Multiobjective Space Search Optimization and Information Granulation in the Design of Fuzzy Radial Basis Function Neural Networks

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1 636 Joural of Electrcal Egeerg & Techology Vol. 7, No. 4, pp. 636~645, ultobjectve Space Search Optmzato ad Iformato Graulato the Desg of Fuzzy Radal Bass Fucto Neural Networks We Huag*, Sug-Kwu Oh ad Hoghao Zhag* Abstract Ths study troduces a formato graular-based fuzzy radal bass fucto eural etworks (FRBFNN) based o multobjectve optmzato ad weghted least square (WLS). A mproved multobjectve space search algorthm (ISSA) s proposed to optmze the FRBFNN. I the desg of FRBFNN, the premse part of the rules s costructed wth the ad of Fuzzy C-eas (FC) clusterg whle the cosequet part of the fuzzy rules s developed by usg four types of polyomals, amely costat, lear, quadratc, ad modfed quadratc. Iformato graulato realzed wth C-eas clusterg helps determe the tal values of the apex parameters of the membershp fucto of the fuzzy eural etwork. To ehace the flexblty of eural etwork, we use the WLS learg to estmate the coeffcets of the polyomals. I comparso wth ordary least square commoly used the desg of fuzzy radal bass fucto eural etworks, WLS could come wth a dfferet type of the local model each rule whe dealg wth the FRBFNN. Sce the performace of the FRBFNN model s drectly affected by some parameters such as e.g., the fuzzfcato coeffcet used the FC, the umber of rules ad the orders of the polyomals preset the cosequet parts of the rules, we carry out both structural as well as parametrc optmzato of the etwork. The proposed ISSA that ams at the smultaeous mmzato of complexty ad the maxmzato of accuracy s exploted here to optmze the parameters of the model. Expermetal results llustrate that the proposed eural etwork leads to better performace comparso wth some exstg eurofuzzy models ecoutered the lterature. Keywords: Fuzzy Radal Bass Fucto Neural Networks (FRBFNN), Improved ultobjectve Space Search Algorthm (ISSA), Iformato Graulato (IG), Weghted Least Squares (WLS). Itroducto Fuzzy Radal Bass Fucto Neural Networks (FRBFNNs) has bee utlzed umerous felds for egeerg, medcal egeerg, ad socal scece [, ]. They are desged by tegratg the prcples of Radal Bass Fucto Neural Networks (RBFNNs) ad vokg the mechasms of formato graulato provded by the Fuzzy C-eas (FC). There are two ma ad coflctg objectves gudg the desg of FRBFNNs. Oe s the accuracy of the etwork (to be maxmzed) ad aother oe s ts complexty (to be mmzed). The objectve of ay effectve learg method s to develop accurate, smple, ad terpretable FRBFNNs. I the 99s, the emphass of eurofuzzy modelg was clearly postoed o accuracy maxmzato. As powerful optmzato tools may scece felds [3, 4], varous evolutoary algorthms have bee proposed to mprove the accuracy of models cludg such as Geetc Algorthms Correspodg Author: Dep. of Electrcal Egeerg, The Uversty of Suwo, Korea. (ohsk@suwo.ac.kr) * School of Computer ad Commucato Egeerg, Taj Uversty of Techology, Cha. (huagwabc@63.com) Receved: Jue 8, ; Accepted: February 7, (GAs), Partcle Swarm Optmzato (PSO), ad Geetc Programmg (GP) [5-7]. Those methods usually help mprove the accuracy, whle the complexty of the model creases as a result of the accuracy maxmzato. Some researchers attempted to smultaeously optmze the accuracy ad the complexty of the fuzzy models [8, 9]. Evdetly, such crcumstaces oe has to become aware of the accuracy-complexty tradeoff. Recetly, the accuracy maxmzato ad complexty mmzato have bee ofte cast the settg of mult-objectve optmzato. A umber of evolutoary algorthms (EAs) have bee developed to solve mult-objectve optmzato problems such as NSGA-II [], IC-OP [], ad so o [, 3]. These EAs are populato-based algorthms, whch allow explorg smultaeously dfferet segmets of the Pareto frot. As a result, multobjectve optmzato techques have bee appled to the desg of fuzzy models strvg for ther hgh accuracy ad sgfcat terpretablty [4-8]. I our prevous study [9], we troduce a FRBFNN based o a mult-objectve space search algorthm. Ths work provdes some ehacemets of the eural etwork. Notceably, some costras of ths eural etwork are as follows: () there s oly a sgle type of all polyomals of

2 We Huag, Sug-Kwu Oh ad Hoghao Zhag 637 the cosequece part of fuzzy rules, so the flexblty ad predctve ablty are lmted; ad () the C-meas method s exploted to costruct fuzzy rules, yet the formato of the dataset s ot adequately abstracted for the desg of fuzzy rules. I ths paper, we troduce a fuzzy radal bass fucto eural etworks based o formato graulato ad mult-objectve optmzato. Whle ths s a exteded work of [9], here we use a modfed FRBFNN ad propose a mproved mult-objectve space search algorthm (ISSA) to optmze ths FRBFNN. The resultg FRBFNN addresses the two lmtatos metoed above. O the oe had, formato graulato ad resultg formato graules themselves become a mportat desg aspect of fuzzy rules. Wth the use of formato graulato, the formato of the dataset s effcetly used for the desg of fuzzy rules. O the other had, stead of the ordary least squares (OLS) that was commoly used FRBFNN, the weghted least squares (WLS) method s exploted here to estmate the coeffcets of the cosequet s polyomals. Wth the use of WLS, FRBFNN exhbts dfferet types of polyomals, whch ca vary from oe rule to aother. x k x k x kl x k [ xk, xk L, xkl ] A A A w k w k w k f ( x, L, x ) k k kl f ( x, L, x ) kl f ( x, L, x ) k kl R : IF x s clued cluster A THEN y = f ( x, x,..., x ) k k k k kl [ xk, xk (a) Wthout use of IG A L, x ] kl { v, L, v } w k f( x k, L, x kl, v) ŷ k. Archtectures ad Learg of the FRBFNN x k x kl A A w k w k f ( x k, L, x kl, v ) f x, L, x, v ) ( k kl ŷ k I covetoal RBFNNs, the output of the RBFNN comes as a weghted sum of the actvato levels of the dvdual RBFs. RBFs realzed the form of some Gaussa fucto are the actvato fuctos of hdde odes. Fuzzy clusterg may be used to determe the umber of RBFs, as well as a posto of ther ceters ad the values of the wdths []. The gradet - method or the least square algorthm s used to realze parametrc learg to deal wth the coclusos of the rules [, ]. I comparso wth Fuzzy ferece systems (FIS), the umber of the RBF uts s equal to the umber of the fuzzy f-the rules preset the FIS. Ulke RBFNNs, FRBFNNs the FC s used to form the receptve felds (RBFs), ad the RBFs do ot assume ay explct fuctoal form (such as Gaussa, ellpsodal, tragular ad others). The FRBFNN has the advatage by provdg a through coverage of the etre put space. Gve the zero-order polyomal used as a local model, the resultg accuracy of the model becomes lmted.. Archtecture of FRBFNN Based o IG ad WLS Ths secto we dscuss the FRBFNN whose archtecture s exteded by IG ad WLS. I essece, formato graules [3-5] are vewed as hghly related collectos of objects (data pots, partcular) draw together by some crtera of proxmty, smlarty, or fuctoalty. Fg. llustrates a comparso of two dfferet types of y FRBFNN. I the IG-based FRBFNN, formato graules (the ceters of dvdual clusters) ad actvato levels (degree of membershps) are determed by meas of the FC. As show Fg., t s clear that the dfferece betwee the IG-FRBFNN ad our prevous FRBFNN [9] arses at the cosequet part of fuzzy rules. The FRBFNN based o IG (see Fg. (b)) ca be represeted form of f-the fuzzy rules R : IF xk s clued cluster A THEN y = f ( x, v ) k k Where R s the th fuzzy rule, =,,, s the umber of fuzzy rules (the umber of clusters), f( xk, v ) s the cosequet polyomal of the th fuzzy rule,.e., a local model represetg put-output relatoshp of the th subspace (local area). w k s the degree of membershp (vz. actvato level) of the th local model whle v =[v v R : IF x s clued cluster A THEN y = f ( x, x,..., x, v) k k k k kl (b) Use of IG Fg.. Comparso of two dfferet types of FRBFNNs ()

3 638 ultobjectve Space Search Optmzato ad Iformato Graulato the Desg of Fuzzy Radal Bass Fucto~ v l ] T s the th prototype. Four types of polyomals, amely a costat type (a zero-order polyomal), lear type (a frst-order polyomal), quadratc type (a secod-order polyomal), ad a modfed quadratc type (a modfed secod-order polyomal) are cosdered as the type of cosequet part of fuzzy rules. Oe of the four types s selected for each sub-space as the result of the optmzato, whch wll be descrbed later ths study. We stressed that the RBFNN usg IG does ot suffer from the curse of dmesoalty (as all varables are cosdered e block). As a result, more accurate ad compact models wth a small umber of fuzzy rules by usg hgh-order polyomals may be costructed. Here we admt the cosequet polyomals to be of oe of the followg forms Type Zero-order polyomal (costat type):. Learg of the cosequet part of the FRBFNN Regardg the learg of the cosequet part of the rules, we cosder two types ths study. Oe s the ordary least squares method (OLS), whle the other s a weghted least squares method (WLS). f ( x,, x, v ) = a j k kl j Type Frst-order polyomal (lear type): f ( x,, x, v ) = a + a ( x V ) + + a ( x V ) j k kl j j j jk k jk (a) Use of OLS Type 3 Secod-order polyomal (Quadratc type): f ( x,, x, v ) = a + a ( x V ) + + a ( x V ) j k kl j j j jk k kj + a ( x V ) + a ( x V ) + j( k+ ) j j( k) k kj + a ( x V )( x V ) + j(k+ ) j j + a ( x V )( x V ) j(( k+ )( k+ )/) k ( k ) j k kj Type 4 odfed secod-order polyomal (odfed Quadratc type): f j( xk,, xkl, v) = aj + aj( x Vj) + + ajk( xk Vkj) + + a ( x V )( x V ) + + a ( x V )( x V ) j( k+ ) j j j( k( k+ )/) k ( k ) j k kj The determato of the umerc output of the model, based o the actvato levels of the rules, s gve the form yˆ = w f ( x,, x, v ) k k k kl = () Next, we cosder the desg of IG-based FRBFNN by meas of WLS. A comparso of OLS ad WLS the desg of IG-based FRBFNN s show Fg.. Both OLS ad WLS ca be used to estmate the coeffcets of the cosequet part of fuzzy rules. However, OLS ca oly deal wth the same polyomal types of cosequece part of fuzzy rules, whle WLS ca hadle the dfferet polyomal types of cosequet part of fuzzy rules. (b) Use of WLS Fg.. Comparso of OLS ad WLS the desg of IGbased FRBFNNs:.. OLS Learg OLS s a well-kow global learg algorthm that mmzes a overall squared error JG betwee output of the model ad the expermetal data. m JG = yk wk f( k ) k= = x v (3) where w k s the ormalzed frg (actvato) level of the -th rule, v s the prototype (cetrod), x k ad y k are put data ad output data, respectvely. The performace dex J G ca be expressed a cocse form as follows

4 We Huag, Sug-Kwu Oh ad Hoghao Zhag 639 G T ( ) ( ) J = Y Xa Y Xa (4) where a s the vector of coeffcets of the polyomal, Y s the output vector of real data, X s matrx whch rearrages put data, formato graules (ceters of each cluster) ad actvato level. I case all cosequet polyomals are lear (frst-order polyomals), X ad a ca be expressed as follows w( x v) w( xl v l) w ( x v ) w ( xl vl) X = wk ( x v) wk ( xl v l ) wk ( x v ) wk ( xl vl ) wm ( xm v) wm ( xlm v l ) wm ( xm v ) wm ( xlm vl ) a = [ a a a a a a ] T l l The optmal values of the coeffcets of the cosequet are determed the form T T a= ( X X) X Y (5) X ( x v) ( xl vl) ( x v ) ( x v ) ( x m v) ( xlm vl ) a = [ a a a ] l l =, l For the local learg algorthm, the objectve fucto s defed as a lear combato of the squared error beg the dfferece betwee the data ad the correspodg output of each fuzzy rule, based o a weghtg factor matrx. The weghtg factor matrx, W, captures the actvato levels of put data to th sub-space. I ths sese we ca cosder the weghtg factor matrx to form a dscrete verso of the fuzzy lgustc represetato for the correspodg sub-space. The coeffcets of the cosequet polyomal of the th fuzzy rule ca be determed a usual maer, amely a = ( X WX ) X WY (8) T.. WLS Learg The WLS method s to determe the coeffcets of the model through the mmzato of the objectve fucto J L. The ma dfferece betwee the WLS ad OLS s the weghtg scheme, whch comes as a part of the WLS ad makes ts focused o the correspodg local model. Notce that the coeffcets of the cosequet polyomal of each fuzzy rule are computed depedetly usg a certa subset of the trag data. These computatos ca be mplemeted parallel ad ths case the overall computg load becomes uaffected by the total umber of the rules. m L = k( k ( xk v )) (6) = k= J w y f The performace dex J T ( Y Xa ) W ( Y Xa ) = L = J L ca be rearraged as / / T / / ( W Y W Xa ) ( W Y W Xa ) = = where a s the vector of coeffcets of th cosequet polyomal (local model), W s the dagoal matrx (weghtg factor matrx) whch volves the actvato levels. X s a matrx whch cludes put data shfted by the locatos of the formato graules (more specfcally, ceters of clusters). I case the cosequet polyomal s Type (lear or a frst-order polyomal), X ad a read as follows W w w wm m m = R, (7) 3. ultobjectve Optmzato of the FRBFNN The objectve of learg method s to develop the FRBFNN satsfyg the crtera of accuracy as well as smplcty (complexty). The complexty of the model ca be expressed through the rate of teracto betwee the local models, a ssue beg closely related wth the socalled overlappg crtero [6]. As far as the structure of the FRBFNN s cocered, there are two compoets to be cosdered (optmzed),.e., the umber of fuzzy rules, ad the order of polyomal each cosequet part. Wth regard to parameter detfcato, there s a adjustable fuzzfcato coeffcet used the FC. These three compoets affect the performace of the FRBFNN ad have to be optmzed. Hece, the goal of multobjectve optmzato of the FRBFNN s to determe a value of the fuzzfcato coeffcet, the umber of fuzzy rules ad the type of polyomal beg used each cocluso part of the rules by smultaeously maxmzg the accuracy ad mmzg the complexty of the model. Here we cosder the structural as well parametrc optmzato realzed wth the framework of a ISSA. We start wth the troducto of ISSA ad the dscuss the arragemet ad terpretato of the solutos ad multobjectve fuctos used for the optmzato of the FRBFNN.

5 64 ultobjectve Space Search Optmzato ad Iformato Graulato the Desg of Fuzzy Radal Bass Fucto~ 3. Improved mult-objectve space search algorthm Geerally, multobjectve optmzato geerates a Pareto frot, whch s the set of o-domated solutos. A soluto s sad to be o-domated f t s mpossble to mprove oe objectve of the soluto wthout worseg at least oe other objectves whch s preseted the problem. I ths secto, we develop a mproved multobjectve space search algorthm (ISSA). We frst troduce the sgle-objectve space search algorthm (SSA), a adaptve heurstc optmzato algorthm whose search method comes wth the aalyss of the soluto space [7, 8]. The dea of the SSA comes from the aalyss of the soluto space. I fact, a precodto should be satsfed whe evolutoary algorthm ca fd the optmal soluto. The precodto s that, most of local areas, a pot (soluto) ad the other pots located the pot s adjacet space have the smlar values of the objectve fucto (ftess values). I other words, most of local areas, a soluto wth better ftess s closer to the optmal soluto. Fg. 3 depcts the comparso of two dfferet optmzato problems usg the geetc algorthm (GA). The search method of SSA s based o the operator of space search, whch geerates two basc steps: geerate ew subspace (local area) ad search the ew space. Regardg the geerato of the ew space, we cosder two cases: (a) space search based o selected solutos (deoted here as Case I), ad (b) space search based o the curret best soluto (Case II). To mprove the performace of SSA, we developed a ew spacer search operator of Case I wth the ad of a orthogoal approach [9]. The salet feature of ths ew operator s to corporate a expermetal desg method called orthogoal desg to the space search operator of SSA. As a result, t ca search the soluto space more effcetly ad t s well suted for parallel mplemetato. The steps that geerate ew solutos usg ths space search operator are as follows: Step. Record the selected solutos for space search a set P m Step. Geerate a ew soluto set P m, where a ew soluto of the set a j s geerated by usg the orthogoal approach. Step.. Set J = log Q (( Q )* + ), where Q s the factor of orthogoal expermets ad we fx t as 5. J k Step.. Set aj = (( ) / Q ) mod Q, where k Q J k =,3,..., J, j = +, =,,..., Q. Q Step.3. Calculate the basc colum of solutos by usg the followg expresso: a + ( s )*( Q ) + t = ( a * t+ a )modq j s j (a) A problem that s adapted to use GA (b) A problem that s ot adapted to use GA Fg. 3. Comparso of two types of optmzato problems k Q where k =,3,..., J, j = +, s =,,..., j, ad Q t =,,..., Q. Step3. Selected solutos wth best ftess from Pm Pm. Wth uderstadg of the mproved SSA, we ca develop a ISSA. As so far, several techques have bee corporated to mult-objectve optmzato algorthms order to mprove covergece to the Pareto frot as well as produce a well-dstrbuted Pareto frot. These techques clude eltsm, dversty operators, mutato operators, ad costrat hadlg. The techque of odomate sort wth the ad of the crowdg dstace [] s used the ISSA. The detals are preseted Table. The operator of the search space s realzed by meas of the followg two basc steps: we geerate a ew subspace (local area) ad realze search there. The o-domated sort s realzed wth the ad of estmato of the crowdg dstace amog solutos the curret soluto set. The termato codto of the ISSA s such that all the solutos the curret populato have the same ftess, or termate after a certa fxed umber of geeratos.

6 We Huag, Sug-Kwu Oh ad Hoghao Zhag 64 Table. The flow of computg of the ISSA BEGIN Italze the soluto set S (populato) Evaluate each soluto the soluto set S Sort all curret solutos S wth the ad of odomato strategy Whle { the termato codtos are ot met } Select the solutos from S Search soluto space (case I) Search soluto space (case II) Sort all curret solutos S wth the ad of odomato strategy Ed whle Report the optmal solutos Ed 3. A Arragemet ad Iterpretato of Solutos I the ISSA, a soluto s represeted as a vector comprsg the fuzzfcato coeffcet, the umber of fuzzy rules ad the type of polyomal stadg the cosequet part for each fuzzy rule as llustrated Fg. 4, The legth of the soluto vector correspods to the maxmal umber of fuzzy rules to be cosdered the optmzato. Fg. 4(b) offers a terpretato of the cotet of the partcle case the upper boud of search space of the fuzzy rule s set to 6. As the umber of rules, ad the orders of the polyomals have to be teger umber, we roud off these values stadg the partcle to the earest teger. The fuzzfcato coeffcet s 3., whle the umber of the rule s 4. The frst local model s of costat type whle the other three local models are lear ad quadratc. 3.3 Objectve Fuctos of the IG-FRBFNN Three objectve fuctos are used to evaluate the accuracy, the complexty ad the terpretablty of a IG- FRBFNN. These three objectve fuctos are mea squared error (SE) or the root mea squared error (RSE), etropy of partto ad the total umber of the coeffcets of the polyomals to be estmated, respectvely. The SE ad RSE servg as the accuracy crtero of the IG-RBFNN are gve as m PI( or E _ PI) = m m * ( y y ), ( RSE) = m * ( y y ). ( SE) = To evaluate the structural complexty of the model, we cosder the etropy of the partto [5]. The etropy of partto reflects a degree of overlap betwee the regos of the fuzzy sets. Cosderg all samples of the trag dataset, the etropy of the partto reads as (9) m H = w log( w ) () j= k= k k (a) As a measure of smplcty, we cosder the overall umber of coeffcets of the local models, whch s computed as N = C, j = f type of local model s costat + l f type of local model s lear form C = ( + l + l l) / + l f type of local model s quadratc form + l+ ( l l)/ f type of local model s modfed quadratc form () (b) Fg. 4. Soluto composto of ISSA ad ts terpretato: (a) Arragemet of the soluto for the ISSA; (b) Example of terpretato of soluto where, C s the umber of coeffcets of the th polyomal ad l stads for the umber of put varables. I a utshell, we fd the Pareto optmal sets ad Pareto frot by mmzg {E, H, N} by meas of the ISSA. Ths leads to easly terpretable, smple, ad accurate fuzzy models.

7 64 ultobjectve Space Search Optmzato ad Iformato Graulato the Desg of Fuzzy Radal Bass Fucto~ 4. Expermetal Studes I ths study, we carred out two cases of multobjectve optmzato. I the frst case, we used the OLS method to estmate the coeffcets of the polyomal. I the secod scearo, the WLS method s used. Table summarzes the lst of parameters ad boudares of the search spaces of solutos used by the ISSA. Table. Lst of the parameters of ISSA ISSA parameters Number of geeratos 5 Number of solutos Number of solutos (case I) 8 Boudary of search space of decso varables Fuzzfcato coeffcet. ~ 5 Number of rules ~ Order of polyomals ~4 (Type) 4. Gas furace process The frst well-kow dataset s tme seres data of a gas furace utlzed by Box ad Jeks. The tme seres data, whch cossts of 96 put-output pars resultg from the gas furace process has bee tesvely studed the prevous lterature [3-34]. The delayed terms of methae gas flow rate ut () ad carbo doxde desty yt () are used as sx put varables wth vector formats such as [ ut ( 3), ut ( ), ut ( ), yt ( 3), yt ( ), yt ( )]. yt () s used as output varable. The gas furace process s parttoed to two parts. The frst 5% of data set s used for the costructo of the fuzzy model. The remag 5% data set, the testg data set, s used to quatfy the predctve qualty of the model. The performace dex s specfed as the SE gve by (9). Fg. 5 llustrates the Pareto frots geerated by meas of the ISSA whe usg WLS ad OLS. Geerally, as t could have bee expected, by creasg the total umber of coeffcets, the accuracy of IG-FRBFNN becomes better. The total umber of coeffcets relates drectly to the terpretablty of the model. The ISSA just provdes the Pareto optmal sets, whch are o-domated solutos. Selectg the optmal soluto wth the Pareto optmal sets s a dfferet from the ISSA. If we place more emphass o the accuracy tha the smplcty (complexty) of the model, the we defe the partcle havg the best accuracy wth the Pareto optmal set to be the best soluto. The performace of the proposed model s compared wth the performace of some other models avalable the lterature; refer to Table 3. Here PI stads for the performace dex of accuracy for trag data ad E_PI meas accuracy for the testg data. Local models of other models have same type of fuzzy rule such as costat or lear form. The proposed model ca have dfferet types of local models. I ths comparso, the proposed model, The umber of coeffcet The umber of coeffcet havg a small umber of rules, shows better accuracy, whle the model leads to dsadvatage of havg a large umber of coeffcets of local model case of selectg the quadratc form. Table 3. Results of selected models (Gas).5 odel Pt PI E_PI No.of rules Pedrycz's model [3].776 Tog's model [3] Xu's model [3].38 5 Smplfed Oh et al.'s o del [33] Lear Smplfed HC+GA [34] Lear Sequetal tug Our model Smultaeous tug Automoble les Per Gallo (PG) Data We cosder the automoble PG data (ftp://cs.uc.edu/ pub/mache-learg-databased/auto-mpg) wth the output beg the automoble s fuel cosumpto expressed mles per gallo. The data set cludes 39 put-output 5 (a) Use of OLS Partto etropy 4 Partto etropy (b) Use of WLS Fg. 5. Pareto frot produced by ISSA 5 6

8 We Huag, Sug-Kwu Oh ad Hoghao Zhag 643 pars (after removg complete staces) where the put space volves 8 put varables. To come up wth a quattatve evaluato of the fuzzy model, we use the stadard RSE performace dex as the oe descrbed by (9). The automoble PG data s parttoed to two separate parts. The frst 35 data pars are used as the trag data set for IG-FRBFNN whle the remag 57 pars are the testg data set for assessg the predctve performace. Fg. 6 llustrates Pareto frots geerated by meas of the ISSA case of usg WLS ad OLS. Table 4 offers the values of the performace dexes of the IG- FRBFNN havg the best accuracy wth Pareto optmal set. The umber of coeffcet The umber of coeffcet (a) Use of OLS 5 Partto etropy Partto etropy (b) Use of WLS Fg. 6. Pareto frot produced by ISSA Table 4. Results of comparatve aalyss (PG) odel No. of rules Order of Polyomal PI E_PI Lgustc model [5] 36 Costat RBFNN [6] 36 Costat Fuctoal RBFNN[6] 33 Costat.4.8 Our model OLS odfed quadratc.67.9 WLS 4 Dfferet types Bosto housg data Last we expermet wth the Bosto housg data set. Ths data set cocers a descrpto of real estate the 5 Bosto area where houses are characterzed by features such as crme rate, sze of lots, umber of rooms, age of houses, etc. ad ther meda prce. The dataset cossts of 56 4-dmesoal data. The performace dex s defed as the RSE as gve by (9). The umber of coeffcet The umber of coeffcet (a) Use of OLS 4 5 Partto etropy 6 8 Partto etropy (b) Use of WLS Fg. 7. Pareto frot produced by ISSA We cosder the Bosto Housg data set, whch s splt to two separate parts. The costructo of the fuzzy model s completed for 53 data pots beg regarded as a trag set. The rest of the data set s retaed for testg purposes. Fg. 7 shows Pareto frots geerated by meas of the ISSA case of usg WLS ad OLS. Table 5 shows the results of comparatve aalyss of the proposed model whe beg cotrasted wth other models. Table 5. Results of comparatve aalyss (Housg) odel PI E_PI No. of rules RBFNN [6] SVR [35] FNN [36] NN FPNN Our model OLS WLS

9 644 ultobjectve Space Search Optmzato ad Iformato Graulato the Desg of Fuzzy Radal Bass Fucto~ 5. Coclusos I ths study, we have troduced a formato graulato-based FRBFNNs by meas of ISSA ad WLS. The proposed ISSA was exploted as a multobjectve optmzato vehcle to carry out the structural as well as parametrc optmzato of the FRBFNN. The OLS ad the WLS learg to estmate the coeffcets of the cosequet parts of the formato graulato-based FRBFNN are preseted ad compared. From the results of the ISSA, we have show that there exsts the accuracyterpretablty tradeoff a detfcato of the FRBFNN. Ackowledgemets Ths work was supported by Natoal Research Foudato of Korea Grat fuded by the Korea Govermet (NRF--D65 & NRF ) ad supported partally by the GRRC program of Gyeogg provce [GRRC SUWON -B, Ceter for U-cty Securty & Survellace Techology]. Refereces [] S. tra, J. Basak, FRBF: A Fuzzy Radal Bass Fucto Network, Neural Comput. Applc., vol., pp. 44-5,. [] F. Behloul, B.P.F. Leleveldt, A.Boudraa, ad J.H.C. Reber, Optmal desg of radal bass fucto eural etworks for fuzzy-rule extracto hgh dmesoal data, Patter Recogto, vol. 35, pp ,. [3] W. Huag, L. Dg, Project-schedulg problem wth radom tme-depedet actvty durato tmes, IEEE Trasactos o Egeerg aagemet, vol. 58, o., pp ,. [4] F.J. L, L.T. Teg, J.W. L, S.Y. Che, Recurret Fuctoal-Lk-Based Fuzzy-Neural-Network-Cotrolled Iducto-Geerator System Usg Improved Partcle Swarm Optmzato, IEEE Tras. Idust. Elect., vol. 56, o. 5, pp , 9. [5] W. Pedrycz, K.C Kwak, Lgustc models as a framework of user-cetrc system modelg, IEEE Tras. Syst., ma cyber. PART A : Systems ad humas, vol. 36, o. 4, pp , 6. [6] W. Pedrycz, H.S. Park, S.K. Oh, A graular-oreted developmet of fuctoal radal bass fucto eural etworks, Neurocomputg, vol. 7, pp , 8. [7] D.R. aryly, A.. Also, S.B. Wllam, S.C. Chrstopher, H.. Jaso, Geetc programmg eural etworks: A powerful boformatcs tool for huma geetcs, Appled Soft Computg, vol. 7, Issue 4, pp , 7. [8] Y. J, Fuzzy modelg of hgh-dmesoal systems: complexty reducto ad terpretablty mprovemet, IEEE Tras. Fuzzy Syst., vol. 8, o., pp.,. [9]. Setes, H. Roubos, GA-based modelg ad classfcato: complexty ad performace, IEEE Tras. Fuzzy Syst., vol. 8, o. 5, pp. 59 5,. [] K. Deb, A. Pratab, S. Agrawal, T. eyarva, A fast ad eltst multobjectve geetc algorthm: NSGA- II, IEEE Tras. Evol. Comput., vol. 6, pp. 8-97,. [] G. Avgad, A. oshaov, Iteractve Evolutoary ultobjectve Search ad Optmzato of Set-Based Cocepts, IEEE Tras. Syst., a cyber.-part B, vol. 38, ol., pp , 8. [] B.Zafer, Adaptve geetc algorthms appled to dyamc multobjectve problems, Appled Soft Computg, vol. 7, Issue 3, pp , 7. [3] S. Chua, Y. Zheyu, S. Zhogzh, Z. Le, A fast mult-objectve evolutoary algorthm based o a tree structure, Appled Soft Computg, vol., Issue, ,. [4] K.T. Parvee, B. Saghamtra, K.P. Sakar, ult- Objectve Partcel Swarm Optmzato wth tme varat erta ad accelerato coeffcets, Iformato Sceces, vol. 77, o., pp , 7. [5] S. asato, G. tsuo, Fuzzy multple objectve optmal system desg by hybrd geetc algorthm, Appled Soft Computg, vol., Issue 3, pp , 3 [6]. Delgado,.P. Ceullar,.C. Pegalajar, ultobjectve Hybrd Optmzato ad Trag of Recurret Neural Networks, IEEE Tras. Syst., a cyber. Part B, vol. 38, ol., pp , 8. [7] C. arco, L. Beatrce,. Fracesco, O reducg computatoal overhead mult-objectve geetc Takag-Sugeo fuzzy systems, Appled Soft Computg, vol., Issue, pp ,. [8] Z.Yog, W. Xao-be, X. Zog-y, H. We-l, O geeratg terpretable ad precse fuzzy systems based o Pareto mult-objectve cooperatve coevolutoary algorthm, Appled Soft Computg, vol., Issue, pp ,. [9] W. Huag, L. Dg, S.K. Oh, Desg of fuzzy radal bass fucto eural etworks wth the ad of multobjectve optmzato based o smultaeous tug, Eghth Iteratoal Symposum o Neural Networks, LNCS 6677, pp ,. [] A. Staao, R. Taglaferr, W. Pedrycz, Improvg RBF etworks performace regresso tasks by meas of a supervsed fuzzy clusterg, Neurocomputg, vol. 69, pp , 6. [] X. Hog, S. Che, A New RBF Neural Network wth Boudary Value Costrats, IEEE Tras. Syst.,

10 We Huag, Sug-Kwu Oh ad Hoghao Zhag [] [3] [4] [5] [6] [7] [8] [9] [3] [3] [3] [33] [34] [35] [36] a Cyber. PART B, vol. 39, ol., pp , 9. C.. Huag, F.L. Wag, A RBF Network wth OLS ad EPSO Algorthms for Real-Tme Power Dspatch, IEEE Tras. Power Syst., vol., o., pp.96-4, 7. F. Hoffma, Combg boostg ad evolutoary algorthms for learg of fuzzy classfcato rules, Fuzzy Sets ad Syst.,vol. 4, pp , 4. W. Pedrycz, K.C. Kwak, Boostg of graular models, Fuzzy Sets ad Syst., vol. 57, pp , 6. W. Pedrycz, P. Ra,, Collaboratve clusterg wth the use of fuzzy C-eas ad ts quatfcato, Fuzzy Sets ad Syst., vol. 59, pp , 8. R.R. Yager, D.P. Flev, Ufed Structure ad Parameter Idetfcato, IEEE Tras. Syst. a Cyber., vol. 3, o. 4, pp. 98-5, 993. W. Huag, L. Dg, S.K. Oh, C.W. Jeog, S.C. Joo, Idetfcato of Fuzzy Iferece System Based o Iformato Graulato, KSII Trasactos o Iteret ad Iformato Systems, vol. 4, o. 4, pp ,. W. Huag, S.K. Oh, L. Dg, H.K. Km, S.C. Joo, Idetfcato of Fuzzy Iferece Systems Usg a ult-objectve Space Search Algorthm ad Iformato Graulato, Joural of Electrcal Egeerg & Techology, vol. 6, o. 6, pp ,. Q. Zhag, Y.W. Leug, A orthogoal geetc algorthm for multmeda multcast routg, IEEE Trasactos o Evolutoary Computato, vol. 3, o., pp. 53-6, 999. W. Pedrycz, A detfcato algorthm fuzzy relatoal system, Fuzzy Sets Syst., vol. 3, pp. 5367, 984. R.. Tog, The evaluato of fuzzy models derved from expermetal data, Fuzzy Sets Syst., vol. 3, pp -, 98. C. W. Xu., Y. Zalu, Fuzzy model detfcato selflearg for dyamc system IEEE Tras. Syst., a cyber., vol. 7, o 4, pp, , 987. S. K. Oh, W. Pedrycz, Idetfcato of Fuzzy Systems by meas of a Auto-Tug Algorthm ad Its Applcato to Nolear Systems, Fuzzy Sets ad Syst., vol. 5, o, pp, 5-3,. B. J. Park, W. Pedrycz, S. K. Oh, Idetfcato of Fuzzy odels wth the Ad of Evolutoary Data Graulato, IEE Proc.-Cotrol Theory ad Applcatos, vol. 48, pp, 46-48,. H.D. Chrs, J.C. Burges, L. Kaufma, A. Smola, V. Vapk, Support vector regresso maches, Adv. Neural Iform. Process Syst. 9 (997) A.F. Gob, W. Pedrycz, Fuzzy modelg through logc optmzato, It. J. Approx. Reaso., vol. 45, pp , We Huag He receved the S degree at school of Iformato Egeerg from East Cha Isttute of Techology, Cha, 6, ad Ph.D. degree at State Key Laboratory of Software Egeerg, Wuha Uversty, Cha,. He s curretly a lecturer wth the School of Computer ad Commucato Egeerg, Taj Uversty of Techology, Taj, Cha. Hs research terests clude evolutoary computato, operatos research, fuzzy system, fuzzy-eural etworks, ad software relablty. Sug-Kwu Oh He receved the B.Sc.,.Sc., ad Ph.D. degrees electrcal egeerg from Yose Uversty, Seoul, Korea, 98, 983, ad 993, respectvely. Durg , he was a Seor Researcher of R&D Lab. of Lucky-Goldstar Idustral Systems Co., Ltd. From 996 to 997, he was a Postdoctoral Fellow wth the Departmet of Electrcal ad Computer Egeerg, Uversty of atoba, Wpeg, B, Caada. He s curretly a Professor wth the Departmet of Electrcal Egeerg, Uversty of Suwo, Suwo, South Korea. Hs research terests clude fuzzy system, fuzzy-eural etworks, automato systems, advaced computatoal tellgece, ad tellget cotrol. He curretly serves as a Assocate Edtor of the KIEE Trasactos o Systems ad Cotrol, Iteratoal Joural of Fuzzy Logc ad Itellget Systems of the KFIS, ad Iteratoal Joural of Cotrol, Automato, ad Systems of the ICASE, South Korea. Hoghao Zhag He receved the S degree at School of Iformato Scece ad Egeerg from Ngbo Uversty, Cha,. He s curretly a teacher wth the School of Taj Uversty of Techology, Taj, Cha. Hs research terests clude etwork securty, trusted etworks ad ext geerato etwork.