Criterion in selecting the clustering algorithm in Radial Basis Functional Link Nets

Transcription

1 WSEAS TRANSACTIONS o SYSTEMS Ag Sau Loog, Og Hog Choo, Low Heg Chi Criterio i selectig the clusterig algorithm i Radial Basis Fuctioal Lik Nets ANG SAU LOONG 1, ONG HONG CHOON 2 & LOW HENG CHIN 3 Departmet of Mathematical Scieces Uiversiti of Sais Malaysia Uiversiti of Sais Malaysia, 118, Pulau Piag MALAYSIA Dellphi1223@yahoo.com Abstract: - K-meas ad k-media clusterig algorithms ca help i the selectio of cetres for the Radial Basis Fuctioal Lik Nets. Radial Basis Fuctioal Lik Nets is used to classify the data. I this paper, we will show the importace of kowig the skewess of the data i decidig to choose betwee k-meas or k- media clusterig algorithm i fidig the cetre of Radial Basis Fuctioal Lik Nets ad we will also show that this iitial selectio criterio will result i the improvemet of efficiecy i terms of speed ad accuracy i data classificatio. Two sets of real data are used to demostrate our results. Key-Words: - Radial Basis Fuctioal Lik Nets; Radial Basis Fuctio Network; k-meas; k-media; clusterig; skewess. 1 Itroductio This paper shows the importace of calculatig the skewess of data ad i makig a choice betwee k- meas ad k-media clusterig method i the cetre selectio of the Radial Basis Fuctioal Lik Nets (). We use the Mardia s skewess to obtai the iformatio about the ature of the data [1]. If the data is skewed, it is better to select k-media as our clusterig algorithm i cetre selectio for Radial Basis Fuctioal Lik Nets. However, if the data is almost symmetrical, the k-meas will be a better choice. There are may clusterig methods to fid the cetre of a Radial Basis Fuctioal Lik Nets like DBSCAN [2], Dyamic Clusterig [3] ad may other methods. However, amog these clusterig methods, k-meas ad k-media algorithms usually give the solutio i the quickest ad more efficiet way. K-meas algorithm is liked to the Radial Basis Fuctio cetre selectio to ehace the performace [4]. The k-meas algorithm self orgaizes to create partitios, which act as the clusters [5]. The data which fall ito ay cluster will be averaged ad fially we will obtai the cetre of the cluster. The k-media will do the same process but istead of averagig, we will use the media i fidig the cetre of every cluster [6]. By usig these clusterig methods, we ca reduce the distaces betwee iput data ad the cetre of the Gaussia fuctio. This will eable the sum of squared error to coverge faster so that the optimum solutio ca be obtaied oce the sum of squared error exceeds the stoppig criteria [7]. K- meas ad k-media algorithms also help i cuttig dow the umber of iput data that are out of the coverage of the feature space regio by clusterig them ito suitable partitios. Radial Basis Fuctioal Lik Nets used by Looey [8] was origiated by Pao [9]. It had bee prove to be more efficiet tha Radial Basis Fuctio Network [1] ad [11]. The core fuctio i Radial Basis Fuctioal Lik Nets is the Gaussia fuctio that had bee prove as uiversal robust approximators [12]. By applyig Radial Basis Fuctio, we ca use the etwork i fuctio approximatio, iterpolatio, classificatio ad patter recogitio [13]. However, i this study, we use two real data sets i classificatio to determie the accuracy ad speed i the performace results. 2 Methodology I this study, we apply K-meas ad K-media clusterig algorithms to obtai the cetres of the hidde odes accordig to the Mardia s multivariate skewess calculatio. The most importat property of Mardia s multivariate skewess calculatio is that it ca idicate the ature of the iput data as to whether the iput data are skewed or early symmetrical [14]. Thus, Mardia s multivariate skewess ca be cosidered as oe of the statistical methods that aalyzes the distributio of the give multidimesioal data. Uder this circumstace, Mardia s multivariate skewess calculatio is useful ISSN: Issue 11, Volume 7, November 28

2 WSEAS TRANSACTIONS o SYSTEMS Ag Sau Loog, Og Hog Choo, Low Heg Chi i gettig the preprocess iformatio o the iput data. Sice our iput data are mostly i multidimesioal form, Mardia s multivariate skewess is used i determiig the ature of distributio for the iput data so that choice of the selectio of cetres betwee K-meas or K-media clusterig algorithm ca be doe objectively. Here, the measuremet of Mardia s multivariate skewess gives the iformatio about the spread of the iput data based o the multivariate ormality so we ca select the clusterig algorithm for fidig the cetre of radial basis fuctioal lik ets accordig to the results obtaied. I other words, the iput data will be tested for ormality to check whether the iput data is skewed or symmetrical. If the data sets are highly skewed, K-media is sufficiet to be chose for computig the cetres from the iput data while if the data sets are symmetrical, K-meas becomes the better choice. The Mardia s multivariate skewess is defied as skewess = 1 - probchi ( kappa1, dfchi ) (1) where probchi is the probability of chi square [15]. The first parameter, kappa1 is defied as kappa1 = * Beta1hat / 6 (2) where is the umber of dataset while Beta1hat is Mardia s sample skewess. The secod parameter dfchi is the degree of freedom for the chi square approximatio of multivariate skewess. dfchi = p * ( p+1 ) * ( p + 2 ) / 6 (3) where p is the dimesio of the dataset. I k-meas algorithm, we must decide the value of k to be used first. Sice the k is priori kowledge, we will set it to be the same as the umber of cetres i the hidde odes of Radial Basis Fuctioal Lik Nets. I order to get the best solutio for k-meas, it is better to ru the k-meas algorithm may times. This will icrease the probability that the data will be classified to the earest cluster ad yield the optimum solutio. I this study, the k-meas algorithm is applied to ru for 1 times ad we will use the result as our cetre of hidde odes i Radial Basis Fuctioal Lik Nets. Geerally, the k-meas fuctio is defied as K N J = x ( j) c j 1 i = i= 1 j (4) ( j where x ) i - c j is the distace betwee oe of the data x ad a cetre of a cluster, c j. i K-media uses the same formula as i Equatio (4) but the differece is that istead of usig averagig, it will take the media of the data to be the cetre for each cluster. Radial Basis Fuctioal Lik Nets is a hybrid or extesio of the Radial Basis Fuctio Network. The basic idea for Radial Basis Fuctio is that the feature space [, 1] is covered M overlappig circular hyper ball regios. I every regio there exists a cotiuous Radial Basis Fuctio that assumes its maximum value at the cetre of the regio but reduces i value to zero whe away from it. I this study, we use Radial Basis Fuctioal Lik Nets to classify multidimesioal data ito their respective classes. The structure of the Radial Basis Fuctio Network cosists of three layers. The first layer is the iput layer. The secod layer will be the hidde layer where the hidde euros are located. The last layer will be the output layer. I this layer, the output is compared the target values to form the value for the sum of squared error. We apply Radial Basis Fuctio Network Gaussia fuctio because it ca perform exact iterpolatio of a set of data poits which is i a high dimesioal space [16]. Exact iterpolatio is the process to map every iput vector exactly to its correspodig target vector [17]. If the data sets which we feed ito the eural etworks are o-liear, it is better if the eural etworks are able to perform a o-liear mappig which ca trasform a o-liear separable classificatio problem ito a liear separable problem [18]. The liear separable problem is easier to be solved tha the o-liear problem [18]. The Radial Basis Fuctio Network which has the Gaussia fuctio as the kerel fuctio possesses the ability to perform the task. Geerally, Radial Basis Fuctio is mappig fuctios, s, as show below: s : R R (5) where is the umber of the dimesio i the iput. From Equatio (5), we observe that the multidimesioal iput R are trasformed ito sigle dimesioal output, R. ISSN: Issue 11, Volume 7, November 28

3 WSEAS TRANSACTIONS o SYSTEMS Ag Sau Loog, Og Hog Choo, Low Heg Chi Whe a set of data which cosists of Q umber of iput vectors x () where =1,, N the correspodig outputs, t, we aim to fid a fuctio g(x) i such a way that g x t N ( ) ( ) = = 1,..., (6) Radial Basis Fuctio has the form of g( x ) = wφ( x x ) = 1,..., N (7) where each data poit x belogs to a basis fuctio ad the basis fuctio takes the form φ( x x ) where φi ( ) is the o-liear fuctio such as the Gaussia fuctio. x x is the distace betwee the iput vectors, x from the cetres of Gaussia fuctio, x. w is the weight i the Radial Basis Fuctio [16]. By havig the similar form as the geeralized liear discrimiat fuctio, we ca write the iterpolatio coditio of Equatio (6) ad Equatio (7) i matrix form as Φw=t (8) where t (t () ), w (w ), ad the square matrix Φ has elemet Φ ' = φ( x x ) [17]. If the iverse matrix for Φ -1 exists, the solutio to Equatio (8) is give by w=φ -1 t (9) If the data poits are differet for a large class of fuctio φ ( ), the matrix Φ is o sigular which also meas the matrix has a iverse. By settig the weights i Equatio (7) to the values give i Equatio (9), the fuctio g(x) represets a cotiuous differetiable surface that passes exactly through each data poit [19]. I our study case, the umbers of hidde odes M, will be determied heuristically depedig o the results. If we use less M tha the umber of traiig sets, we will get a uder traiig situatio. If M is too large, the computatio o-lie of ukow vectors will be slower ad will build up extraeous error. Therefore we eed to choose the M a larger set tha the iput data, ad try to reduce or icrease the M util the data are well classified. A RBF defied o N-dimesioal feature vectors x is ( δ ( m) ) ( q) ( m) 2 2 y = exp x v / 2 m= 1,..., M m (1) where q is the dataset umber, x is a iput vector, v ( m) is the cetre of m th hidde odes while the spread of the receptive field is δ. y ( m) will be maximum if the distaces betwee x ad v ( m) equals to zero. The δ will determie how far the spread of the circular disk that covers the iterest bouded regio i the feature space [1]. The activatio fuctio used i Radial Basis Fuctioal Lik Nets is the Gaussia fuctio. Oce the iput vector gets ear to the cetre, it will start to activate the Gaussia fuctio, y = f(x). I Radial Basis Fuctioal Lik Nets, there are weights i betwee the iput layer hidde layer ad also the hidde layer the output layer. Both weights are adjusted i a steepest descet way. The iput weights will ifluece the iput vector to get closer to the cetre while the output weights will move the output, z towards the target value, t. ( The iput weights, w j) are updated as ( q ) ( q ) ( q ) ( 2 η 1 / ) ( j j ) q= Q j= J (11) k+ 1 k j = j + ( 1, ) ( 1, ) w w M t z x where is the iput umber, η 1 is the learig rate, ( q Q is the umbers of dataset, z ) j is the output, ad k is the iteratio umber The output weights, u m ( j ) are updated as ( q ) ( q ) ( q ) ( 2 η 2 / ) ( j j ) q= Q j= J (12) m k+ 1 k mj = mj + ( 1, ) ( 1, ) u u M t z y where η 2 is the learig rate. The differeces betwee the observed value, z ad the target value, t will be calculated as the sum of squared error. Our aim is to obtai the miimum total sum squared of error. The output z ad the total sum squared of error E are defied as ( 1/ M) ( 1/ N) z = u y + w x (13) (q) (q) (q) j ( m= 1, M ) mj m ( = 1, N ) j ( { ( M) E= t ( 1/ u y + (q) (q) (q=1,q) (j=1,j) j ( m= 1, M ) mj m (q) ( 1/ N) }) 2 = ( = 1, N ) w x (14) j I the Radial Basis Fuctioal Lik Nets used, the spread parameter will be fixed as it will be easier to compare the results usig k-meas ad k-media i fidig the cetres for the hidde odes i the hidde layer. O the other had, the cetres obtaied from differet clusterig methods are used ISSN: Issue 11, Volume 7, November 28

4 WSEAS TRANSACTIONS o SYSTEMS Ag Sau Loog, Og Hog Choo, Low Heg Chi as the cetre of each hidde euro to ru whe traiig the data. 3 Results I this sectio, we show the criterio i selectig the clusterig algorithm i Radial Basis Fuctioal Lik Nets based o two data sets. The ature of the data sets is idetified ad umbers of attribute are stated. We explai the classes for each data set. Oce we idetified the multivariate skewess of each data set, the we decide which clusterig algorithm to be the better choice for selectio of cetres i Radial Basis Fuctioal Lik Nets. The, we feed the data sets i Radial Basis Fuctioal Lik Nets to show the performace of data traiig i term of accuracy ad speed. Basically, we separate the data sets ito two subgroups which are the testig data set ad the traiig data set. The traiig data sets are used to trai the Radial Basis Fuctioal Lik Nets while the test sets are used to show the accuracy of the type of method beig applied for the selectio of cetres. To determie whether the data sets are correctly classified or wrogly classified, a rule is set. The strict rule is that ay output values which fall betwee.4 ad.6 is cosidered wrogly classified [8]. This rule gives a tolerace space for the output to be grouped ito its class ad the rule also filters out those output which do ot belog to ay class. By applyig the rule, we ca calculate the umber of traiig misses i the Radial Basis Fuctioal Lik Nets whe applyig the differet methods i selectig the cetres. Results are orgaized i tables form for compariso purposes. We show the results i the type of method used, umber of iteratios for the method used, ad the testig error ad traiig misses i Table 2, Table 3, Table 5 ad Table 6. Next, we the commet o the performaces of the differet types of method used i Radial Basis Fuctio Network ad based o the umber of iteratios usig the sum of squared error ad the umber of traiig misses. The sum of squared error i each category for the umber of iteratios is discussed i terms of speed. The umber of traiig misses is discussed to show the accuracy of the results obtaied. We summarize the results ad give the overall performaces of the traiig by Radial Basis Fuctioal Lik Nets. We also state the importace of Mardia s multivariate skewess measuremet i determiig the clusterig method which is more suitable to be used i selectio of cetres for the Radial Basis Fuctioal Lik Nets. The first iput data is the geological oil exploratory data set ad this data set cosists of 4 dimesios [8]. Each dimesio represets a feature ad all the features aalyzed will lead to the determiatio of which exploratios to be made (for example, for oil or coal). There are 69 data poits i the geological oil exploratory data set. The first te data poits will be used as the testig set. The rest we will put ito the traiig set to see the results. The secod data set is the fittig cotact leses ad this data set cosists of 24 data poits four attributes [2]. The data set is oise free, highly simplified problem, ad each data poit is complete ad correct. The applicatio of this data ca be foud i [21]. The first attribute i the data set is the age of the patiet while the secod attribute is the spectacle prescriptio. The third attribute idicates whether the patiet is astigmatic or o-astigmatic whereas the fourth attribute is the tear productio rate of the patiet. The patiets are categorized ito three classes ad these classes are fitted hard cotact leses, fitted soft cotact leses ad ot fitted cotact leses. The patiet should be classified ito oe of the classes accordig to their attributes. Table 1 The skewess measuremet for datasets Dataset Geological Oil exploratory Fittig cotact leses Skewess Testig Traiig x 1-9 Testig Traiig 1 ISSN: Issue 11, Volume 7, November 28

5 WSEAS TRANSACTIONS o SYSTEMS Ag Sau Loog, Og Hog Choo, Low Heg Chi Table 2 Results of traiig the data for testig based o the 1 data poits i the Geological Oil Exploratory data set (skewed) Table 3 Results of traiig the data based o the 59 data poits i the Geological Oil Exploratory data set (skewed) Type of methods used radom cetre selectio k-meas k-media No. of iteratios Testig error Traiig misses Type of methods used radom cetre selectio k-meas k-media No. of iteratios Testig error Traiig misses ISSN: Issue 11, Volume 7, November 28

6 WSEAS TRANSACTIONS o SYSTEMS Ag Sau Loog, Og Hog Choo, Low Heg Chi Fig. 1 Iteratio versus total sum of squared error for geological oil exploratory testig data set (skewed) Table 4 Results of traiig the data for testig based o the 8 data poits i the Fittig Cotact Leses data set (symmetrical) T o t a l S u m o f S q u a r e d E r r o r Iteratio versus Total Sum of Squared Error for the Geological Oil Exploratory Testig Data Set (Skewed) Iteratio Radom K-meas K-media Types of method used radom cetre selectio No. of iteratios Testig error Traiig misses Fig. 2 Iteratio versus total sum of squared error for Geological Oil Exploratory traiig data set (skewed) k-meas Iteratio versus Total Sum of Squared Error for the Geological Oil Exploratory Traiig Data Set (Skewed) T o t a l S u m o f S q u a r e d E r r o r k-media Iteratio i ' Radom K-meas K-media ISSN: Issue 11, Volume 7, November 28

7 WSEAS TRANSACTIONS o SYSTEMS Ag Sau Loog, Og Hog Choo, Low Heg Chi Table 5 Results of traiig the data based o the 16 data poits i the Fittig Cotact Leses data set (symmetrical). Fig. 3 Iteratio versus total sum of squared error for the Fittig Cotact Leses testig data set (symmetrical) Types of method used radom cetre selectio No. of iteratios Testig error Traiig misses 9 T o t a l S u m o f S q u a r e d E r r o r Iteratio versus Total Sum of Squared Error for Fittig Cotact Leses Testig Data Set (Symmetrical) Iteratio k- meas Radom K-meas K-media Fig. 4 Iteratio versus total sum of squared error for the Fittig Cotact Leses traiig data set (symmetrical) Iteratio versus Total Sum of Squared Error for Fittig Cotact Leses Traiig Data Set (Symmetrical) k-media T o t a l S u m o f S q u a r e d E r r o r Iteratio i ' Radom K-meas K-media ISSN: Issue 11, Volume 7, November 28

8 WSEAS TRANSACTIONS o SYSTEMS Ag Sau Loog, Og Hog Choo, Low Heg Chi 4 Discussios Table 1 shows the value of skewess for both the data sets, amely the oil exploratory data ad the fittig cotact leses data. The skewess values rage from to 1. If the skewess value is ear to 1, the data is almost symmetrical but if the values tur out to be closer to, this meas the data set is skewed. It is obvious that the the skewess of the datasets has its ifluece i decidig which clusterig method is best used to fid the cetres. For the geological oil exploratory testig data set, the skewess is which is almost i the middle of ad 1. Sice the data set is oly used for testig purpose, therefore we apply both clusterig algorithms to verify which oe is better to be used for selectio of the Gaussia cetres. For the traiig data set of the geological oil exploratory, the value of skewess is x 1-9 ad it is almost. Theoretically, the data set is highly skewed ad K-media will be better i selectig the Gaussia cetres. Besides, the fittig cotact leses testig data set takes as the value of the skewess. That meas the testig data set is more symmetrical. For the fittig cotact leses traiig data set, the skewess is 1 ad the data set is symmetrical. Whe the data set is more symmetrical as i the case of the fittig cotact leses data set, k-meas algorithm is the best choice for selectig the Gaussia cetre. Table 2, Table 3, Table 4 ad Table 5 show the effect of the preprocess calculatio o skewess i selectig the clusterig algorithm. For the observatios i the testig data based o geological oil exploratory dataset ad fittig cotact leses data set, the total sum of squared error reduces very quickly mostly because the data set to be traied is small. We set the stoppig criterio to be.1 to halt the traiig of each data set. The value of the stoppig criterio is small which also meas that the sum of squared error reach a level where the output values are close to the target values ad we stop the traiig process of data classificatio for result evaluatio. If the sum of squared error is less tha.1, the the iteratio will stop immediately ad our aim is achieved. Overall, the sum of squared error of the traiig coverges i speed. Accuracy of the traiig is ot oly based o total sum of squared error but also based o the umber of traiig data missed. We set a strict tolerace to determie the output data that will be cosidered misclassified or traiig miss. Table 2 shows that k-media is the best amog the three methods after 1 iteratios. This is the followed by K-meas ad the radom selectio method. We ca also see the comparative performace of these methods i Fig. 1. The sum of squared error of the test data set usig three differet methods coverges i speed. Besides, we foud that there is o traiig miss for the two data sets due to the efficiecy of the powerful tool, Radial Basis Fuctioal Lik Nets, beig used. Therefore, we will justify which clusterig method to use based o the results that achieve the lowest sum of squared error. Applicatios of K-meas ad K-media clusterig clearly improve the performace of the Radial Basis Fuctioal Lik Nets but the choice of whether to use K-meas or K- media depeds o the skewess of the data. By the way, we ca see cosistet result i the geological oil exploratory traiig for both testig ad traiig data set. We observe that k-media coverge fastest i Table 3 ad this is show i Fig. 2. As the data is much skewed as i the case of the geological oil exploratory data i Table 3, K-media outperforms other methods such as K-meas ad radom selectio. Table 2 ad 3 also show that for both testig ad traiig data, K-meas performs better tha radom selectio of cetres because K- meas reduces the umber of outliers i our iput data so that each output has a better chace to get closer to the cetre of the Gaussia fuctio. Table 4 shows that the k-meas has the lowest sum of squared error after 1 iteratios i the ru for testig amog the three methods. All the methods have o traiig miss ad the tred of the total sum of squared error is show i Fig. 3. Table 5 shows that k-meas outperforms other methods for a more symmetrical data set. Therefore, if the data is more symmetrical the skewess of Mardia ear to 1 as i the fittig cotact leses data set, k-meas will perform better (Fig. 4). We have reduced the distaces betwee iput data from the cetre of the Gaussia Fuctio i both models by usig these clusterig methods so that the sum of squared error coverges faster ad the optimum solutio ca be obtaied oce the sum of squared error exceeds the stoppig criteria. K- meas ad K-media algorithms also help i reducig the umber of iput data that are out of the coverage feature space regio by clusterig them ito suitable partitios. I other words, the outlier ca be reduced ad this will certaily icrease the power of traiig speed ad accuracy. If we use radom cetres for the iput vectors, we might lose the best positio for the cetres which represet the distributio of iput data. ISSN: Issue 11, Volume 7, November 28

9 WSEAS TRANSACTIONS o SYSTEMS Ag Sau Loog, Og Hog Choo, Low Heg Chi 5 Coclusio Applicatio of k-meas ad k-media clusterig clearly improve the performace of the Radial Basis Fuctioal Lik Nets i terms of speed ad accuracy but the choice of whether to use k-meas or k- media depeds o the skewess of the data. Sice the preprocessig of the iput data usig Mardia s multivariate skewess ca show the details of the iput data, we ca choose the clusterig algorithm accordig to the type of data. By usig the clusterig algorithm determied by Mardia s multivariate skewess i selectio of cetres of the Gaussia fuctio, we ca see the differece i the performace. K-meas are better tha K-media for symmetrical data while K-media outperforms the former whe the give iput data is skewed. Both algorithms are useful i helpig the Radial Basis Fuctioal Lik Nets i shorteig the traiig iteratio ad improvig the accuracy of the outcomes. As the iitial selectio of cetres have great impact o the performace of the eural etworks, the choice of the differet types of clusterig algorithms become a very importat issue. The measure of skewess is just oe of the tools to determie the selectio betwee K-meas ad K- media but it may ot be relevat if we use other clusterig methods. Hece, our future research is to fid out more relatioships betwee the iput data ad properties of the clusterig algorithms so that we are able to make a big step forward i improvig the performace of Radial Basis Fuctioal Lik Nets. Besides, there is still room to improve the efficiecy of Radial Basis Fuctioal Lik Nets. The determiatio of the optimum umber of hidde odes, M, i the Radial Basis Fuctioal Lik Nets remais a challege. The type of activatio fuctios i the radial fuctioal lik ets ca also be ivestigated for the possibility to be replaced by a more stable ad efficiet fuctio which is able to boost the performace of the eural etwork. Furthermore, several importat parameters such as the spreads ad the weights which lie i betwee the layers ca be studied to create more advaced Radial Basis Fuctioal Lik Nets. The spreads which cotrol the size of the circular disks covered o the feature space of iterest are importat to make sure all the iput data are uder their ifluece. The rages of the iitial weights are also importat so that a set of efficiet iitial weights ca be obtaied to icrease the speed of traiig. Ackowledgemets This work was supported i part by U.S.M. Fudametal Research Grat Scheme (FRGS) o. 23/PMATHS/ Refereces: [1] K. V. Mardia, Measures of multivariate skewess ad kurtosis applicatios, Biometrika, Vol.57, No.3, 197, pp [2] M. Ester, H. P. Kriegel, J. Sader, ad X. W. Xu, A desity-based algorithm for discoverig clusters i large spatial database oise, Proceedigs of the 2 d Iteratioal Coferece o Kowledge Discovery ad Data Miig (KDD-96), 1996, pp [3] J. T. Tou, Dyoc-a dyamic optimal cluster seekig techique, Iteratioal Joural of Computig ad Iformatio Scieces, Vol.8, 1974, pp [4] A. Cichocki ad R. Ubehaue, Neural etworks for optimizatio ad sigal processig, Joh Wiley ad Sos, New York, [5] J. Macquee, Some methods for classificatio ad aalysis of multivariate data, Proceedigs of the 5 th Berkeley Symposium o Probability ad Statistics Vol.1, 1967, pp , Uiversity of Califoria, Berkeley. [6] L. Mico ad J. Ocia, A approximate media search algorithm i o-metric spaces, Patter Recogitio Letters, Vol.22, 21, pp [7] J. K. Sig, D. K. Basu ad M. Nasipuri, Improved k-meas algorithm i the desig of RBF eural etworks, Proceedigs of the Coferece o Coverget Techologies for Asia-Pasific Regio, Teco, Vol.2, 23, pp , Bagalore, Idia. [8] C. G. Looey, Radial Basis Fuctioal Lik Nets ad fuzzy reasoig, Neurocomputig Vol.48, No.(1-4), 22, pp [9] Y. H. Pao, G. H. Park ad D. J. Sobajic, Learig ad geeralizatio characteristic of the Radom Vector Fuctioal Lik Net, Neurocomputig, Vol.6, 1994, pp [1] C. G. Looey, Patter Recogitio usig Neural Networks, Oxford Uiversity Press, New York, [11] S. L. Ag, H. C. Og ad H. C. Low, Compariso betwee modified Radial Basis Fuctioal Lik Nets ad Radial Basis Fuctio, Proceedigs of the 2 d Idoesia- Malaysia-Thailad Growth Triagle (IMT-GT), Regioal Coferece o Mathematics, ISSN: Issue 11, Volume 7, November 28

10 WSEAS TRANSACTIONS o SYSTEMS Ag Sau Loog, Og Hog Choo, Low Heg Chi Statistics ad Applicatios, Vol.4, 26, pp [12] J. T. H. Lo, Multilayer perceptros ad Radial Basis Fuctio are uiversal robust approximators, Neural Networks Proceedigs, IEEE World Cogress o Computatioal Itelligece, Vol.2, 1998, pp [13] M. Sgarbi, V. Cola ad L. M. Reyeri, A compariso betwee weighted Radial Basis Fuctios ad Wavelet Networks, Proceedigs of the 6 th Europea Symposium o Artificial Neural Network, 1998, pp , Bruges, Belgium. [14] K. V. Mardia, Assessmet of Multiormality ad the Robustess of Hotellig s T2 Test, Applied Statistics, Vol.24, No.2, 1975, pp [15] R. Khattree ad D. N. Naik, Applied Multivariate Statistics SAS Software, SAS Istitute Icorporated, Cary, North Carolia, USA, [16] M.J.D. Powell, Radial Basis Fuctios for Multivariable Iterpolatio: A Review, Claredo Press, Oxford, [17] C. M. Bishop, Neural Networks for Patter Recogitio, Oxford Uiversity Press, New York, [18] S. Hayki, Neural Networks. A Comprehesive Foudatio, Pretice Hall, New Jersey, [19] C. A. Micchelli, Iterpolatio of Scattered Data: Distaces Matrices ad Coditioally Positive Defiite Fuctios, Costructive Approximatio, Vol.2, 1986, pp [2] C. Blake, Repository of machie learig databases, Uiversity of Califoria, Irvie, Dowloadable at [21] I. H. Witte ad B. A. MacDoald, Usig Cocept Learig for Kowledge Acquisitio, Iteratioal Joural of Ma-Machie Studies, Vol.27, 1988, pp ISSN: Issue 11, Volume 7, November 28