Clustering Text Documents: An Overview

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1 Clusterng Text Documents: An Overvew Radu CREȚULESCU*, Danel MORARIU*, Lucan VINȚAN* * "Lucan Blaga" Unversty of Sbu, Romana, Engneerng Faculty, Eml Coranst. no. 4, Sbu, Phone , Fax , e-mal {radu.kretzulescu, danel.moraru, lucan.vntan}@ulbsbu.ro Abstract: Clusterng s an mportant process n text mnng used for gropng documents based on ther contents n order to extract knowledge. In data mnng process, clusterng and cluster analyss can be used as stand-alone applcaton for data dstrbuton analyss, n order to observe certan characterstcs of each cluster (group). In ths paper we wll present some requrements for clusterng algorthms, and then we wll present a possble taxonomy of the commonly used clusterng algorthms. In the last secton we wll dscuss the external and nternal technques for clusterng valdaton. Key words: Text Mnng, Unsupervsed leanng, Clusterng, Cluster Valdaton 1. Introducton In the recent years, sgnfcant ncreases n usng the Web and the mprovements of the qualty and speed of the Internet have transformed our socety nto one that depends strongly on the nformaton. The huge amount of data that s generated by ths process of communcaton represents mportant nformaton that accumulates daly and that s stored n form of text documents, databases etc. The retrevng of ths data s not smple and therefore the data mnng technques were developed for extractng nformaton and knowledge that are represented n patterns or concepts that are sometmes not obvous. As mentoned n [5, 9], machne learnng software provdes the basc technques for data mnng by extractng nformaton from raw data contaned n databases. The process usually goes through the followng steps: transform the data nto a sutable format data cleanng deducton or conclusons on the extracted data. Machne learnng technques are dvded nto two sub domans: supervsed learnng and unsupervsed learnng. Under the category of unsupervsed learnng, one of the man tools s data clusterng. Ths paper attempts to provde taxonomy of the most mportant algorthms used for clusterng. For each algorthm category, we selected the most common verson of ths entre famly. Below we present algorthms used n context of document clusterng.. Unsupervsed versus supervsed learnng In supervsed learnng, the algorthm receves data (the text documents) and the class label for the correspondng classes of the documents (called labeled data). The purpose of supervsed learnng s to learn the concepts that correctly classfy documents for gven classfcaton algorthm. Based on ths learnng the classfer wll be able to predct the correct class for unseen examples. Under ths paradgm, t s also possble the appearance of the over-fttng effects. Ths wll happen when the algorthm memorzes all the labels for each case. The outcomes of supervsed learnng are usually assessed on a dsont test set of examples from the tranng set examples. Classfcaton methods used are vared, rangng from tradtonal statstcal approaches, neural networks to kernel type algorthms [6]. The qualty measure for classfcaton s gven by the accuracy of classfcaton. In unsupervsed learnng the algorthm receves only data wthout the class label (called unlabeled data) and the algorthm task s to fnd an adequate representaton of data dstrbuton. The central pont of ths paper s to present the clusterng as a key aspect n unsupervsed learnng. Some researchers have combned unsupervsed and supervsed learnng that has emerged the concept of sem-supervsed learnng [4]. In ths approach s appled ntally an unknown data set n order to make some assumptons about data dstrbuton and then ths hypothess s confrmed or reected by a supervsed approach.

2 3. Cluster Analyss Cluster analyss s an teratve process of clusterng and cluster valdaton facltated by clusterng algorthms and cluster valdaton methods. Cluster analyss ncludes two maor aspects: clusterng and cluster valdaton. Clusterng refers to groupng obects accordng to certan crtera. To acheve ths goal researchers have developed many algorthms [5, 13, 14, 15]. Snce there are no general algorthms that can be appled to all types of cases t becomes necessary to apply a valdaton mechansm so that the user wll fnd an algorthm sutable for ts partcular case. Cluster analyss s a process of dscovery through exploraton. It can be used to dscover structures wthout the need for nterpretaton [13]. The valdaton of the clusters becomes a cluster qualty evaluaton process. 3.1 Defntons Def. 4.1: Clusterng s the process of groupng physcal or abstract obects nto classes of smlar propertes. [14] A cluster s a collecton of obects that are smlar to each other from the same group and are dssmlar to obects that are form dfferent groups. Def. 4. Conceptual clusterng s a process of groupng where the obects n a group wll form a class only f they can be descrbed by one concept. Ths approach dffers from conventonal clusterng where the dssmlarty measure s based on mathematcal dstances. The basc dea for clusterng, whch s the startng pont for both prevous presented approaches, s to fnd those clusters wth hgh nter-cluster smlarty and very small ntra-cluster smlarty. The man drectons of clusterng research are focused more on dstance-based cluster analyss. Several algorthms have been developed such as parttonal algorthms lke k-means and k-medods, or herarchcal algorthms. Further nvestgatons have demonstrated the usefulness of other approaches such as algorthms based on suffx trees [0, 7], the bo-nspred algorthms, such as those based on the behavor of ants n search of food and creaton of cemeteres [1, 7, 17], partcles swarms (partcle swarm optmzaton) [1] and ontologes [3, 1]. 3.. General requrements for clusterng algorthms Typcal requrements for clusterng algorthms n data mnng and text documents [9] are: a. Scalablty of the algorthms: Many clusterng algorthms work very well wth small data sets. However, large databases contanng mllons of obects and usng a large sample of these sets would lead to nconclusve results. b. Ablty to use dfferent types of attrbutes: Many clusterng algorthms use numercal data as nput. In some cases t s necessary to apply clusterng algorthms on strng data, bnary data, ordnal data or a combnaton thereof. c. Clusters of arbtrary shape recognton: Many algorthms fnd clusters usng geometrc dstances such as Eucldean dstance or Manhattan dstance. These algorthms, however, tend to fnd sphercal clusters wth smlar sze and densty. In realty, clusters can be n any form. d. Settng nput parameters based on mnmal knowledge of the feld: Many clusterng algorthms requre certan parameters such as the number of clusters that need to be determned (e.g. k-means). The result of clusterng can be sgnfcantly dfferent dependng on the nput parameters set. The nput parameters for data sets contanng hgh dmensonal obects are dffcult to be determned. e. Ablty to use data that contan nose: Many real data sets contanng mssng data, unknown or outlner. Some algorthms are senstve to such data and clusterng qualty becomes poor. f. Insenstvty to the order of data processng: Some clusterng algorthms may produce dfferent results for dfferent order of the nputs (e.g., Sngle Pass, BIRCH- Balanced Iteratve Reducng and Clusterng). g. Hgh dmensonalty: Data may contan a lot of dmensons and attrbutes. Many clusterng algorthms fal to produce good results even for data wth small dmensons. The human eye fals to assess the qualty of a cluster by up to three dmensons. h. Interpretablty and usablty: Users want that the clusterng results shall be useful, nterpretable and understandable. 4. Data used n clusterng 4.1 Data representaton A data matrx represents for the clusterng algorthm a matrx wth n obects and each has p attrbutes. The representaton of the data wll be a matrx of n p attrbutes. x11... x1... x1 p x1... x... x p (4.1) xn1... xn... xnp 4. Dssmlarty Matrx Ths matrx wll be a N x N square matrx and contans the dssmlarty measures of all pars of obects for the clusterng. Snce d(, ) = d (, ) and d(, ) = 0 we obtan the followng dssmlarty matrx 0 d(,1) 0 d(3,1) d(3, ) 0 (4.) dn (,1) dnn (, 1) 0 where d(, ) s the measure of dssmlarty between two obects. Obects are clustered accordng to the smlarty between them or by ther dssmlarty.

3 4.3 Dssmlarty versus smlarty The dssmlarty d(, ) s a postve number close to 0 when the documents and are close to each other and ncreases when the dstance between and ncreases. The dssmlarty between two obects can be obtaned by subectve evaluaton made by experts based on drect observaton or can be calculated based on correlaton coeffcents. The smlarty s(, ) s a postve number close to 0 when the two documents are not smlar and ncreases when the documents are more smlar. If the smlarty and the dssmlarty coeffcents are n the range [0, 1] the followng equaton s true: d (, ) 1 s (, ) (4.3) 4.4. Common formula used for dssmlarty To calculate the dssmlarty of obects we calculate the dstance between every two obects. The dstances must satsfy the followng propertes: d (, ) 0 non-negatvty d (,) 0 d (, ) d(,) Symmetry d(, ) d(, h) d( h, ) Trangle rule Commonly used formulas for dstance are: a. Eucldean dstance: p E(, ) ( k k) (4.4) k 1 d x x x x b. Manhattan dstance d ( x, x ) x x (4.5) Ma k k k1 c. Mnkowsk dstance p q M k k k 1 p q d ( x, x ) ( x x ) (4.6) d. Cosne dstance dcos p p xk x k k1 k1 p ( xk xk ) k 1 e. Canberra dstance d CAN n k 1 (4.7) xk xk (, ) (4.8) x x k 5. A possble Taxonomy for clusterng algorthms Arrangng data n dfferent groups can be made usng dfferent strateges. Based on ths, the clusterng strategy can be dvded nto several categores: technques based on data parttonng, clusterng technques based on k herarchcal methods [5], methods based on attrbutes order, methods based on densty, grd-based methods and methods based on models. [9]. 5.1 Parttonng Methods If n obects must be grouped nto k groups, then a parttonng method constructs k parttons of the obects, each partton s represented by a cluster wth k n. The clusters are formed takng nto account the optmzaton of a crteron functon. Ths functon expresses the dssmlarty between the obects, so that the obects that are grouped nto a cluster are smlar and obects from dfferent clusters are dssmlar. For ths type of groupng method the clusters must satsfy two condtons: Each cluster must contan at least one obect; Each obect must be ncluded n a sngle cluster. The basc dea of ths type of methods s that the algorthm ntally starts wth a gven number k groups representng the number of parttons (clusters) and then t apples a parttonng method that recalculates the k clusters. The method also uses an teratve relocaton technque that attempts to mprove the parttonng by movng obects from one group to another. The movng crteron needs to respect the condton that the obects of the same cluster are smlar (close). There are dfferent crtera to evaluate the qualty of clusters. These wll be presented n Secton 6. We can dentfy three dfferent types of parttonng algorthms: K-Means clusterng algorthms; for ths type of algorthm each cluster s represented by the average value of all obects from the same cluster. K-Medods clusterng algorthms: for ths type of algorthm each cluster s represented by the obects whch are closest to the center (medod) of the cluster. Probablstc algorthms for clusterng: a probablstc method assumes that the data come from a mxture of populatons whose dstrbutons and probabltes must be determned. These algorthms can be used n collectons of small to medum data when the resulted clusters can be found n sphercal forms K-Means algorthm K-Means algorthm [10], [11], [18] s the most popular algorthm used n scentfc and ndustral applcatons. The name of the algorthm comes from the representaton of k-clusters C whose weghts (means) c are calculated as the centrod of the ponts whch are grouped n cluster C. The value for the parameter k s set at the start of the algorthm and represent the number of cluster that want to be obtaned. The smlarty between the tems from the same cluster C s calculated n relaton to the centrod. The error crteron used s defned n the formula k E( C) x c (5.1) 1 x C Where x s the gven entry obect and c the centrod of C.

4 The scope s to acheve a greater ntra-cluster smlarty and very small nter-cluster smlarty so that we get k clusters as compact and smultaneously separated as possble. It s obvous that ths type of algorthm works wth numercal varables. The algorthm performs the followng steps: 1. Generate k random centers n n-dmensonal space, whch represents the ntal centrods.. For each obect compute the dstance between t and all the centrods, then the obect s assgned to the closest centrod for computng the dstance there are several formulas presented n secton Once all obects have been properly assgned to the centrods, the postons of the k centrods are recalculated as the center of all samples assgned to each centrod ndvdually. 4. Repeat steps and 3 untl the centrods no longer change ther postons. 5. The obects assgned to the centrods represent the contents of the fnal clusters K-Medods method K-medods algorthm s an adaptaton of k-means algorthm. Instead of computng the centrod of each group, the algorthm chooses a representatve element called medod for each cluster at each teraton. The medod for each group s calculated by fndng an tem n the cluster that mnmzes the sum: C d (, ) (5.) Where C s the cluster that contans the obect and d(,) the dstance between obect and obect. There are two advantages by usng exstng obects as cluster centers. Frst, a medod can descrbe usefully a cluster. Second, t s not necessary to calculate the dstance between the obects for each teraton of the algorthm. The dstances can be computed once and then saved n a so called dstances matrx. The steps of the k-medods algorthm can be summarzed as follows: 1. Choose k obects random to be the orgnal medods of the clusters.. Assgn each obect to the nearest cluster assocated to the medod. 3. Recompute the postons for the k-medods. 4. Repeat steps and 3 untl the medods doesn t change. Kaufman and Rousseeuw presented n [16] the PAM (Partton Around Medods) algorthm whch s an teratvely mplementaton of the k-medods algorthm. The K-Medods algorthm s more robust n terms of nose comparng to the k-means algorthm. Snce k- Medods s workng drectly wth medods t s not so much nfluenced by the outlners lke the centrod calculaton nfluences the k-means algorthm. However n terms of computatonal cost the k-medods algorthm has a hgher cost because for a sngle teraton the computatonal cost s O(k(n-k) ). The K-medods algorthm s effcent on small data sets. On large data sets, due to ts hgh computatonal cost, the algorthm becomes qute slow. For applyng the PAM algorthm to large data sets Rousseew and Kaufman (1990) [16] developed the algorthm CLARA (Clusterng Large Applcatons) who reles on the PAM algorthm whch s appled only to a sample of data extracted from a gven set. CLARA wll dscover n ths reduced set the k- medods. The complexty of each teraton s O(kS +k(kn)) where S s the data set sze chosen (number of samples), k the number of clusters and n the total number of ponts. In [15,, 5] s proposed to choose the value S = 40+k. 5. Herarchcal methods These types of methods provde a herarchcal structure of the set of obects. There are two approaches of herarchcal methods: Agglomeratve -These methods have a "bottom-up" approach. At the begnnng of the algorthm each obect s a cluster, and then n the next steps the clusters are merged together based on smlarty measures creatng a herarchcal structure untl all clusters are oned nto a sngle cluster or untl another stoppng condton (gven number of clusters, tme, etc.) s reached. Dvsve - These methods have a "top-down" approach. Frst all obects are consdered to be contaned n a sngle cluster, then after successve teratons each cluster s dvded nto smaller clusters untl each obect s a cluster or untl a stoppng condton s reached. A maor problem of herarchcal clusterng algorthms s that once the dvson or mergng step has been made t cannot be canceled. Ths ssue also represents a maor advantage of ths type of calculatons due to reducton algorthms avodng calculatng the varous combnatons of possbltes. AGGLOMERATIVE ALGORITHMS Dependng on the method of computng the smlarty between the clusters n the herarchcal agglomeratve approaches there can be dstngushed more models of ths category of clusterng algorthms [19]: Sngle Lnk The smlarty between two clusters s gven by the mnmum dstance between the most smlar two documents contaned n those clusters. sm( A, B) mn sm( a, b) (5.3) aa, bb where A and B are the clusters to be merged and a, b are documents wth aa and bb. Complete lnk The smlarty between two clusters s gven by the smlarty of the most dssmlar documents from the two clusters. Ths s equvalent to choosng pars of clusters for mergng wth the smallest dameter. sm( A, B) max sm( a, b) (5.4) aab, B Average lnk The smlarty between two clusters s calculated as the average sums of dstances between each element from the frst cluster and all elements from the second cluster.

5 1 sm( A, B) sm( a, b) A B aa, bb (5.5) Centrod Lnk The smlarty between two clusters s calculated as the dstance between the cluster centrods. sm( A, B) CA CB (5.6) Ward's method Ward's method [31] attempts to mnmze the Sum of Squares (SS) of any two (hypothetcal) clusters that can be formed at each step: (5.7) ( A, B) x mab x ma x mb AB A B na nb mamb na nb Where m s the centrod of cluster and n s the number of obects n cluster. If the two pars of cluster centers wll have dentcal dstances usng the Ward's method then the smaller clusters wll be merged. DIVISIVE ALGORITHMS The dvsve herarchcal clusterng approach s conceptually more complex than the agglomeratve. If the frst step of an agglomeratve approach can have n( n1) possbltes to merge two obects then n the 1 dvsve approach there are n 1possbltes to dvde data nto two groups. Ths number s consderably hgher than that n case of agglomeratve methods. Kaufmann and Rousseeuw have presented n [15] a dvsve clusterng algorthm DIANA (Dvsve Analyss). Algorthms lke BIRCH (Balanced Iteratve Reducng and Clusterng) proposed n [8] and CURE (Clusterng Usng Representatves) proposed n [8] combnes the parttonng methods wth the herarchcal ones. 5.3 Method based on the order of words Suffx Tree Clusterng (STC) The algorthm presented n [7] does not requre a vector representaton of obects (documents) lke the algorthms presented n the prevous sectons. The STC algorthm uses the Suffx Tree Document Model (STDM) to represent the documents. Ths algorthm wll take nto account the words order, and creates clusters based on words or groups of consecutve words from the documents. Thus can be a maor advantage because t takes account of word order n sentences. A suffx tree of a document d s a compact tree contanng all suffxes s of that document. In our case a suffx s a strng consstng of one or more words. Rules for buldng the suffx tree: 1. Each nternal node other than root must have at least two chldren and each edge leavng a node s labeled wth a nonempty substrng n.. Any two edges that start from the same node cannot start wth the same word. Suffx tree constructon Let d = w 1 w w 3... w m be the representaton of a document as a sequence of words n same order from as n the document wth 1, m and S a set of n documents. The suffx tree for a set S contanng n strngs, each of them havng the length m n s a tree that contans exactly one root node and Σm n leaf nodes. Snce each node contans at least two chldren t wll contan the common part of at least two suffxes. Each leaf of the tree can represent one or more documents and all edges from the root to the leaf nodes represent an entre document or a substrng from a document. Two documents wth many common nodes tend to be smlar. Selectng the base nodes After creatng the suffx tree some nodes wll become clusters. The basc rules for a node to become a cluster are: a. Each node n the suffx tree that has at least two chldren s the consdered a basc node and receves a score S(B) accordng to equaton (5.8) b. After orderng the nodes n descendng order based on the obtaned score the algorthm retans the frst k nodes (a gven threshold). These basc nodes wll contnue to be used n the next step. c. Nodes that receve a score lower than predetermned threshold are elmnated. Formulas used: Let B be the number of documents contaned n cluster B and P the number of words n the sentence P. A cluster score s calculated as SB ( ) B f( P) (5.8) where the weghtng functon for the length of a sentence s 0.5, P 1 f( P) P, P 6 6, P 6 ( 5.9) Ths functon penalzes the sentences whch contan a sngle word, s lnearly ncreasng for sentences contanng two to sx words and become constant for sentences longer than sx words. The smlarty between two clusters s calculated: B B B B 1, SIM ( B, B ) B B 0, otherwse (5.10) where α s a chosen threshold. In [7] threshold for α s 0.5 and n [6] t s 0.8. Regardng the formula for calculatng the smlarty we can choose to use the Jaccard's coeffcent, whch gves us the degree of smlarty between two nodes. The hgher the value the greater s the smlarty between nodes (the value s n the nterval [0, 1] - wth 1 for dentcal nodes):

6 J B, B B B B B (5.11) SOM neural networks. It may also be ncluded n ths category, algorthms that rely on ants behavor and partcle swarms. Combnng the base nodes a. The smlarty between two base nodes s calculated accordng to formula (5.10). b. If the smlarty between two nodes exceeds a certan threshold then they wll merge and after mergng the resulted node wll become a cluster. 5.4 Densty-based methods A topologcal space can be decomposed nto ts connected components. Startng from ths pont of vew we can defne a cluster as a related component that s constructed n the drecton where the densty s hgher. Ths s the reason why the densty based algorthms are able to dscover clusters of arbtrary shapes. Also constructng towards hgher densty the algorthm protects tself aganst nose (outlner). However, these methods have the dsadvantage that f a cluster conssts of two adacent areas of dfferent densty but larger than a gven threshold the result s not very nformatve [9]. Important algorthms of ths method nclude: DBSCAN (Densty-Based Spatal Clusterng wth Nose Algorthm) [], whch produces clusters wth a threshold densty. OPTICS (Orderng Ponts to Identfy the Clusterng Structure) [] extends the DBSCAN algorthm and computes the dstances ε less gven threshold ε wth 0 ε ε. The only dfference from the OPTICS algorthm DBSCAN s that the obects are not assgned to a cluster but the algorthm saves the order n whch obects were processed. 5.5 Grd- based methods Grd-based methods are methods that quantfy the representaton of the obect space nto a fnte number of cells that form a grd structure. Once ths representaton s made a clusterng algorthm s appled. The great advantage s that the processng tme depends only on the number of cells n each dmenson of the transformed space and not by the number of obects contaned n the cells. Typcal algorthms for ths type of method are STING (Statstcal Informaton Grd) [3], DENCLUE (DENsty-based CLUstErng), Clque, MAFIA (MAxmal Frequent Itemset Algorthm) 5.6 Model - based methods Methods based on models are assumng that there s a model for each cluster and they try to fnd data that ft best wth the model. A model-based algorthm can dscover clusters by constructng a densty functon that reflects the spatal dstrbuton of the data. However these algorthms can lead to an automatc dscovery of clusters, when the numbers of clusters are based on standard statstcal methods takng nto account the nose. Algorthms that fall nto ths category are: AutoClass (Automatc Classfcaton), COWEB, Class. Also n ths category are used as a model and algorthms such as 6. Cluster valdaton Each clusterng algorthm appled on the same set of data wll group the data n dfferent ways dependng on the smlarty metrc used. Ths makes analyss for the effcency of clusterng algorthms very dffcult. To evaluate the performance or qualty of clusterng algorthms, obectve measures must be establshed. There are three types of qualty measures: a. external, when there s an a pror knowledge about the clusters (we have pre-labeled data); b. nternal, whch have no nformaton about clusters; c. relatve assessng group dfferences between dfferent solutons. External measures are appled to both classfcaton and clusterng algorthms, whle nternal and relatve measures are appled only to the clusterng algorthms. EXTERNAL VALIDATION MEASURES External valdaton measures requre pre-labeled data sets for clusterng analyss. Because clusterng data can be done from many ponts of vew and the labels may vary from these ponts of vew, the comparson s made to the group contanng the most data n a specfc category. Some of the external evaluaton measures: Precson s the percentage of retreved documents whch are really relevant to that category. The value for the precson s n the nterval [0,1], wth 1 as best. For a cluster C and a known class S we compute the precson: C S precson( C, S ) C (6.1) Recall s the percentage of documents that are relevant to that category and are ndeed grouped nto that category. The value for the recall s n the nterval [0,1], wth 1 as best. C S recall( C, S ) S (6.) Accuracy s the percentage of documents that are correctly grouped nto categores dependng on the labels of documents (needs labeled documents). Fmeasure s a measure that combnes precson and recall ste and s calculated accordng to formula (6.3) precza( C, S ) recall( C, S ) Fmeasure( C, S ) precza( C, S ) recall( C, S ) (6.3) For each class t wll select only the cluster wth the hghest Fmeasure. In the fnal Fmeasure for overall measure of a clusterng soluton s weghted by the sze of each cluster. INTERNAL VALIDATION MEASURES In [31] are presented four such metrcs: Compactness Ths measure expresses how smlar

7 are the data from a gven cluster n C compactness( C) ( c x ) 1 (6.4) Where C s the current cluster, n c the number of elements n the cluster, c the cluster centrod and x s an element of the cluster. In other words, these metrc measures how "close" the documents wthn a cluster are. The value s n the nterval [0, ) and as the lower the better s the measure Separabllty Ths measure between the clusters expresses how dssmlar the clusters are. separablty mn( dst( c, c )) (6.5) Thus for each cluster, the nearest cluster s determned. Based on ths metrc we seek clusters that maxmze ths value, so that the clusters are very dssmlar. Balance - The balance s computed as the clusters are formed and expresses how well-balanced the formed clusters are. (6.6) Where n s the total number of documents, n the number of documents n the cluster I (the formula takes nto consderaton the bggest formed cluster), k s the number of formed clusters. Ths measure can take values n the nterval [0,1], the maxmum value 1 s acheved when all clusters have the same number of documents. A value close to 0 s obtaned when the number of documents contaned n clusters vares greatly. 7. Conclusons C, 1 n/ k balance max ( n ) 1, k Clusterng n text documents s an unsupervsed learnng method of classfyng documents wth a certan degree of smlarty usng dfferent metrcs based on dstance. Bascally, the clusterng algorthm must dentfy (must fnd) the clusters n whch documents are grouped and/or some patterns (rules) that separate one group from another group. There s no predefned taxonomy. It s establshed when the clusterng algorthm run to the set of documents. Cluster evaluaton s an next mportant task. The formulas presented for nternal and external evaluaton of clusters can gve a measure for evaluate the qualty of clusterng process. Stll t s very hard to compare dfferent clusterng results. 8. Acknowledgment Ths work was partally supported by CNCSIS- UEFISCSU, proect number PN II-RU code PD_670/ References [1] Abraham, A., Ramos, V. - "Web Usage Mnng Usng Artfcal Ant Colony Clusterng and Genetc Programmng" Proc. of the Congress on Evolutonary Computaton (CEC 003), Canberra, pp , IEEE Press. 003 [] Ankerst, M., Breunng, M., Kregel, H.-P., Sander, J.,- Oprcs: "Orderng ponts to dentfy the clusterng structure" In. Proc ACM-SIGMOD, Int. Conf. Management of Data, Phladelpha, 1999 [3] Batet, M., Valls, A., Gbert, K. - "Improvng classcal clusterng wth ontologes", In Proc. IASC 008, Yokohama, 008 [4] Bennett, K. P., Demrz A., - "Sem-supervsed support vector machnes." In M. S. Kearns, S. A. Solla, and D. A. Cohn, edtors, Advances n Neural Informaton Processng Systems, pages , Cambrdge, MA, MIT Press. [5] Berkhn, P., - "A Survey of Clusterng Data Mnng Technques", Kogan, Jacob;Ncholas, Charles; Teboulle, Marc (Eds.) Groupng Multdmensonal Data, Sprnger Press, pp. 5-7 (006) [6] Burges, C. J. C., - "A tutoral on support vector machnes for pattern recognton." Data Mnng and Knowledge Dscovery, ():11-167, [7] Dorgo, M., Bonabeau, E., Theraulaz, G., - "Ant Algorthms and stgmergy" Future Generaton Computer Systems Vol.16, 000. [8] Guha, S. Rastog, R., Shm, K., - "CURE: an effcent clusterng algorthm for large databases." n Proceedngs of the 1998 ACM SIGMOD nternatonal conference on Management of data (SIGMOD '98), AshutoshTwary and Mchael Frankln (Eds.). ACM, New York, NY, USA1998 [9] Han, J., Kamber, M., - "Data Mnng: Concepts and Technques", Morgan Kaufmann Publshers, 001 [10] Hartgan, J. A., Wong, M., Algorthm as 136: A k- means clusterng algorthm, Journal of the Royal Statstcal Socety. Seres C (Appled Statstcs), London, 1979 [11] Hartgan, J. A., - "Clusterng Algorthms", New York: John Wley & Sons, Inc, 1975 [1] Hotho, A., Staab, S. Stumme,G., - "Ontologes Improve Text Document Clusterng", IEEE Internatonal Conference on, p. 541, Thrd IEEE Internatonal Conference on Data Mnng (ICDM'03), 003 Implementaton, Morgan Kaufmann Press, 000 [13] Jan, A., K., Dubes, R.,C. - "Algorthms for Clusterng Data", Prentce Hall, Englewood Clffs, NJ [14] Jan, A. Murty, M. N., Flynn, P. J., - "Data Clusterng: A Revew", ACM Computng Surveys, Vol. 31(3), pp (1999) [15] Kaufman, L. and Rousseeuw, P.J. - "Fndng Groups n Data: An Introducton to Cluster Analyss", Wley-Interscence, New York (Seres n Appled Probablty and Statstcs), 1990 [16] Kaufman, L., Rousseeuw, P. J., - "Clusterng by means of medods, n Statstcal Data Analyss based on the L, Norm", edted by Y. Dodge, Elsever/North- Holland, Amsterdam, 1987

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