FINDING INITIAL PARAMETERS OF NEURAL NETWORK FOR DATA CLUSTERING

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1 FINDING INITIAL PARAMETERS OF NEURAL NETWORK FOR DATA CLUSTERING ABSTRACT Suneetha Chttnen 1 and Raveendra Babu Bhogapath 2 1 R.V.R. & J.C. College of Engneerng, Guntur, Inda. Chttnen_es@yahoo.com 2 VNR Vgnana Jyoth Insttute of Engneerng & Technology, Hyderabad, Inda. rbhogapath@yahoo.com K-means fast learnng artfcal neural network (K-) algorthm begns wth the ntalzaton of two parameters vglance and tolerance whch are the key to get optmal clusterng outcome. The optmzaton task s to change these parameters so a desred mappng between nputs and outputs (clusters) of the K- s acheved. Ths study presents fndng the behavoral parameters of K- that yeld good clusterng performance usng an optmzaton method known as Dfferental Evoluton. DE algorthm s a smple effcent meta-heurstc for global optmzaton over contnuous spaces. The K- algorthm s modfed to select wnnng neuron (centrod) for a data member n order to mprove the matchng rate from nput to output. The experments were performed to evaluate the proposed work usng machne learnng artfcal data sets for classfcaton problems and synthetc data sets. The smulaton results have revealed that optmzaton of K- has gven qute promsng results n terms of convergence rate and accuracy when compared wth other algorthms. Also the comparsons are made between K- and modfed K-. KEYWORDS Optmzaton, Clusterng, Dfferental Evoluton, Artfcal neural network 1. INTRODUCTION Optmzaton s a procedure used to make a clusterng mechansm as effectve as possble usng mathematcal technques. It s nothng but fndng an alternatve wth the hghest achevable performance under the gven constrants. Clusterng s the process of organzng objects n to groups, whose members are smlar n some way. A cluster s therefore a collecton of objects whch are smlar to each other and dssmlar to the objects belongng to other clusters. In the context of machne learnng, clusterng s an unsupervsed task. The nput for clusterng problem s the set of objects, where each object s descrbed by ts relevant attrbutes. The desred output s the parttonng set of objects nto dsjont groups by mnmzng the objectve functon. Clusterng problems are of two types. In hard clusterng, an object s assgned to only one cluster. In fuzzy clusterng, an object s assgned to more than one cluster wth some membershp degree. Artfcal neural network composed of nter-connected artfcal neurons that mmc the propertes of bologcal neural networks. The major applcatons of artfcal neural networks appled to clusterng n dfferent domans are mage segmentaton for groupng smlar mages, n data mnng dvdng data set nto dfferent groups, n bo-nformatcs groupng genes whch are DOI : /jaa

2 havng smlar characterstcs. In ths paper, clusterng s carred out usng Modfed k-means fast learnng artfcal neural network. Evoluton of fast learnng artfcal neural networks begns wth the two basc models proposed by Tay and Evans: I and II. I model s the modfcaton of Adaptve Resonance Theory (ART 1) developed by Carpenter and Grossberg [6]. The desgn of II was based on Kohonen LVQ network wth Eucldan dstance as smlarty metrc [7].The later mprovement n the II s to take contnuous value n nput [8]. Later K-means calculatons were ncluded n the algorthm. Ths leads to the development of K- [9].Later K was modfed to nclude data pont reshufflng whch resolves the data sequence senstvty that creates stable clusters [10]. The most popular and well known algorthm known as Genetc algorthms, developed by John Holland n 1975, are a model or abstracton of bologcal evoluton based on Charles Darwn's theory of natural selecton [2].Later the bologcal crossover and mutaton of chromosomes are smulated n the algorthm by Holland and hs collaborators to mprove the qualty of solutons over successve teratons [3][24]. Dfferental evoluton was frst popularzed by R. Storn and K. Prce n 1997[1]. It s vector based evolutonary algorthm that s hghly effcent for global optmzaton problems [12-15]. It can be consdered as further development of genetc algorthm. Storn and Prce n 1997 stated that DE was more effcent than smulated annealng and genetc algorthms [1][18][19]. When compared to GA, DE s very attractve due to ts smple desgn and good performance. The mplementaton of DE s easy because smple mathematcal operators can be appled to evolve the populaton wth few control parameters [1]. The most mportant features of DE notced by several researchers are [11] fast convergence, speed, stronger stablty, easy to realze and easly adaptable for nteger and dscrete optmzaton. The same work s carred wth GA also. When GA s combned wth these algorthms, t was notced that GA s effcent for data sets of small sze.but when sze of the data set s large; GA requres more executon tme than DE to perform data clusterng [24]. In DE, all solutons have the same chance of beng selected as parents wthout dependence on ther ftness value. The better one of new soluton and ts parent wns the competton provdng sgnfcant advantage of convergng performance over genetc algorthms (GA) [11]. 2. K-MEANS FAST LEARNING ARTIFICIAL NEURAL NETWORK K- s an atomc module, sutable for creatng scalable networks that mmc the behavor of bologcal neural systems.k- s used to cluster data effcently and consstently wth the mnmal varaton n cluster centrods and s not senstve to the data presentaton sequences (DPS) caused by randomly orderng the arrval of data[10] Archtecture K- conssts of two layers, nput layer and output layer. The set of nodes n the nput layer s determned based on the dmenson of data. Input nodes are drectly connected to output nodes by a set of weghts. The number of nodes n the output layer grows as new groups formed n the clusterng process. The dynamc formaton of output nodes yeld K- s a self-organzed clusterng ANN model. The archtecture of the K- s shown n fgure 1. 58

3 Fgure 1. K- archtecture Algorthm: Step 1 Intalze the network parameters.. Step 2 Present the pattern to the nput layer. If there s no output node, GOTO step 6 Step 3 Determne all possble matches output node usng Eq. (1). d D δ 2 = 1 ( x ) n w j 2 ρ (1) 1, a > D[a]= 0, a 0 0 Step 4 Determne the wnner from all matches output nodes usng Eq. (2) Wnner = d mn wj = 1 ( ) x 2 (2) Step 5 Match node s found. Assgn the pattern to the match output node. Go to Step 2 Step 6 Create a new output node. Perform drect mappng of the vectors. nput vector nto weght Step 7 After completng a sngle epoch, compute clusters Centrod.If centrod ponts of all clusters unchanged, termnate Else GO TO Step 2. Step 8 Fnd closest pattern to the centrod and re-shuffle t to the top of the dataset lst, Go to Step 2. Note: ρ s the Vglance Value, δ s the tolerance for th feature of the nput space, and W j s the cluster center of the correspondng output node denotes the weght connecton of j th output to the th nput, X represent the th feature. K- algorthm conssts of two phases. Pror to clusterng process, step 1 called ntalzaton phase, s needed to determne vglance and tolerance. The computaton complexty depends on sze of the data set and dmensonalty of data set. The value of vglance s usually between 0.5 to 1.Snce the ρ value determnes the rato between the number of features to be wthn tolerance and the total number of features, the clusterng formaton becomes more strngent as the vglance settng approaches 1.Ths parameter was adopted from [26]. The tolerance settng of features s the measurement of attrbutes dsperson, and thus computaton s 59

4 performed for every feature of the tranng data at the ntal stage. Several methods were nvestgated to deal wth tolerance settng [10]. The formula s gven by the eq. (3). = max dff + mn dff 2 (3) Where mn dff Mnmum dfference n attrbute values for attrbute. Ths s the dfference between the smallest and second smallest values of the attrbute. max dff Maxmum dfference n attrbute values for attrbute. Ths s the dfference between the smallest and largest values of the attrbute. In the second phase, clusterng process begns wth the presentaton of data patterns n sequence. Step 3 nvolves two types of computaton called tolerance flterng and vglance screenng. Tolerance flterng shows local varaton n the nput features.e. t determnes whether the feature s wthn the acceptable tolerance or not. It s mplemented usng dscrmnatng functon D[a] n step 3.The postve value of D[a] ndcates that the partcular attrbute s wthn the bounds of the correspondng attrbute of the cluster n queston. Vglance screenng shows global varaton n the nput features.e. t determnes the number of features wthn the localzed nput space that have passed tolerance tunng. Compettve learnng begns from step 4.After satsfyng the screenng crtera, there are stll chances of exemplar mapped to more than one cluster. The fnal decson of ownershp s decded by the WTA property of compettve learnng. Eventually the cluster centrod bearng strongest weght wll absorb the exemplar n to ts cluster. The wnnng neuron s chosen based on the closest dstance from the data pont to centrod. The exemplar s assgned as the new cluster member of the selected wnner centrod n step 5. In step 6, f the new exemplar does not meet the screenng crtera, a new output node s created to represent new cluster. After a sngle epoch, cluster stablty s checked. New cluster centrods are calculated n step 7 and the data patterns closest to the centrods s determned.re-shufflng the seed ponts closest to the cluster centrods to the top of the lst creates new lst of patterns for the next teratve clusterng process n step Modfcatons n the K- Algorthm The mprovements n the K- algorthm are used to select the best matchng unt for the formaton of consstent clusters. A step 4 of the K- algorthm s modfed as follows: Step 4 Determne the wnner from all matched output nodes usng the followng crtera: If same match s found 60

5 Else n Wnner = mn ( w j x ) 2 (4) = 0 n δ 2 ( w x ) 2 = j max 0 Wnner = n (5) In the modfed K-, a data pont successfully satsfed the screenng process and well suted as a member of more than one cluster wth dfferent match values to the centrods.the data pont s assgned as a new cluster member of centrod f t has maxmum matched value out of all matches. The wnner centrod s the one wth the hghest matched value for a data pont even f t s not closest to the data pont. But n the orgnal K- algorthm, a data pont successfully satsfed the screenng process and well suted as a member of more than one cluster, s assgned as a new cluster member of the selected wnner centrod.the centrod closest to the data pont s the Wnner centrod because t has same match values for the other centrods. 3. DIFFERENTIAL EVOLUTION The dfferental evoluton (exploraton) [DE] s a novel parallel drect search method, whch utlzes NP parameter vectors as a populaton for each generaton G.DE can be categorzed nto a class of floatng-pont encoded, evolutonary optmzaton algorthms. Currently, there are several varants of DE. The DE varant used throughout ths nvestgaton s the DE/rand/1/bn scheme [1][17]. The dfferental evoluton (DE) algorthm s a populaton-based optmzaton method that smulates natural evoluton combned wth a mechansm to generate multple search drectons based on the dstrbuton of solutons (vectors) n the current populaton. The DE algorthm s man strategy s to generate new ndvduals by calculatng vector dfferences between other randomly selected ndvduals of the populaton. Man Steps of the DE Algorthm 1) Defnton of encodng scheme 2) Defnton of selecton crteron (ftness functon) 3) Creaton of populaton of chromosomes 4) Evaluaton Repeat Mutaton Crossover Selecton Untl (termnaton crtera are met). In DE, a soluton vector s randomly created at the start. Ths populaton s successfully mproved by applyng mutaton, crossover and selecton operators. Mutaton and crossover are used to generate new vectors (tral vectors), and selecton then are used to determne whether or not the new generated vectors can survve the next teraton [1]. The operators and equatons are descrbed n

6 3.1. Defnton of encodng scheme Intal populaton s a set of NP D dmensonal parameter vectors known as genomes or chromosomes. Each soluton vector conssts of two components representng vglance and tolerance parameters of K. Bnary encodng s used for vglance and floatng pont encodng s used for tolerance. Each parameter s of length d n a soluton vector. For example, consder a data set conssts of 5 features. The representaton of a soluton vector s shown below. Bts of Vglance parameter Feature Tolerances(δ) Vglance(ρ) Fg. 2 structure of a soluton vector So the total length of a soluton vector s ((2*D) +1) where D s the number of features n a data set. For a feature, tolerance s the dfference between maxmum value and mnmum value wthn the feature. In the above example, f the feature to be matched, the correspondng bt s 1 else the correspondng bt s 0.So for four features the correspondng bt s 1. So vglance s 4/5= Defnton of ftness functon The man goal of clusterng algorthms s maxmzng the between group varance and mnmzng wthn group varance. Any cluster valdty measure can be used as ftness functon. In ths paper, a new cluster valdty measure called CS measure proposed by Chou n 2004 s used as ftness functon to evaluate the results of clusterng and to select best vectors n the populaton. Also CS measure deals wth clusters of dfferent denstes and/or szes and s more effcent than other cluster valdty ndces. The CS measure s a functon of the rato of the sums of wthn cluster scatter to between-cluster separaton and smlar to DB or DI ndces. If the clusters are compact then wthn cluster scatter s mnmum and between-cluster separaton s maxmum for well separated clusters. So the CS measure s defned as follows: CS ( K ) = 1 K 1 max d X, X K q = 1 N x c X C q K 1 mn d m m K =, j 1 j K, j (6) = K 1 max d X, X q = 1 N x c X C q K = mn d m m, j 1 j K, j (7) Where k- the number of clusters 62

7 X X j d, - s the dstance metrc between any two data ponts X and X j m m j d, - s the dstance metrc between any two centrods m and m j The cluster centrod s calculated by takng the average of all data ponts that les n a cluster. It s denoted by m. m = 1 n x j c x j (8) The ftness functon s the CS measure s combned wth the ms-classfcaton rate.the error rate s computed by comparng cluster structure produced by K- and externally suppled class labels. Error rate of a soluton vector s the rato of the number of samples msclassfed and the total number of samples. (9) Where mc s the no.of ms-classfed samples total s the total number of samples Eq. (10) gves the ftness functon of a soluton vector ( X f ( X 1 )= CS ( K ) * errorrate + eps ) based on the result of K-. Where CS (K) - CS measure calculated over K parttons errorrate - ms-classfcaton rate of eps - small bas term equal to 2x 10-6 and prevents the denomnator of R.H.S from beng equal to zero Mutaton errorrate = mc total x100 To create V,G+1 for each th member of the current populaton (also called the target vector), three other dstnct parameter vectors, say the vectors X r1,g, X r2,g, and X r3,g are pcked up randomly from the current populaton. For a soluton vector, both bnary mutaton and floatng pont mutaton s appled to create tral vector Bnary mutaton In the bnary mutaton, logcal OR operator s used to add a vector to the weghted dfference between the two vectors. Dfference of two vectors s done by Exclusve- OR operaton. Multplcaton s performed by logcal AND operaton.so mutaton operaton for bnary vectors defned as follows: (10) For each bnary target vector X,G =1,2,3,..., NP (populaton Sze) A bnary mutant vector s produced by V,G+1 = X r1,g OR F AND ( X r2,g XOR X r3,g ) (11) 63

8 Floatng- pont mutaton DE generates new parameter vectors by addng the weghted dfference between two populaton vectors to a thrd vector. Ths operaton s called mutaton. The mutated vector s parameters are then mxed wth the parameters of another predetermned vector, the target vector, to yeld the socalled tral vector. For each target vector X,G =1,2,3,..., NP (populaton Sze). A mutant vector s produced by V, G+1 = X r1,g + F (X r2,g X r3,g ) (12) Where, r1, r2, r3 {1, 2,..., NP} are randomly chosen and must be dfferent from each other. F s the scalng factor, whch controls the magntude of the dfferental varaton of (x r2, G x r3, G). NP s sze of the populaton, whch remans constant durng the search process [1] Crossover The parent vector s combned wth the mutated vector to produce a tral vector through a crossover operaton whch combnes components from current element and mutant vector accordng to a control parameter CR Two types of crossover schemes can be used for DE famly of algorthms. In ths work, bnomal crossover s performed on each of the D varables whenever a randomly pcked number between 0 and 1. If (rnd j CR) or j = r n U j,g+1 = U j,g+1 End Else U j,g+1 = X j,g Where j = 1, 2,..., D; rnd j random number generated n the nterval [0 1]; CR s crossover constant [0, 1]; r n (1, 2,..., D) s the randomly chosen ndex; D represents the number of dmensons of a vector Selecton Selecton s the process of decdng the better one by comparng the performance of tral vector and target vector. The newly formed tral vector s evaluated to decde whether t wll survve to the next generaton or not. If the tral vector produces a better ftness value.e. lower value of objectve functon then t s coped to next generaton otherwse target vector s passed nto next generaton [1]. The pseudo code for selecton s as follows: 64

9 If f (U, G+1 ) f (X, G ) X, G+1 = U, G+1 Else X,G+1 = X,G End 4. K- AND DIFFERENTIAL EVOLUTION The K- s competent clusterng algorthm that provdes consstent clusters wth a short number of epochs. To partton a data set, K- needs optmal values of Vglance and Tolerance. To ntalze these parameters, a large search space s explored and best parameter values are determned usng the dfferental evoluton method. The encodng scheme s dscussed n secton 3.1. For each soluton vector, K- runs and exts when stable centrods formed. The results of K- are evaluated usng ftness functon dscussed n secton 3.2.Durng a generaton; new ndvduals are created usng the dfferental evoluton operators mutaton and crossover dscussed n secton 3.3 and 3.4.The ftness values of tral vectors s compared wth the ftness values of target vectors usng selecton explaned n secton EXPERIMENTAL RESULTS 5.1. Data sets used The data sets used to carry ths work are lsted n Table 1.It gves detaled descrptons of both artfcal data sets and synthetc data sets. The artfcal data sets are downloaded from UCI machne learnng repostory [27]. Table 1. Data sets descrpton. S.No. Data Set sze #of features # of clusters 1 Irs (class 1-50, class 2-50, class 3-50) 2 Wne (class 1-59,class 2-70, class 3-49) 3 New Thyrod (class 1-150,class 2-35, class 3-30) 4 Haberman (class 1-227,class 2-79) 5 Glass (class1-70,class 2-76, class 3-17,class 4-0,class 5-13, class 6-9, class 7-29) 6 Synthetc data set (class 1-500,class ) 7 Synthetc data set (class 1-500,class 2-500) 8 Synthetc data set (class 1-500,class 2-500) 9 Synthetc data set (class 1-250, class 2-150, class ) 10 Synthetc data set (class 1-150,class 2-150, class ) 11 Synthetc data set (class 1-100,class 2-150, class 3-100) 12 Synthetc data set 7 13 Synthetc data set (class1-150, class 2-100,class3-100,class 4-50, class 5-50) 4(class 1-100,class 2-150,class 3-200, class 4-80) 65

10 5.2. Scatter plots for data sets Fgure 2 shows the scatter plots of 8 synthetc data sets Comparson of Modfed K- and K- The results of modfed K- and K- were shown n two separate tables (table 2 and table 3) due to large sze. The followng secton gves detaled nformaton about them Results of DE appled to modfed K- The tables 2 and 3 shows average values of neural network parameters when DE runs about 100 runs and each run conssts a maxmum of 20 generatons usng both K- and Modfed K-. Table 2. Mean values of neural network parameters usng DE and Modfed K- 66

11 Results of DE appled to K- S.No Table 3. Mean values of neural network parameters usng DE and K- Data set Avg. no of clusters Avg. error rate Avg. ftness Avg. tolerance Avg.vgl ance 1 Irs ,0.4657,0.5569, Wne ,0.3196,0.3752,0.3378, ,0.2596,0.2985,0.3448, ,0.3413,0.3139,0.3429, New thyrod ,0.2742,0.2539, Glass ,0.0841,0.0644,0.0405, ,0.0824,0.1306,0.0398, Haberman ,0.0492, Syndata , Syndata , Syndata , Syndata ,3.4220,3.6010,3.6698, ,3.1782,2.9961, Syndata ,3.2593,3.5499,3.5297, ,3.1813,3.3929, Syndata6 3.89(Actual number s 3) ,1.7823,2.0226,2.0770, , ,2.0789, Syndata ,3.8599,3.8655,3.0607, ,4.1074,3.8569,3.3853, ,3.5799,3.7644, Syndata8 4.75((Actual ,9.0517,10.054,10.901, number s 4) ,11.417,9.6887,9.3184, , The results also compare both the versons of K- combned wth optmzaton algorthm called DE to determne best values of parameters vglance and features tolerance. For most of the data sets used n ths work, t was observed that modfed K- s effcent than orgnal K- n terms of error rate and ftness. Both algorthms were able to dscover number of clusters actually present n the data set shown n the frst column of the above two tables. The performance of modfed K- s better than the orgnal K- n terms of average error rate obtaned over 100 runs. In each generaton, the values of both tolerance and vglance values are recorded and the mean value s taken for both parameters at the end whch were shown n table 2 and table Results Based on maxmum vglance Table 4 shows maxmum vglance and correspondng tolerance values obtaned by DE for 100 runs. 67

12 Table 4. Comparson between modfed K- and K- based on maxmum vglance S.No Data set Algorthm Maxmum vglance(ρ) Number of clusters Features tolerance (δ) Error rate (%) 1 rs Modfed K- 0.75(3/4) ,0.6788,0.5608, K (3/4) ,0.3327,0.5247, Wne Modfed K (9/13) ,0.3101,0.2929,0.3157,0.3823, ,0.2822,0.4702,0.4058,0.3388, ,0.4672, K (9/13) ,0.2005,0.2188,0.4898,0.4101, ,0.3386,0.3851,0.3528,0.3982, ,0.4017, New Modfed K- 0.8(4/5) ,0.2167,0.4847,0.3654, Thyrod K- 0.8(4/5) ,0.4982,0.2404,0.3778, Glass Modfed K , ,0.0205, ,0.0189,0.0911, ,0.0191, K ,0.0938,0.1048,0.0613,0.0269, ,0.2059, , Haberman Modfed K (1/3) ,0.0517, K (1/3) ,0.0355, Syndata1 Modfed K- 0.5(1/2) , K- 0.5(1/2) , Syndata2 Modfed K- 0.5(1/2) , K- 0.5(1/2) , Syndata3 Modfed K- 1(2/2) , K- 1(2/2) , Syndata4 Modfed K- 1(8/8) ,5.3906,5.0819,4.3982,3.1581, ,4.9923, K- 1(8/8) ,2.3999,3.7814,5.1546,5.4707, , , Syndata5 Modfed K- 1(8/8) ,4.1205,4.6169,5.1569,2.6966, ,4.7333, K- 1(8/8) ,2.8447,5.1821,4.7824,5.6415, ,5.3356, Syndata6 Modfed K- 0.75(6/8) 4(Actual ,2.3158,2.6881,0.8054,1.9819, number s 3) ,1.7463, Syndata7 Modfed K- 13 Syndata8 Modfed K- K- 0.75(6/8) 4 (Actual number s 3) (10/12) 4(Actual number s 5) ,1.4159,2.4478,2.8806,1.3764, ,1.0430, ,3.4565,3.9432,4.0429,5.9774, ,4.3376,0.4221,6.3798,5.6094, , K (10/12) ,6.2038,3.4297,3.1956,5.0315, ,3.7991,2.8922,6.3156,4.8360, , (8/10) 5(actual number s 4) K- 0.8(8/10) 6(actual number s 4) , ,8.6679,12.899, ,6.855,9.6163,13.619,13.702, , , , , , , , , , The error rate was shown for maxmum vglance because vglance tells about the number of features that are wthn ther tolerance out of total features les n the data set. The values n bold shows mnmum error rate obtaned by the two varatons of K-.It was observed that modfed K- performs well for rs, wne, new thyrod, Glass, etc.it was shown that, the performance of both algorthms.e. the error rate s same for all well separated data sets 68

13 (syndata1, syndata2, syndata 4 and syndata5). For overlapped data sets syndata 7 and syndata 8 K- s effcent due to low error rate for maxmum vglance observed over all 100 runs. For most of the data sets the maxmum vglance value s same wth dfferent feature tolerances and ms-classfcaton rate Results Based On Mnmum Error Rate Table 5 shows the mnmum error rate obtaned for each data set over all 100 runs. For each data set, the number of clusters formed, tolerances of each feature and vglance value s recorded for mnmum error rate. For rs data set, the vglance parameter value s same for both the algorthms wth dfferent tolerance values. But ms-classfcaton rate s mnmzed when modfed K- s used. In the case of wne, vglance value s small ( e. 6 features matched out of 13) wth low ms-classfcaton rate. But usng K-, the vglance value s e. 9 features matched out of 13, but error rate s hgh. In the smlar way the analyss can be done for the other data sets also. For a data set, the obtaned feature tolerances and vglance values can be used for the K- algorthms to determne groups n a data set. Table 5. Comparson between modfed K- and K- based on mnmum error rate 69

14 6. CONCLUSIONS In ths work, the optmzaton algorthm s appled to neural network to set ntal parameters called vglance (ρ) and tolerance (δ). From the above results, t was observed that the number of output nodes generated (centrods) depends on these parameters. The nodes created n output layer gve the number of clusters present n a data set. So usng effcent values of the parameters found usng DE, optmal clusterng s acheved. Also orgnal K- algorthm s modfed to determne the best matchng unt out of the total matched nodes to mprove accuracy. The above results show that modfed K- s effcent to dscover actual number of clusters presents n a data set wth the best values of vglance and tolerance les n a soluton vector. In ths paper, the comparson s also made between orgnal K- and modfed K-.The above results show that the percentage of tuples ms-classfed (error rate) s mnmum usng modfed K- compared to orgnal K-. In some cases, the orgnal K- s profcent than modfed K-. So n future, modfed K- can be mproved for hghly overlapped data sets. In future, the algorthm can be extended to apply for medcal magng applcatons. 70

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