzzy lustering methods, some reently develoed algorithms have been onsidered and their results on some real world datasets are resented in Table 11.The erformane of these methods is exressed in terms of ureness ratio whih is the average ureness of lusters after luster labeling that is based on maximum number of samle lasses in eah luster. Along with FPCM algorithms, some other lustering algorithms that run on these datasets are FCM, PCM [8], α-ut FCM (AFCM) [18], entroy based fuzzy lustering (EFC) [19], fuzzy mixture of Student s t fator analyzers (FMSFA) [20], and fuzzy rinial omonent analysis guided robust k-means (FPRk) [21]. The IFPCM algorithm maintains areiable erformane omared to

14 14 Advanes in Fuzzy Systems Table 8: Clustering results of IFPCM and FPCM algorithms. Dataset Iris Thyroid Eoli Caner Glass Vowel Wine Vehile WDBC Ionoshere Sonar IFPCM FPCM Atual lass label number of luster number of luster

15 Advanes in Fuzzy Systems 15 Table 9: Comutational osts of IFPCM and FPCM algorithms. Dataset CPU time (in seonds) IFPCM FPCM Iris Thyroid Eoli Caner Glass Vowel Wine Vehile WDBC Ionoshere Sonar Table 10: Clustering results of IFPCM algorithm on multilass datasets. Dataset Atual lass label Thyroid Eoli Number of luster other methods in terms of ureness ratio although this is not true for lustering auray. The seifiation of threshold t in IFPCM algorithm for effetiveness measure is more intuitionisti and less data deendent in nature. 6. Conlusion In this aer, we have roosed IFPCM and IVIFPCM algorithms to luster IFSs and IVIFSs, resetively. Both the algorithms are develoed by integrating onets of FPCM, IFSs, IVIFSs, and basi distane measures. In interval valued intuitionisti fuzzy environments, the lustering algorithm has membershi and nonmembershi degrees as intervals rather than exat numbers. The algorithms overome roblems involved with membershi values of objets to eah luster by generalizing degrees of membershi of objets to eah luster. This is ahieved by extending membershi and nonmembershi degrees with hesitany degree. The algorithms also rovide information about membershi and tyiality degrees of samles to all lusters. Exeriments on bothrealworldandsimulateddatasetsshowthatifpcm has some notable advantages over FPCM. IFPCM algorithm is simle and flexible. It generates valuable information and rodues overlaed lusters where instanes have different membershi degrees in aordane with different real world aliations. The algorithm has relatively lower omutational omlexity. It also takes into aount inherent unertainty in information atured by IFSs whih beomes ruial for suess of some lustering tasks. The evaluation of the algorithm is erformed through luster validity measures. The lustering auray of the algorithm is determined by lassifiation datasets with labeled atterns. IFPCM maintains areiable erformane omared to other methods in terms of ureness ratio although this is not true for lustering auray. For multilass datasets there is a hane for exloring fewer lusters than lasses. This is handled by Glass Vowel Vehile dereasing value of threshold for effetiveness measure. The seifiation of threshold is more intuitionisti and less data deendent in nature. For an unknown dataset, IFPCM must omute luster auray measure for all otential lusters. A sudden dro in values should be onsidered as stoing riterion whereby the number of lusters is determined whih an be exlored. The different drawbaks of FPCM are

16 16 Advanes in Fuzzy Systems Table 11: Comarative analysis of IFPCM algorithm with other Fuzzy lustering methods in terms of Pureness ratio. Fuzzy lustering method Datasets Parameters Iris Wine Thyroid Caner Sonar IFPCM t = FPCM t = FCM [7] =,m= PCM [8] AFCM [18] EFC [19] FMSFA [20] FPRk [21] =,m=2 ε = =,m=2 ε = β = γ = m I =,q=1 λ = k =, λ = 0.1 β = effetively handled by ossibilisti fuzzy C means (PFCM) model roosed by Pal et al. in Our future work entails develoment of IFSs framework for PFCM. Conflit of Interests The author delares that there is no onflit of interests regarding the ubliation of this aer. Referenes [1] R.Duda,P.Hart,andD.Stork,Pattern Classifiation,JohnWiley & Sons, New York, NY, USA, 2nd edition, [2]A.K.JainandR.C.Dubes,Algorithms for Clustering Data, Prentie-Hall, Englewood Cliffs, NJ, USA, [] A. K. Jain, M. N. Murty, and P. J. Flynn, Data lustering: a review, ACM Comuting Surveys, vol. 1, no., , [4] H. Frigui, Simultaneous lustering and feature disrimination with aliations, in Advanes in Fuzzy Clustering and Feature Disrimination with Aliations, , John Wiley & Sons,NewYork,NY,USA,2007. [5] B.S.Everitt,S.Landau,andM.Leese,Cluster Analysis,Oxford University Press, Oxford, UK, [6] W. Pedryz, Knowledge Based Clustering, JohnWiley&Sons, Hoboken, NJ, USA, [7] J. C. Bezdek, Pattern Reognition with Fuzzy Objetive Funtion Algorithms,PlenumPress,NewYork,NY,USA,1981. [8] R. Krishnauram and J. M. Keller, A ossibilisti aroah to lustering, IEEE Transations on Fuzzy Systems,vol.1,no.2, , 199. [9] M.R.Anderberg,Cluster Analysis for Aliations, Aademi Press, New York, NY, USA, [10] F. D. A. T. de Carvalho, Fuzzy -means lustering methods for symboli interval data, Pattern Reognition Letters,vol.28,no. 4, , [11] G. Beliakov and M. King, Density based fuzzy -means lustering of non-onvex atterns, Euroean Journal of Oerational Researh,vol.17,no., ,2006. [12] M. Barni, V. Caellini, and A. Meoi, Comments on a ossibilisti aroah to lustering, IEEE Transations on Fuzzy Systems,vol.4,no.,.9 96,1996. [1] H. Timm, C. Borgelt, C. Doring, and R. Kruse, Fuzzy luster analysis with luster reulsion, in Proeedings of the Euroean Symosium in Intelligent Tehnologies, Tenerife, Sain, [14] H. Timm and R. Kruse, A modifiation to imrove ossibilisti fuzzy luster analysis, in Proeedings of the IEEE International Conferene on Fuzzy Systems (FUZZ-IEEE 02), , Honolulu, Hawaii, USA, May [15] H. Timm, C. Borgelt, C. Doring, and R. Kruse, An extension to ossibilisti fuzzy luster analysis, Fuzzy Sets and Systems,vol. 147,no.1,. 16,2004. [16] D. E. Gustafson and W. C. Kessel, Fuzzy lustering with a fuzzy ovariane matrix, in Proeedings of the IEEE Conferene on Deision and Control inluding the 17th Symosium on Adative Proesses, , San Diego, Calif, USA, January [17] N. R. Pal, K. Pal, and J. C. Bezdek, A mixed -means lustering model, in Proeedings of the 6th IEEE International Conferene on Fuzzy Systems,vol.1,.11 21,Barelona,Sain,July1997. [18] M.-S. Yang, K.-L. Wu, J.-N. Hsieh, and J. Y. Hsieh, Alha-ut imlemented fuzzy lustering algorithms and swithing regressions, IEEE Transations on Systems, Man, and Cybernetis Part B: Cybernetis, vol. 8, no., , [19] J. Yao, M. Dash, S. T. Tan, and H. Liu, Entroy-based fuzzy lustering and fuzzy modeling, Fuzzy Sets and Systems, vol. 11, no.,.81 88,2000. [20] S. Chatzis and T. Varvarigou, Fator analysis latent subsae modeling and robust fuzzy lustering using t-distributions, IEEE Transations on Fuzzy Systems, vol.17,no., , [21] K. Honda, A. Notsu, and H. Ihihashi, Fuzzy PCA-guided robust k-means lustering, IEEE Transations on Fuzzy Systems,vol.18,no.1,.67 79,2010. [22] K. T. Atanassov, Intuitionisti fuzzy sets, Fuzzy Sets and Systems,vol.20,no.1,.87 96,1986. [2] K. T. Atanassov and G. K. Gargov, Intuitionisti fuzzy logi, Comuting Researh Aademy of Bulgarian Sienes,vol.4,no.,. 9 12, [24] K. Atanassov and C. Georgiev, Intuitionisti fuzzy rolog, Fuzzy Sets and Systems,vol.5,no.2, ,199.

17 Advanes in Fuzzy Systems 17 [25] E. Szmidt and J. Karzyk, Intuitionisti fuzzy sets in grou deision making, Notes on Intuitionisti Fuzzy Sets,vol.2,no.1,.15 2,1996. [26] S.K.De,R.Biswas,andA.R.Roy, Analiationofintuitionisti fuzzy sets in medial diagnosis, Fuzzy Sets and Systems,vol. 117, no. 2, , [27] K. T. Atanassov, New oerations defined over the intuitionisti fuzzy sets, Fuzzy Sets and Systems, vol. 61, no. 2, , [28] K. T. Atanassov, Intuitionisti Fuzzy Sets: Theory and Aliations,vol.5ofStudies in Fuzziness and Soft Comuting,Physia, Heidelberg, Germany, [29] K. Atanassov and G. Gargov, Interval valued intuitionisti fuzzy sets, Fuzzy Sets and Systems, vol. 1, no.,. 4 49, [0] H. Bustine, F. Herrera, and J. Montero, Fuzzy Sets and Their Extensions: Reresentation, Aggregation and Models, Physia, Heidelberg, Germany, [1] H. M. Zhang, Z. S. Xu, and Q. Chen, Clustering aroah to intuitionisti fuzzy sets, Control and Deision, vol. 22, no. 8, , [2]Z.S.Xu,J.Chen,andJ.J.Wu, Clusteringalgorithmfor intuitionisti fuzzy sets, Information Sienes, vol. 178, no. 19, , [] Z. Xu, Intuitionisti fuzzy hierarhial lustering algorithms, Journal of Systems Engineering and Eletronis,vol.20,no.1, , [4] E. Szmidt and J. Karzyk, Distanes between intuitionisti fuzzy sets, Fuzzy Sets and Systems, vol. 114, no., , [5] Z. Xu, Some similarity measures of intuitionisti fuzzy sets and their aliations to multile attribute deision making, Fuzzy Otimization and Deision Making, vol.6,no.2, , [6] P. Burillo and H. Bustine, Entroy on intuitionisti fuzzy sets and on interval-valued fuzzy sets, Fuzzy Sets and Systems,vol. 78,no.,.05 16,1996. [7] Z. S. Xu, Some similarity measures of intuitionisti fuzzy sets and their aliations to multile attribute deision making, Fuzzy Otimization and Deision Making,vol.6,no.2, , [8] N. R. Pal, K. Pal, J. M. Keller, and J. C. Bezdek, A ossibilisti fuzzy -means lustering algorithm, IEEE Transations on Fuzzy Systems,vol.1,no.4, ,2005. [9] K. Ito and K. Kunish, Lagrange Multilier Aroah to Variational Problems and Aliations, SIAM Advanes in Design and Control, SIAM, Philadelhia, Pa, USA, [40] E. N. Nasibov and G. Ulutagay, A new unsuervised aroah for fuzzy lustering, Fuzzy Sets and Systems, vol. 158, no. 19, , [41] A. Asunion and D. J. Newman, UCI Mahine Learning Reository, Deartment of Information and Comuter Siene, University of California, Irvine, Calif, USA, [42] J. Alalá-Fdez, A. Fernández, J. Luengo et al., KEEL datamining software tool: data set reository, integration of algorithms and exerimental analysis framework, Journal of Multile-Valued Logi and Soft Comuting, vol.17,no.2-, , [4] R. Kohavi and F. Provost, Glossary of terms, Mahine Learning, vol. 0, , 1998.

18 Journal of Advanes in Industrial Engineering Multimedia Hindawi Publishing Cororation htt:// The Sientifi World Journal Hindawi Publishing Cororation htt:// Alied Comutational Intelligene and Soft Comuting International Journal of Distributed Sensor Networks Hindawi Publishing Cororation htt:// Hindawi Publishing Cororation htt:// Hindawi Publishing Cororation htt:// Advanes in Fuzzy Systems Modelling & Simulation in Engineering Hindawi Publishing Cororation htt:// Hindawi Publishing Cororation htt:// Submit your manusrits at htt:// Journal of Comuter Networks and Communiations Advanes in Artifiial Intelligene Hindawi Publishing Cororation htt:// Hindawi Publishing Cororation htt:// International Journal of Biomedial Imaging Advanes in Artifiial Neural Systems International Journal of Comuter Engineering Comuter Games Tehnology Hindawi Publishing Cororation htt:// Hindawi Publishing Cororation htt:// Advanes in Advanes in Software Engineering Hindawi Publishing Cororation htt:// Hindawi Publishing Cororation htt:// Hindawi Publishing Cororation htt:// International Journal of Reonfigurable Comuting Robotis Hindawi Publishing Cororation htt:// Comutational Intelligene and Neurosiene Advanes in Human-Comuter Interation Journal of Hindawi Publishing Cororation htt:// Hindawi Publishing Cororation htt:// Journal of Eletrial and Comuter Engineering Hindawi Publishing Cororation htt:// Hindawi Publishing Cororation htt://