Comparison of BPA and LMA methods for Takagi - Sugeno type MIMO Neuro-Fuzzy Network to forecast Electrical Load Time Series

Transcription

1 Comarson of BPA and LA methods for akag - Sugeno tye IO euro-fuzzy etwork to forecast Eectrca Load me Seres [Fex Pasa] Comarson of BPA and LA methods for akag - Sugeno tye IO euro-fuzzy etwork to forecast Eectrca Load me Seres Fex Pasa Eectrca Engneerng Deartment, Petra Chrstan Unversty Surabaya - Indonesa E-ma: fex@etra.ac.d ABSRAC hs aer descrbes an acceerated Backroagaton agorthm (BPA that can be used to tran the akag-sugeno (S tye mut-nut mut-outut (IO neuro-fuzzy network effcenty. Aso other method such as acceerated Levenberg- arquardt agorthm (LA w be comared to BPA. he tranng agorthm s effcent n the sense that t can brng the erformance ndex of the network, such as the sum squared error (SSE, ean Squared Error (SE, and aso Root ean Squared Error (RSE, down to the desred error goa much faster than that the sme BPA or LA. Fnay, the above tranng agorthm s tested on neuro-fuzzy modeng and forecastng acaton of Eectrca oad tme seres. Keywords: S tye ISO neuro-fuzzy network, acceerated Levenberg-arquardt agorthm, acceerated Backroagaton agorthm, tme-seres forecastng IRODUCIO hs aer rooses two extended-tranng agorthms for a mut-nut mut-outut (IO akag- Sugeno tye neuro-fuzzy (F network and neurofuzzy aroach for modeng and forecastng acaton of eectrca oad tme seres. he neurofuzzy aroach attemts to exot the merts of both neura network and fuzzy ogc based modeng technques. For exame, the fuzzy modes are based on fuzzy f-then rues and are to a certan degree transarent to nterretaton and anayss. Whereas, the neura networks based mode has the unque earnng abty. Here, S tye IO neuro-fuzzy network s constructed by mutayer feedforward network reresentaton of the fuzzy ogc system as descrbed n secton, whereas ther tranng agorthms are descrbed n secton. Smuaton exerments and resuts are shown n Secton 4, and fnay bref concudng remarks are resented n secton 5. EURO-FUZZY SYSES SELECIO FOR FORECASIG he common aroach to numerca data drven for neuro-fuzzy modeng and dentfcaton s usng akag-sugeno (S tye fuzzy mode aong wth contnuousy dfferentabe membersh functons such as Gaussan functon and dfferentabe oerators for constructng the fuzzy nference mechansm and defuzzyfcaton of outut data usng the weghted ote: Dscusson s mected before December, st 7, and w be ubced n the urna eknk Eektro voume 8, number arch 8. average defuzzfer. he corresondng outut nference can then be reresented n a mutayer feedforward network structure. In rnce, the neuro-fuzzy network s archtecture used here s dentcay to the mut nut snge outut archtecture of AFIS [], whch s shown n Fgure a. ang ntroduced nuts outut archtecture wth akag- Sugeno tye fuzzy mode wth two rues. he euro- Fuzzy mode AFIS ncororates a fve-ayer network to mement a akag-sugeno tye fuzzy system. Fgure a shows the roosed mode of AFIS and t can rocess a arge number of fuzzy rues. It uses the east mean square method to determne the near consequents of the akag- Sugeno rues. As contnuty from AFIS structure, feedforward mut nut mut outut s roosed by Pat and Babuška [] and Pat and Poovc [] as shown n Fgure b. nuts Layer Layer Layer A Z A B B Degree of fufment Z R : If s A and R : If s A and b Degree of fufment Of rues S S Z b Z b Y S Layer 4 Y S s B hen Y = w w w S Z b Y S Z b Y s B hen Y = w w w Layer 5 Fgure a. AFIS archtecture wth akag-sugeno tye fuzzy mode wth two rues, Inuts, Outut f

2 urna eknk Eektro Vo. 7, o., Setember 7: - 9 n n nuts n G G n G G n Z Z Degree of fufment b Degree of fufment Of rues Z b Z b y y y m y m m oututs Fgure b. Fuzzy system IO feedforward akag- Sugeno-tye eura-fuzzy network F mode as shown n Fgure b s based on Gaussan membersh functons. It uses S tye fuzzy rue, roduct nference, and weghted average defuzzyfcaton. he nodes n the frst ayer cacuate the degree of membersh of the numerca nut vaues n the antecedent fuzzy sets. he roduct nodes (x reresent the antecedent conuncton oerator and the outut of ths node s the corresondng to degree of fufment or frng strength of the rue. he dvson sgn (, together wth summaton nodes (, on to make the normazed degree of fufment ( z b of the corresondng rue, whch after mutcaton wth the corresondng S rue consequent ( y, s used as nut to the ast summaton art ( at the defuzzyfed outut vaue, whch, beng crs, s drecty comatbe wth the actua data. Forecastng of tme seres s based on numerca nut-outut data. Demonstratng ths to the F networks, a S tye mode wth near rues consequent (can be aso sngeton mode for a seca case, s seected []. In ths mode, the number of membersh functons ( to be memented for fuzzy arttonng of nut to be equa to the number of a ror seected rues. In the next secton of ths chater, acceerated BPA and LA w be aed to acceerate the convergence seed of the tranng and to avod other nconvenences. euro mementaton of Fuzzy Logc System he fuzzy ogc system (FLS consdered n Fgure b, can be easy reduced to a ISO-F network by settng number oututs m =, t means ths structure s smar wth Fgure a. akag-sugeno (S tye fuzzy mode, and wth Gaussan membersh f f m functons (GFs, roduct nference rue, and a weghted average defuzzfer can be defned as (-(4 (see [4]. f = = y h ( where, y = W W x W x Wn x ( n z h ( z b =, and b = z = n ( ( x c = µ = x ; x = G G ex σ ( (4 he corresondng th rue from the above fuzzy ogc system (FLS can be wrtten as R : IF x s G AD AD xn s Gn HE (5 y = W Wx W nxn. where, x wth =,,, n; are the n system nuts, f wth =,,, m; are ts m oututs, and G wth =,,, n and =,,, m are the Gaussan membersh functons of form (4 wth the corresondng mean and varance arameters c and σ resectvey and wth y as the outut consequent of the th rue. It must be remembered that the Gaussan membersh functons G actuay reresent ngustc terms such as ow, medum, hgh, very hgh, etc. he rues as wrtten n (5 are known as akag- Sugeno rues. Fgure b shows that the FLS can be reresented as a three ayer feedforward network. Because of the neuro mementaton of the akag-sugeno-tye FLS, ths fgure reresents a akag-sugeno-tye of IO neuro-fuzzy network, where nstead of the connecton weghts and the bases n neura network, we have here the mean c and aso the varance o σ arameters of Gaussan membersh functons, aong wth W, W arameters from the rues consequent, as the equvaent adustabe arameters of the network. If a these arameters of F network are roery seected, then the FLS can correcty aroxmate any nonnear system based on gven data. RAIIG ALGORIH FOR FEEDFORWARD EURAL EWORK he IO Feedforward F network that s reresented n Fgure b can generay be traned usng sutabe tranng agorthms. Some standard tranng agorthms are Backroagaton Agorthm

3 Comarson of BPA and LA methods for akag - Sugeno tye IO euro-fuzzy etwork to forecast Eectrca Load me Seres [Fex Pasa] (BPA and Levenberg-arquardt Agorthm (LA. BPA, the smest agorthm for neura network ( tranng, s a suervsed earnng technque used for tranng artfca neura networks. It was frst descrbed by Pau Werbos n 974, and further deveoed by Davd E. Rumehart, Geoffrey E. Hnton and Ronad. Wams n 986. hs agorthm s a earnng rue for mut-ayered neura networks, credted to Rumehart. It s most usefu for feed-forward networks (networks wthout feedback, or smy, that have no connectons that oo. he term s an abbrevaton for "backwards roagaton of errors". BPA s used to cacuate gradent of error of the network wth resect to the network's modfabe weghts. hs gradent s amost aways used n a sme stochastc gradent descent agorthm to fnd weghts that mnmze the error. In forecastng acaton, ure BPA, has sow convergence seed n comarson to other second order tranng []. hat s why ths agorthm needs to be mroved usng momentum and modfed error ndex. Aternatvey, the more effcent tranng agorthm, such as Levenberg-arquardt agorthm (LA can aso be used for tranng the IO Feedforward systems []. BPA and Acceerated BPA d Let us assume that data ars nut-outut x,, wth x as the number of nuts and d as the number of oututs n IO matrx observaton, s gven. he goa s to fnd a FLS ( f n equatons 6-8, so x that the erformance, Sum Squared Error (SSE, a the equatons exaned by Pat and Poovc ([],.4-.45, can be defned as wth SSE ( e =. 5 =. 5 E E, (6 = E, E are error transose and errors for each e n coumn vector from FLS. For tota erformance outut ( m = ( SSE SSE SSE SSE ota = SSE = m, (7 s mnmzed. e = f ( x d or for convenence e = f d (8 he robem s, how BPA works to adust arameters ( W, from the rues consequent and the mean o W c and varance τ arameters from the Gaussan membersh functons, so that SSE s mnmzed. Furthermore, the gradent steeest descent rue for tranng of feedforward neura network s based on the recursve exressons: W ( k = W ( k η ( SSE W (9 W k = W k η SSE W ( c ( ( ( ( k = c ( k η ( SSE c ( k = σ ( k η ( SSE σ σ ( ( Where SSE s the erformance functon at the k th teraton ste and W ( k, W ( k, c ( k, σ ( k th are the free arameters udate at the next k teraton ste, the startng vaues of them are randomy seected. Constant ste sze η or earnng rate shoud be chosen roery, usuayη <<, =,,, n (n = number of nuts, =,,, m (m=number of oututs, and =,,, (=number of Gaussan membersh functons. ow, et us construct agan (- based on Fgure b, It s sure that SSE deends on outut network f, therefore erformance functon SSE, deends on W, ony through (4, and aso deends o W on c, τ ony through (-(5 f = = y h y = W W x W x W h z ( n n (4 ( z b =, and b = z = = n x c ex = σ herefore, the corresondng chan rues are ( SSE W = ( SSE f ( f y ( y W ( W = ( SSE f ( f y ( y W x (5 (6 (7 SSE (8 ( SSE c = SSE z ( z c = = = ( SSE f ( f z ( z c ( SSE σ = SSE z ( z σ = = = z ( SSE f ( f z ( σ (9 (

4 urna eknk Eektro Vo. 7, o., Setember 7: - 9 And equatons (7 ( can be wrtten as ( SSE W = ( f d ( z b ( ( SSE W = ( f d ( z b x ( ( SSE c = A ( z b ( x c ( σ ( ( ( SSE σ = A z b x c ( σ For smcty, we substtute the term A wth A = ( ( y f f d = n = W W x f f = = ( (4 (5 ( d By substtutng equaton (-4 nto the (9-, udated rues for free arameters can be wrtten as W k = W k η f d z b (6 ( ( ( ( ( k = W ( k ( f d ( z b x ( ( { ( ( } k = c k η A h x c σ ( ( { ( ( } k = σ k η A h x c σ W c η (7 (8 σ (9 where h term s the normazed degree of fufment of the th rue. h = z b, wth b z = = ( BPA tranng agorthm for S-tye IO networks s reresented from equatons (6 to (, equvaent wth near fuzzy rues from S y = Wo W x W x Wn xn. ( In case of BPA, to avod the ossbe oscaton n fna hase of tranng, very sma earnng rate η s chosen. herefore, BPA tranng needs a arge number of tranng eochs. For acceeraton of BPA, momentum (adatve verson of earnng rate and modfed error ndex for erformance functon udate can be aed. Addtona momentum (mo can be seen n equaton (-5 wth momentum constant s usuay ess than one ( mo <. W W c ( k = W ( k ( mo {( f d ( z b } η ( mo W ( k ( k = W ( k ( mo { ( f d ( z b } η ( mo W ( k ( k = c ( k ( mo A z ( x c ( σ mo x { } η (4 c ( k ( k = σ ( k η ( mo A z ( x c ( σ { } σ (5 mo σ ( k o mrove the tranng erformance of feedforward, addtona modfed error ndex verson shoud be added to the modfed BPA wth momentum, as roosed by aosong et a. [5]. hs aroach can be seen n equatons (6 4. SSE e avg m ( r eavg ( w = 5 γ e ( w = r =. (6 e r r = ( w (7 hus the new erformance ndex can be wrtten as SSEnew ( w = SSE( w SSEm( w (8 where SSE( w the unmodfed error erformance as s defned n (7 and varabe w reresents the network free arameter vector n genera. he corresondng gradent now becomes to SSE SSE ( w = ( er ( w ( er ( w m SSE r = ( ( e e ( w ( w = er ( w new r = avg ( r (9 γ (4 ( w = ( er ( w γ ( er ( w eavg r= ( e ( w r (4 he constant gamma γ shoud be defned roery. In ractce, gamma aso sma, γ < Acceerated LA o acceerate the convergence seed on neuro-fuzzy network tranng that haened n BPA, the Levenberg-arquardt agorthm (LA was roosed and roved [4]. If a functon ( w V s meant to mnmze wth resect to the arameter vector w usng ewton s method, the udate of arameter vector w s defned as: w = [ V ( w ] V ( w (4 w k = w k (4 ( ( w From equatons (4-4, V ( w s the Hessan matrx and V ( w s the gradent of V ( w. If the functon V ( w s taken to be SSE functon as foows: V r r =! ( w =.5 e ( w (44 hen the gradent of V ( w and the Hessan matrx V ( w are generay defned as: V ( w = ( w e( w (45 4

5 Comarson of BPA and LA methods for akag - Sugeno tye IO euro-fuzzy etwork to forecast Eectrca Load me Seres [Fex Pasa] V ( w = ( w ( w er ( w er ( w r = where the acoban matrx ( w as foows e ( w e ( w e ( w ( w e = e ( w e ( w e ( w ( w e ( w e ( w L L L (46 (47 From (47, t s seen that the dmenson of the acoban matrx s (, where s the number of tranng sames and s the number of adustabe arameters n the network. For the Gauss- ewton method, the second term n (46 s assumed to be zero. herefore, the udate equatons accordng to (4 w be: w = w w w e w (48 [ ( ( ] ( ( ow et us see the LA modfcatons of the Gauss- ewton method. w = [ ( w ( w µ I ] ( w e( w (49 where dmenson of I s the ( dentty matrx, and the arameter µ s muted or dvded by some factor whenever the teraton stes ncrease V w. or decrease the vaue of ( Here, the udated equaton accordng to (4 [ ] ( ( ( = w( k ( w ( w µ I w e w (5 w k hs s mortant to know that for arge µ, the agorthm becomes the steeest descent agorthm wth ste sze µ, and for sma µ, t becomes the Gauss-ewton method. For faster convergence reason and aso to overcome the ossbe tra at oca mnma and to reduce oscaton durng the tranng [6], ke n BPA, a sma momentum term mo (ractcay n eectrca oad forecastng, addng mo around 5% to % w gve better resuts aso can be added, so that fna udate (5 becomes w( k = w( k [ ( w ( w µ I ] ( w e( w (5 mo ( w( k w( k Furthermore, ke n BPA, aosong et a [5] aso roosed to add modfed error ndex (EI term n order to mrove tranng convergence. Smar to equatons (9-4, the corresondng gradent wth EI can now be defned by usng a acoban matrx as: [ ( e ] SSEnew( w = ( w e( w γ e( w avg (5 where e ( w s the coumn vector of errors, e avg s sum of error of each coumn dvded by number of tranng, whe γ s a constant factor, γ << has to be chosen aroratey. ow, comes to the comutaton of acoban atrces. hese are the most dffcut ste n mementng LA. We descrbe a sme technque to comute the acoban matrx from the Backroagaton resuts, by takng nto account equaton (46, the gradent V W can be wrtten as ( SSE ( W ( S W = { z b} ( f d V Where (5 f and d are the actua outut of the akag-sugeno tye IO and the corresondng desred outut from matrx nut-outut tranng data. And then by comarng (5 to (45, where the gradent V ( w s exressed wth the transose of the acoban matrx muted wth the network's error vector V ( w = ( w e( w (54 then the acoban matrx, the transose of acoban matrx for the arameter W of the F network can be wrtten by ( W ( z b ( W [ ( W ] = [ z b] = (55 (56 wth redcton error of fuzzy network e ( f d (57 But f the normazed redcton error on F network s consdered, then nstead of equatons (55 and (56, the equatons w be ( W ( z [ W ] [ z ] = (59 ( W ( = (6 hs s because the normazed redcton error of the IO-F network s e normazed f d (6 ( ( b By usng smar technque and takng nto account equaton (54, the transose of acoban matrx and acoban matrx for the arameter W of the F network can be wrtten as 5

6 urna eknk Eektro Vo. 7, o., Setember 7: - 9 ( W ( z b x ( W [ ( W ] = [( z b x ] = (6 (6 Aso, by consderng normazed redcton error from (6, equatons (6-6 then become: W = z x (64 ( ( ( W [ ( W ] = [ z x ] (65 ow, comes to the comutaton of the rest arameters c andτ. Usng the smar equaton to comute the term A n equaton (5, the equaton has to be reorganzed as foows D y f (66 ( By combnng equatons (57 and (66, the term A whch s exaned before can be rewrtten as A ( D e = ( D e D e Dm em m = L (67 Let us defne the terms D and e as A D e = ( D e D e L Dm em (68 Wth e as the same amount of sum squared error that can be found by a the errors e from the IO network. ( e e e em = L (69 Where, =,,,, ; corresondng to as number of tranng data. From (68, the term D can be determned as D = ( A e (7 hs can aso be wrtten n matrx form usng seudo nverse as ( Ε ( Ε Ε D = Α (7 he terms E (s the equvaent error vector, D and A are matrces of sze (x, (x and (x resectvey. ow matrx A can be reaced wth scaar roduct of e and D A = D e (7 Wth equaton (7, we can wrte equatons ( and (4 as SSE c = D e z b x c σ (7 ( ( ( ( ( ( ( ( ( SSE σ = D e z b x c ( σ (74 ow, by consderng normazed equvaent error ke n (6, takng nto account the equaton (54 and comarng t resectvey wth (7 and (74, the transosed acoban matrx, the acoban for the arameters c andτ can be comuted as: ( c ( ( D z x c σ ( c = [ ( c ] = D z ( x c ( σ ( ( σ D z x c ( σ = (75 { } (76 = (77 ( = [ ( σ ] = D z ( x c ( σ σ (78 It s to be noted that normazed redcton error s consdered for comutaton of acoban matrces for the free arameters W andw. eanwhe normazed equvaent error has been consdered for the comutaton of transosed acoban matrces and ther acoban matrces resectvey for the free arameters mean c and varanceτ. odeng of non-near dynamcs data tme-seres he tme seres wth ead tme redcts the vaues at tme ( t L, based on the avaabe tme seres data u to the ont t. o forecast the eectrca oad tme seres usng the neuro-fuzzy aroach, the tme seres data = {,,,, } can be rearranged n a IO - IO ke structure. IO stands for the tme seres s reresented n Inut and Outut form. For the gven tme seres modeng and forecastng acaton the IO neuro-fuzzy redctor to be deveoed s suosed to oerate wth four nuts ( n = 4 and three oututs ( m = ony. ow f the samng nterva and the ead tme of forecast both are taken to be tme unt, t means f tme unt means hour (ths s ony an exame, because another tme unt can equa to 5 mnutes, so ths IO system was takng 4 hours as an nut data and redcted hours n advance, then for each t 4, the nut data n ths case reresent a four dmensona vector and outut data a three dmensona vector as descrbed beow. I = [ ( t, ( t, ( t, ( t ], (79 [ ( t, ( t, ( ] O = t (8 hus, n order to have sequenta outut n each row the vaues of t run as 4, 7,,, (-6; and so on, so that the IO matrx w roose an sort-term and offnebatch mode scenaro, whch s ook ke n equaton (8. In ths equaton, the frst four coumns of the IO matrx reresent the four nuts to the network whereas, ast three coumns reresent the outut from the neuro-fuzzy network. IO = (8 6

7 Comarson of BPA and LA methods for akag - Sugeno tye IO euro-fuzzy etwork to forecast Eectrca Load me Seres [Fex Pasa] EPERIES AD RESULS OF BPA AD LA he tranng and forecastng erformances, whch roduced by F-network, are ustrated n Fgures a-c and a-c where the IO matrx s shown n equaton (8. here are 45 eectrca oad data from eectrca oad data used n tranng and forecastng, for whch 5 data are for tranng and the remanng 95 data are for forecastng. Because the frst arameters are random, the nta SSE n BPA starts from and the fna SSE tran becomes.87 by usng eochs. In the other case of LA, the nta SSE starts wth 558. and goes to fna SSE tran=9.764 usng ony eochs. SSE (Log SSE tranng vs Eochs F Outut - Actua F Error F ranng and Forecastng (Red vs Actua (Back, Error (Bue ranng Forecastng me Fgure c. ranng and Forecastng Performance of S-tye IO F network wth BPA, n= 4 Inuts, m= oututs, =5, Eochs=, Lead me d=, Learnng Rate η =., Wdness Factor WF=.5, Gamma γ =., omentum mo =.5, Fna SSEtota=8.664 wth 45 data ranng and Forecastng SSE ranng vs Eochs Eochs Fgure a. Grahc SSE vs. Eochs of S-tye IO F networks wth BPA, n= 4 Inuts, m= oututs, =5, Eochs= SSE (Log Eectrca Load tranng, F Outut (Red vs Actua (Back, Error (Bue F Outut - Actua Eochs Fgure a. SSE tranng vs. Eochs of S-tye IO F networks wth LA, n= 4 Inuts, m= oututs, =5, Eochs= F ranng Error me Fgure b.. ranng Performance of S-tye IO F network wth BPA, n= 4 Inuts, m= oututs, =5, Eochs=, Lead me d=, Learnng Rate η =., Wdness Factor WF=.5, Gamma γ =., omentum mo =.5, Inta SSE= , Fna SSEtran=.87, wth out of 5 Data ranng Fgure b, ustrates the roosed acceerated LA whch brngs the erformance smar to the roosed BPA usng ony eochs, ndcatng hgher convergence seed n comarson wth the BPA. In addton, the erformances n both Fgures and shows that the tranng has sma oscaton because of the mementaton of Wdness Factor (WF whch ony aow.% of oscaton. From Fgure a, t can be seen that LA has much faster tranng comared to BPA. LA needs ony around eochs to acheve the erformance, whereas BPA needs eochs. See abe for detaed comarson between these two methods. 7

8 urna eknk Eektro Vo. 7, o., Setember 7: - 9 F Outut - Actua 5 5 Eectrca Load ranng, F Outut (Red vs Actua (Back, Error (Bue he resut shows that SSE, SSE and SSE ndcate the sum squared error at outut, and, resectvey, of the akag-sugeno tye IO F network, and SSE ndcates the summng of a SSE vaues whch are contrbuted by a oututs. he smar defnton s aso aed to SE, SE, SE, RSE, RSE, and RSE. F ranng Error me Fgure b. ranng Performance of S-tye IO F network wth LA, n= 4 Inuts, m= oututs, =5, Eochs=, Lead me d=, Learnng Rate LVmu µ =, Wdness Factor WF=., Gamma γ =.5, omentum mo =.5, Inta SSE= 558., Fna SSEtran=9.764, wth out of 5 Data ranng F Outut - Actua 5 5 F ranng and Forecastng (Red vs Actua (Back, Error (Bue ranng Forecastng abe. Comarson between BPA and LA on Eectrca Load tranng and forecastng erformance of akag-sugeno-tye IO-F network wth 4 Inuts and Oututs Descrton ranng Data ( to 5 Eectrca Load ranng and Forecastng ( to 45 Performance usng BPA (ormazed Data Inta SSE= o of Eochs = SSEtran =.87 SSE =.998 SSE = SSE = SEtran =.58 SE =.48 SE =.6 SE =.6 RSE =.69 RSE =.8 RSE =.6 SSEtota = SSE =. SSE =.478 SSE = 4.67 SEtota =.6 RSEtota =.8 Performance usng LA (ormazed Data Inta SSE= 558. o of Eochs = SSEtran = SSE =.8777 SSE =.68 SSE = 5.75 SEtran =.4 SE =.5 SE =.7 SE =.9 RSE =.59 RSE =.4 RSE =.5 SSEtota =.786 SSE =.5575 SSE = 6.64 SSE =.7 SEtota =.9 RSEtota =.6 F error me Fgure c. ranng and Forecastng Performance of S-tye IO F network wth LA, n= 4 Inuts, m= oututs, =5, Eochs=, Lead me d=, Learnng Rate for LA µ =, Wdness Factor WF =., Gamma γ =.5, omentum mo =.5, Fna SSEtota=.786 abe demonstrates the dfference of usng BPA and LA n S-tye IO-F network. ote aso that SSE = Sum Squared Error, SE = ean Squared Error, and RSE = Root ean Squared Error as shown n equaton (8. ( en ( en ( en n= SSE =. 5, SE =, RSE = n= n= (8 COCLUSIO REARKS In the aer neuro-fuzzy aroaches wth two tyes of tranng agorthms have been resented for shortterm forecastng of eectrca oad. Performance resuts from Secton 4 roved that the traned F network usng LA s found to be very effcent n modeng and redcton of the varous nonnear dynamcs, comared to BPA. An effcent tranng agorthm based on combnaton of LA wth addtona modfed error ndex extenson (EI and adatve verson of earnng rate (momentum have been deveoed to tran the akag-sugeno tye mut-nut mut-outut (IO and mut-nut snge-outut (ISO euro-fuzzy network, mrovng the tranng erformance. Severa ssues need to be addressed n the future works whch ncudes many transarency and nterretabty of generated fuzzy mode (rues. For the atter ssue set theoretc smarty measures [7] shoud be comuted for each ar of fuzzy sets and the fuzzy sets whch are hghy smar shoud be merged together nto a snge one. 8

9 Comarson of BPA and LA methods for akag - Sugeno tye IO euro-fuzzy etwork to forecast Eectrca Load me Seres [Fex Pasa] REFERECES [] ang SR., AFIS: Adatve network Based Fuzzy Inference System, IEEE rans. On SC., (: , 99. [] Pat AK, Babuška R., Effcent tranng agorthm for akag-sugeno tye euro-fuzzy network, Proc. of FUZZ-IEEE, ebourne, Austraa, vo. : 58-54,. [] Pat AK, Poovc D., onnear combnaton of forecasts usng A, FL and F aroaches, FUZZ-IEEE, :566-57,. [4] Pat AK, Poovc D., Comutatona Integence n me Seres Forecastng, heory and Engneerng Acatons, Srnger, 5. [5] aosong D, Poovc D, Schuz-Ekoff, Oscaton resstng n the earnng of Backroagaton neura networks, Proc. of rd IFACIFIP, Begum, 995. [6] Pasa Fex, Forecastng of Eectrca Load usng akag-sugeno tye IO euro-fuzzy network, aster hess, Unversty of Bremen, 6. [7] Setnes, Babuška R, Kaymark U., Smarty measures n Fuzzy rue base smfcaton, IEEE ransacton on System, an and Cybernetcs, vo.8:77-775,