Evaluation of Fuzzy Labor Market by Fuzzy Neural Network

Transcription

1 Australan Journal of Basc and Appled Scences, 5(6): 66-85, 0 ISSN Evaluaton of Fuzzy abor Market by Fuzzy Neural Network S. Gavdel, M. Otad, M. Mosle Departent of Econocs, Froozkoo Branc, Islac Azad nversty, Froozkoo, Iran Departent of Mateatcs, Froozkoo Branc, Islac Azad nversty, Froozkoo, Iran Abstract: In ts paper, a novel ybrd etod based on fuzzy neural network for estate fuzzy coeffcents (paraeters) of fuzzy supply and deand labor functon wt fuzzy output and fuzzy nputs, s presented. Here a neural network s consdered as a part of a large feld called neural coputng or soft coputng. Moreover, n order to fnd te approxate paraeters, a sple algort fro te cost functon of te fuzzy neural network s proposed. Fnally, we llustrate our approac by soe nuercal exaples,specally for Iran labor arket. Key words: abor arket; Fuzzy neural networks; Fuzzy nuber; earnng algort; Fuzzy regresson. INTRODCTION General nput deand functons generated by consderng te fr's proft-axzng decson can be stated for te two-nput case as =(P, w, r) Were s te labor deand, P s te product prce, w s te labor prce (wage) and r s te captal prce. Oter way for nput deand functons generate s Constant Output Deand Functons wt an analyss of cost nzaton Cowell (004) and Henderson (985), called Separd's lea, wc uses te q teore to sow tat te constant output deand functon for can be found sply by partally dfferentatng total costs (C) wt respect to w. General labor supply functon generated by axzaton utlty functon of consupton and lesure wt constant personalty budget lne, t s wrtten by followng: =(p, w, A, e) Were s te labor supply, p s te product prce and t as postve effect on, w s te wage and ts relatonsp s postve, A are personalty factors suc as age, educaton, gender and etc. e s te resdual ter. In te econoy to sow te relatonsp of two or ore varables anly s used regresson. For exaple, n te followng equaton, fluctuatons n Y are descrbed by X and X. In ost cases A 0, A and A are estated wt te OS tecnque, Y = A AX A X. 0 Mostly of eprcal researc for estaton deand and supply of labor are econoetrcal approac. Heckan and MaCurdy (980) consder a fxed effects Tobt odel to estate a lfecycle odel of feale labor supply. Verbon (980) apples te SR procedure wt one-way error coponents to a set of four labor deand equatons. Conway and Knesner (99) wo used te Panel Study of Incoe Dynacs to study te senstvty of ale labor supply functon estates to ow te wage s easured. Mofftt (993) llustrates s estaton etod for te lnear fxed effects lfecycle odel of labor supply usng repeated cross-sectons fro te S Current Populaton Survey (CPS). Pesaran and St (995) consder te proble of estatng a dynac panel data odel wen te paraeters are ndvdually eterogeneous and llustrate ter results by estatng ndustry-specfc K labor deand functons. Jorgenson.W (008) tey represent labor deand for eac of 35 ndustral sectors of te.s. econoy as a response to te prces of productve nputs: labor, captal, and nteredate goods and servces. In addton,labor deand s drven by canges n tecnology. Nuercal soluton of a deand and supply of labor s obtaned now n a natural way, by extendng te exstng classcal etods to te fuzzy case, we generalzed a nuercal etod presented for approxatng deand and supply of labor. Correspondng Autor: M. Otad, Departent of Mateatcs, Froozkoo Branc, Islac Azad nversty, Froozkoo, Iran E-al: otad@aufb.ac.r; Tel.:

2 In ts paper Y, X and X ave been consdered fuzzy varables and te coeffcents A 0, A and A ave been estated wt OS tecnques. In ts paper Frst, estated coeffcents of regresson wt tree explanatory fuzzy varables and ten a nuercal exaple are entoned for t. Fnally, we are estated supply and deand labor wt te assupton of fuzzy varables. Te concept of fuzzy nubers and fuzzy artetc operatons were frst ntroduced by Zade (975), Dubos and Prade (978). We refer te reader to Kaufann and Gupta (985) for ore nforaton on fuzzy nubers and fuzzy artetc. Regresson analyss s of te ost popular etods of estaton. It s appled to evaluate te functonal relatonsp between te dependent and ndependent varables. Fuzzy regresson analyss s an extenson of te classcal regresson analyss n wc soe eleents of te odel are represented by fuzzy nubers. Fuzzy regresson etods ave been successfully appled to varous probles suc as forecastng (Cang 997; Cen and Wang 999; Tseng and Tzeng 00) and engneerng (a and Cang 994). Tus, t s very portant to develop nuercal procedures tat can approprately treat fuzzy regresson odels. Modarres et al. (005) proposed a ateatcal prograng odel to estate te paraeters of a fuzzy lnear regresson Y = AX A X A X, n n X were j and A, A,, An, Y are syetrc fuzzy nubers for =,,...,, j=,,..., n. Recently, Mosle et al. (00) proposed fuzzy neural network to estate te paraeters of a fuzzy lnear regresson Y = AX A X A X, n n X were j and A, A,, An, Y are fuzzy nubers for =,,...,, j=,,..., n. Isbuc et al. (995) proposed a learnng algort of fuzzy neural networks wt trangular fuzzy wegts and Hayas et al. (993) fuzzfed te delta rule. Buckley and Esla (997) consder neural net solutons to fuzzy probles. Te topc of nuercal soluton of fuzzy polynoals by fuzzy neural network nvestgated by Abbasbandy et al. (006), conssts of fndng soluton to polynoals lke axax ax = a x a0 a n n 0,,, an were and are fuzzy nubers, and fndng soluton to systes of s fuzzy polynoal equatons (Abbasbandy et al. 008). In ts paper, we frst propose an arctecture of fuzzy neural network (FNN) wt fuzzy wegts for real nput vectors and fuzzy targets to fnd approxate coeffcents to fuzzy lnear regresson odel Y = A AX A X, 0 n n were ndexes te dfferent observatons, X, X,, X E, all coeffcents and are fuzzy n nubers. Te nput-output relaton of eac unt s defned by te extenson prncple of Zade (975). Output fro te fuzzy neural network, wc s also a fuzzy nuber, s nuercally calculated by nterval artetc (Alefeld and Herzberger 983) for fuzzy wegts and real nputs. Next, we defne a cost functon for te level sets of fuzzy outputs and fuzzy targets. Ten, a crsp learnng algort s derved fro te cost functon to fnd te fuzzy coeffcents of te fuzzy lnear and nonlnear regresson odels. Te proposed algort s llustrated by soe exaples n te last secton. Prelnares: In ts secton te basc notatons used n fuzzy calculus are ntroduced. We start by defnng te fuzzy nuber. Defnton : A fuzzy nuber s a fuzzy set u: I =[0,] suc tat. u s upper se-contnuous.. u(x) = 0 outsde soe nterval [a, d].. Tere are real nubers b and c, a<b<c<d, for wc. u(x) s onotoncally ncreasng on [a, b],. u(x) s onotoncally decreasng on [c, d], Y 67

3 3. ux.( )=, b xc Te set of all te fuzzy nubers (as gven n defnton ) s denoted by E. An alternatve defnton wc yelds te sae E s gven by Kaleva (987), Mng Ma and Fredan (999). Defnton : A fuzzy nuber u s a par ( uu, ) of functons ur () and ur (), 0 r, wc satsfy te followng requreents:. ur () s a bounded onotoncally ncreasng, left contnuous functon on (0, ] and rgt contnuous at 0.. ur () s a bounded onotoncally decreasng, left contnuous functon on (0, ] and rgt contnuous at 0.. ur () ur (),0r. A crsp nuber r s sply represented by u( )= u( )= r,0. Te set of all te fuzzy nubers s denoted by E. A popular fuzzy nuber s te trangular fuzzy nuber u =( u, ul, ur) were u denotes te odal u >0 >0 value and te real values and represent te left and rgt fuzzness, respectvely. Te l ebersp functon of a trangular fuzzy nuber s defned by: x u, u ul xu, ul u x u( x)=, u x u ur, ur 0, oterwse. Its paraetrc for s u( )= u u ( ), u( )= u u ( ). u r l r Trangular fuzzy nubers are fuzzy nubers n R representaton were te reference functons and R are lnear. Te set of all trangular fuzzy nubers on s called... Operatons on Fuzzy Nubers: We brefly enton fuzzy nuber operatons defned by te extenson prncple (Zade 975). Snce nput vector of feedforward neural network s fuzzfed n ts paper, te operatons we use n our fuzzy neural network are fuzzfed by eans of te extenson prncple as follows: A B( z)= ax{ A( x) B( y) z = x y}, ( z)= ax{ ( x) ( y) z = xy}, AB A B ˆFZ ( z)= ax{ ( x) z = f ( x)}, f ( Net) Net 68

4 were A, B and Net are fuzzy nubers, (.) * denotes te ebersp functon of eac fuzzy nuber, v s te nu operator, and f(.) s a contnuous actvaton functon (suc as f(x)=x) nsde unts of our fuzzy neural network. Te above operatons of fuzzy nubers are nuercally perfored on level sets. Te -level set of a fuzzy nuber X s defned by [ X] ={ x ( x) } for 0<, X 0 (0,] [ X ] and [ X ] = [ X ]. Snce level sets of fuzzy nubers becoe closed ntervals, we denote by [ X] =[[ X],[ X] ], X [ X ] were [ X ] and [ ] are te lower and te upper lts of te -level set, respectvely. Fro nterval artetc (Alefeld 983), te above operatons on fuzzy nubers are wrtten for -level sets as follows: A= B [ A] =[ B] for 0<, () [ AB] =[[ A] [ B],[ A] [ B] ], () [ AB. ] = [[ A],[ A] ].[[ B],[ B] ] = [ n{[ A].[ B],[ A].[ B],[ A].[ B],[ A].[ B] }, (3) ax{[ A].[ B],[ A].[ B],[ A].[ B],[ A].[ B] }], f ([ Net] ) = f ([[ Net],[ Net] ]) = [ f ([ Net] ), f ([ Net] )], (4) were f s an ncreasng functon. In te case of 0 [ A] [ A], (3) can be splfed as [ AB. ] =[ n{[ A].[ B],[ A].[ B] }, ax{[ A].[ B],[ A].[ B] }]. (5) Te result of a fuzzy addton of trangular fuzzy nubers s a trangular fuzzy nuber agan. So we only ave to copute te followng equaton: ( a, a, a ) ( b, b, b )=( a b, a b, a b ) l r l r l l r r (6) Consderng te fuzzy ultplcaton, soe coputatonal expense probles can be nvestgated. Te result of a fuzzy ultplcaton s a fuzzy nuber n R representaton, but t s dffcult to copute te new functons and R because tey are not necessarly lnear. We approxate ts fuzzy ultplcaton suc tat t coputes a trangular fuzzy nuber too. Ts fuzzy ultplcaton s Denoted by (Feurng 995). Ts fuzzy 69

5 ultplcaton s based on te extenson prncple but s a bt dfferent fro te classcal fuzzy ultplcaton. We copute our operaton by te followng equaton: ( a, a, a )*( ˆ b, b, b )=( c, c, c ) 3 (7) wt l r l r l r c = a. b, c = c c, c = c c, l r c := n( a. b, a. b, a. b, a. b ) c := ax( a. b, a. b, a. b, a. b ), were a = a a l and a = a ar. a and a denote te left and rgt lts of te support of fuzzy te nuber a. Te use of tese fuzzy operatons as soe advantages: Te dstrbutvty of tese operatons s retaned. Ts s very portant for our teoretcal exanatons. Te coputatonal expense s acceptable. Te dea of fuzzy sets s preserved even f a fuzzy nuber s caracterzed by only tree values. We descrbe te classcal defnton of dstance between fuzzy nubers (Feurng 995): Defnton 3: Te appng dˆ : FZ ˆ FZ ˆ s defned by dˆ( A, B )= ax ( a b, a b, a b ), l r B b b b ˆd ˆFZ FZ ˆ dˆ were A =( a, a, a ) and =(,, ). It can be proved tat s a etrc on and so (, ) becoes a etrc space. l r. Input-output Relaton of Eac nt: et us fuzzfy a two-layer feedforward neural network wt n nput unts and one output unt. Input vectors, targets, connecton wegts and bas are fuzzfed (.e., extended to fuzzy nubers). In order to derve a crsp learnng rule, we restrct fuzzy wegts, fuzzy nputs and fuzzy target wtn trangular fuzzy nubers. Te nput-output relaton of eac unt of te fuzzfed neural network can be wrtten as follows: Input unts: O =, O = X, =,,,, j =,, n. 0 j j (8) Output unt: Y = f( Net ), Net = W O * ˆW O * ˆW, =,,,, 0 n n X W were j s a fuzzy nput and j s te fuzzy wegt (see fgure ). (9) (0).3 Calculaton of Fuzzy Output: Te fuzzy output fro eac unt n Eqs.(8)-(0) s nuercally calculated for level sets of fuzzy nputs and fuzzy wegts. Te nput-output relatons of our fuzzy neural network can be wrtten for te -level sets: Input unts: 70

6 Fg. : Fuzzy neural network for approxatng fuzzy lnear regresson. [ O ] = [ X ], =,,, j =,, n. j j () Output unt: [ Y] = f([ Net] ), n [ Net ] = [ O ].[ W ], =,,,. j j j =0 Fro Eqs.()-(3), we can see tat te -level sets of te fuzzy output Y () (3) s calculated fro tose of te fuzzy nputs and fuzzy wegts. Fro Eqs.()-(38), te above relatons are wrtten as follows wen te - level sets of te fuzzy nputs aj 's are nonnegatve,.e., 0 [ Xj] [ Xj ] for all, j 's: Input unts: [ O ] =[[ O ],[ O ] ]=[[ X ],[ X ] ], =,,,, j =,, n. j j j j j (4) Output unt: [ Y ] = [[ Y ],[ Y ] ] = [ f ([ Net ] ), f ([ Net ] )], (5) were f s an ncreasng functon. [ Net ] = [ O ].[ W ] [ O ].[ W ], j j j j ja jb [ Net ] = [ O ].[ W ] [ O ].[ W ], =,,,, j j j j jc jd (6) (7) 7

7 for [ W ] [ W ] 0, were a ={ j [ O ] 0}, b={ j [ O ] <0}, c ={ j [ O ] 0}, j j d ={ j [ O ] <0}, a b={0,, n} and. j j j c d ={0,, n} 3 Te near Regresson Model: We ave postulated tat te dependent fuzzy varable Y, s a functon of te ndependent fuzzy varables j X, X,, Xn n f : E. More forally Y = f( X, X,, X ) n were ndexes te observatons. Te objectve s to estate a fuzzy lnear regresson (FR) odel, express as follows: Y = A A * ˆ X A * ˆ X A * ˆ X. 0 n n Wen n f : E E, we gt do t by eye-fttng te lne tat looks best to us. nfortunately, dfferent people wll draw dfferent lnes and t would be nce to ave a foral etod for fndng te lne tat would consstently provde us wt te best lne possble. Wat would a "best possble lne" look lke? Intutvely, t would see to ave to be a lne tat ft te data well. Tat s, te dstance of te lne fro te observatons sould be as sall as possble. et A0, A,, An (8) denote te lst of regresson coeffcents (paraeters). A0 s an optonal ntercept paraeter and A,, An are wegts or regresson coeffcents correspondng to X,, X n. Ten fuzzy lnear regresson s gven by Y = A A * ˆ X A * ˆ X, 0 n n were ndexes te dfferent observatons and A, A,, An A0, A,, An (9) are fuzzy nubers. We are nterested n fndng 0 of fuzzy lnear regresson suc tat approxates for all =,,...,, closely enoug accordng to soe nor PP,.e., Y Y n P[ Y ] [ Y ] P and n P[ Y ] [ Y ] P, [0,]. (0) Terefore, n dˆ( Y, Y ) for all =,,,. () Ten, t becoes a proble of optzaton. A FNN 3 (fuzzy neural network wt fuzzy nput, output sgnals and fuzzy wegts) soluton to Eq.(9) s gven n fgure. Te nput neurons ake no cange n ter nputs and te nput sgnals nteract wt te wegts, so te nput to te output neuron s A A * ˆ X A * ˆ X 0 n n and te output, n te output neuron, equals ts nput, so 7

8 Y = A A * ˆ X A * ˆ X. 0 n n How does te FNN 3 solve te fuzzy lnear regresson? Te tranng data are {(, X,, X n),,(, X,, Xn)} { Y,,, Y } for nputs and target (desred) outputs are. We proposed a learnng algort fro te cost functon for adjustng fuzzy nuber wegts. Followng secton 4, we proposed a learnng algort suc tat te network can approxate te fuzzy A0, A,, An of Eq. (9) to any degree of accuracy. 3. earnng Fuzzy Neural Network: Consder te learnng algort of te two-layer fuzzy feedforward neural network wt nputs and one output as sown n fgure. et te -level sets of te target output [ Y] =[[ Y],[ Y] ], =,, n, Y, =,, be denoted were Y ( ) sows te left-and sde and Y ( ) te rgt-and sde of te -level sets of te desred output. A cost functon to be nzed s defned for eac -level sets as follows: [ EW (, W,, W)] =[ EW (, W,, W)] [ EW (, W,, W)], were 0 n 0 n 0 n [ EW ( 0, W,, Wn)] = ([ Y] [ Y] ), = () (3) [ EW ( 0, W,, Wn)] = ([ Y] [ Y] ). = Te total cost functon for te nput-output par ( x, Y ) s obtaned as e= [ E( W, W,, W )]. 0 n (4) Hence [ EW (, W,, W)] n 0 desred and te coputed output, and denotes te error between te left-and sdes of te -level sets of te [ (,,, )] EW0 W W n of te -level sets of te desred and te coputed output. In te researc of neural networks, te nor P,P s often defned as follows: denotes te error between te rgt-and sdes 73

9 ˆ n [ EW ( 0, W,, Wn)] = ([ Y] [ Y] ) = ( [ Oj * Wj] [ Y] ), = = j=0 (5) ˆ n [ EW ( 0, W,, Wn)] = ([ Y] [ Y] ) = ( [ Oj * Wj] [ Y] ). = = j=0 Clearly, ts s a proble of optzaton of quadratc functons wtout constrans tat can usually be solved by gradent descent algort. In fact, denotng EW ( ) EW ( ) [ EW ( )] =([ ],,[ ] ), T W0 Wn EW ( ) EW ( ) [ EW ( )] =([ ],,[ ] ), T W0 Wn n order to solve equaton (36), assue k teratons to ave been done and get te k t teraton pont W k. REMARK. Snce te equatons (5) are quadratc functons, supposng and 0 [ O ] [ O ] j j and 0 [ W ] [ Wj] for =,,, j =0,, n j, we rewrte tese as follows: [ EW ( )] = ( [ X * ˆ W] [ Y] ) n j j = j=0 n n = [( [ Xj] [ Wj] ) [ Y] [ Xj] [ Wj] ([ Y] ) ] = j=0 j=0 = {( ([ ] ) )([ ] ) ( ([ ] ) )([ ] ) X W X W = = ( ([ Xn] ) )([ Wn ] ) ( = = [ X ] [ X ] )[ W] [ W ] X X3 W W3 X Xn W Wn = = ( [ ] [ ] )[ ] [ ] ( [ ] [ ] )[ ] [ ] 74

10 X X3 W W3 X X4 W W4 = = ( [ ][ ])[ ][ ] ( [ ][ ])[ ][ ] ( [ X ] [ X ] )[ W ] [ W ] = n n X, n Xn Wn Wn X Y W = = ( [ ][ ])[ ][ ]} ( [ ][ ])[ ] ( [ X] [ Y] ) [ W ] ( [ Xn] [ Y] )[ Wn] ([ Y] ) = = = T T = ([ W])[ Q][ W] ([ B])[ W] [ C], were [ Q] =[( q )], j =([ b ],,[ b ] ), T 0 n = ([ Y] ), = Xk Xkj k = = [ ] [ ], wt [ q ] =[ q ] and [ b] = [ X ] [ Y ]. We ave j j k = k k [ EW ( )] =[ Q] [ W] [ B], (6) and 75

11 [ EW ( )] = ( [ X * ˆ W] [ Y] ) n j j = j=0 n n = [( [ Xj ][ Wj ]) [ Y ] [ Xj ][ Wj ] ([ Y ])] = j=0 j=0 = {( ([ ] ) )([ ] ) ( ([ ] ) )([ ] ) X W X W = = ( ([ Xn] ) )([ Wn ] ) ( = = [ X ] [ X ] )[ W] [ W ] X X3 W W3 X Xn W Wn = = ( [ ] [ ] )[ ] [ ] ( [ ] [ ] )[ ] [ ] X X3 W W3 X X4 W W4 = = ( [ ][ ])[ ][ ] ( [ ][ ])[ ][ ] ( [ X ] [ X ] )[ W ] [ W ] = n n X, n Xn Wn Wn X Y W = = ( [ ][ ])[ ][ ]} ( [ ][ ])[ ] ( [ X] [ Y] )[ W ] ( [ Xn] [ Y] )[ Wn] ([ Y] ) = = = T T = ([ W])[ Q][ W] ([ B])[ W] [ C], were 76

12 [ Q] =[( q )], j =([ b ],,[ b ] ), T 0 n = ([ Y] ), = Xk Xkj k = = [ ] [ ], wt [ q ] =[ q ] and [ b] = [ X ] [ Y ]. We ave j j k = k k [ E( W)] =[ Q] [ W] [ B], To fnd te statonary pont of [ EW ( )] =([ EW ( )],[ EW ( )] ) T [ EW ( )] =[ EW ( )] =0@(0,0,,0). [ ] te statonary pont can be obtaned as follows: (7), we sould put Wen Q and [ Q] are postve defnte atrces, * [ ] = ([ ] W ) [ ] Q B, (8) * [ ] = ([ ] W ) [ ] Q B. Te Hessan atrces at ts pont are * * [ EW ( )] = [ ( EW ( ))] = EW ( ) EW ( ) EW ( ) [ ] [ ] [ ] EW ( ) EW ( ) EW ( ) EW ( ) W WW WnW [ ] [ ] W WnW [ ] [ ] WWn Wn W= W * =[ Q], and 77

13 * * [ EW ( )] = [ ( EW ( ))] = EW ( ) EW ( ) EW ( ) [ ] [ ] [ ] EW ( ) EW ( ) EW ( ) [ ] [ ] [ ] EW ( ) EW ( ) EW ( ) [ ] [ ] [ ] W WW WnW W W W Wn W Wn W W Wn Wn W= W * =[ Q], [ Q] wc are postve defnte atrces because [ Q ] and are postve defnte. Fro optzaton teory, we known tat of te proble. * * * [ ] =([ ],[ ] )=( [ ] [ ] W, [ ] [ ] W W Q B Q B ), s te unque soluton REMARK. Te above etod s not very convenent n applcatons. Now we consder ts explct scee. Snce [ EW ( )] =[ Q] [ W] [ B] and [ E( W)] =[ Q] [ W] [ B], ten [ E( W )] =[ Q] [ W ] [ B] and [ E( W )] =[ Q] [ W ] [ B]. We know tat ( et al. 003). k k k k ([ EW ( )] ) [ EW ( )] =0, ([ EW ( )] ) [ EW ( )] =0, T T k k k k terefore we ave (Isbuc et al. 995). ([ Q]([ W ] [ ][ E( W )]) [ B])([ T Q][ W ] [ B])=0 and k k k k ([ Q] ([ W ] [ ] [ E( W )] ) [ B] ) T ([ Q] [ W ] [ B] ) = 0. k k k k Rearrangng te, we ave: ([ EW ( )] [ ][ Q][ EW ( )])[ T EW ( )] =0, and k k k k ([ EW ( )] [ ][ Q][ EW ( )])[ T EW ( )] =0. k k k k Fro tese equatons, we can easly get an expresson for [ ] and [ ] : k k T ([ EW ( k)] ) [ EW ( k)] [ ] = ([ EW ( )])[ Q ][ EW ( )] k T k k (9) and 78

14 T ([ EW ( k)] ) [ EW ( k)] [ k] =. T ([ EW ( )] ) [ Q] [ EW ( )] k k (30) `Substtutng tese nto equatons (Isbuc et al. 995; Isbuc and N 00; Ruelart et al. 986), we obtan W = W W, k k k W = E( W ) W, k k k k (3) were k ndexes te nuber of adjustents, k =([ k],[ k] ) s a learnng rate and α s a constant oentu ter (a postve real nuber). We can also obtan slar relatons for [ Xj ] [ Xj ] 0 and [ Wj] [ Wj], 0 =,,...,, j=0,..., n, and oter cases. 4 Te Nonlnear Regresson Model: We ave postulated tat te dependent fuzzy varable Y, s a functon of te ndependent fuzzy varables X, X,, Xn n f : E E. More forally Y = f( X, X,, X ) n were ndexes te observatons. Te objectve s to estate a fuzzy nonlnear regresson (FNR) odel, express as follows: Y = A * ˆ X * ˆ X * ˆ * ˆ X. A A A n 0 n (3) A0, A,, An X,, X were ndexes te dfferent observatons, are fuzzy nubers and are fuzzy nubers. et A0, A,, An denote te lst of regresson coeffcents (paraeters). 0 s an optonal A,, An X,, X ntercept paraeter and are wegts or regresson coeffcents correspondng to. By te above equaton, we ave ny = n( A ) A * ˆ nx A * ˆ nx, 0 n n were n s an ncreasng functon. Fro nterval artetc [?], te above operatons on fuzzy nubers are wrtten for -level sets as follows. Input unts: O =, O = X, j = 0,,, n, =,,. 0 j j Output unt: n[ Y ] =[ n[ Y ], n[ Y ] ]=[ f ([ Net ] ), f ([ Net ] )], were f ( x)= x s an ncreasng functon. A n n (33) 79

15 [ Net ] = [ O ].[ W ] [ O ].[ W ], j j j j ja jb [ Net ] = [ O ].[ W ] [ O ].[ W ], =,,,, j j j j jc jd (34) (35) [ W ] [ W ] for, were [[ A ],[ A ] ] = [ e, e ], a ={ j [ Oj ] 0}, b={ j [ O j ] <0}, j j 0 0 [ W ] [ W ] c ={ j [ O ] 0}, d ={ j [ O ] <0}, a b={0,, n} and c d ={0,, n}. j j We are nterested n fndng A0, A,, An of fuzzy nonlnear regresson suc tat approxates Y for all =,,,, closely enoug accordng to soe nor P,P,.e., n P[ ny ] [ ny ] P and n P[ ny ] [ ny ] P, [0,]. Terefore, Y (36) n dˆ( ny, ny ) for all =,,, n. (37) Ten, t becoes a proble of optzaton. A FNN 3 for solvng Eq. (3) s gven n fgure. 5 Coparson wt Oter Metods: Ts study would not be copleted wtout coparng t wt oter exstng etods. Soe coparsons are as follows: Mosle et al. (00) ave consdered te fuzzy regresson were x Y = A Ax A x A x 0 n n s a crsp and y s a fuzzy nuber, but n ts paper, te nput and output are bot fuzzy nubers. For ore detals see exaple 6.. In ts paper, f we take te fuzzy nputs as fuzzy ponts, we wll ave and n ts [ x] =[[ x],[ x] ] case, te content of te present paper reduces to tat of (Mosle et al. 00). Fro te pont of vew of predcton, we ave done ts coparson between ts paper and (Kao et al. 003; Tanaka et al. 989) n exaple Nuercal Exaples: In ts secton we ave two exaple, one of te for sow useful ts tecnque and anoter s nuercal exaple fro Iran labor arket data: Exaple 6.: Kao et al. (003) and Tanaka et al. (989) used an exaple to llustrate ter regresson odel, n tat te explanatory varable s crsp and te responses are trangular fuzzy nubers. Tat exaple as fve sets of te ( x, Y ) observatons, see table. For eac fuzzy nubers, we use -cuts = 0,0.,,, were we calculate te error of eac fuzzy output by e Y Y Y Y = ([ ] [ ] ) ([ ] [ ] ), Y and total error by Eq. (4). 80

16 Fg. : Fuzzy neural network for approxatng fuzzy nonlnear regresson. Table : Nuercal data and te estaton errors for exaple 6.. Independent Response Errors n estaton varable varable Tanaka Kao Neural network (8.0,.8,.8) (6.4,.,.) (9.5,.6,.6) (3.5,.6,.6) (3.0,.4,.4) Total error In te coputer sulaton of ts exaple, we use te followng specfcatons of te learnng algort. () Nuber of nput unts: unts. () Nuber of output unts: unt. (3) Stoppng condton: K=8 teratons of te learnng algort. Te tranng starts wt () = (,0.5,0.5) and W () = (0.3,0.3,0.). Applyng te proposed etod to W 0 te approxate soluton of proble (9). In sybols, te fuzzy neural network odel s: Y FNN = (4.9499,.8399,.8398) (.7, 0.6, 0.60) x. In te study of Tanaka et al. (989), te results of te Mn proble of =0 s used for coparson. Te fuzzy regresson odel s: YT = (3.850,3.850,3.850).00 x. In te study of Kao et al. (003), te results of te fuzzy regresson odel s: Y K = x (0.8,.3,.3). To copare te perforance of tese tree etods n estaton, we apply to calculate te errors n estatng te observed responses. Table sows te errors n estatng te fve observaton for tese tree etods. Te total error of te fuzzy neural network etod s 79.08, wc s obvously better tan te total error of calculated fro te Kao etod and te total error of calculated fro te Tanaka etod. Fgure 3 depcts te estatons of tese tree etods. 8

17 Fg. 3: Estatons of te neural network, Tanaka and Kao etods. Exaple 6.: abor Deand and Supply of Iran: In ts exaple we used supply and deand data of Iran labor arket. Table and 3 are sown deand an supply data. Consder te fuzzy data for a dependent fuzzy varable Y tat t s total eployent n Iran and two ndependent fuzzy varables x and x tat ndcated GDP and wage respectvely n table. Also n table 3, Y s labor force partcpaton rate and x and x ndcated sae n table. sng tese data, develop an estated fuzzy regresson equaton ny n A A ˆ nx A ˆ nx A ˆ ny = ( 0) * * n 3*, n. In te coputer sulaton of ts exaple, we use te followng specfcatons of te learnng algort. () Nuber of nput unts: 3 unts. () Nuber of output unts: unt. (3) Stoppng condton: K=5 teratons of te learnng algort. Te tranng starts wt W0() = (,0.5,0.5), W() = (0.5,0.5,0.5), W() = ( 0.5,0.5,0.5) and W 3 () = (0.5,0.5,0.5). Applyng te proposed etod to te approxate soluton of proble (33). In sybols, te fuzzy neural network odel s: 8

18 Table : Inputs and output data for labor deand estate n exaple 6.. Year X X 988 (808.5,0.75,.5) (9050.6,,) (6874.5,0.5,0.5) 989 (950.6,.,.6) (964.,,.5) (9765.5,.5,.5) 990 (8538.7,.,.4) (06.4,,0.75) ( ,.5,) 99 ( ,.,0.75) (778.4,,0.5) ( ,.75,3) 99 (548.5,.5,) (859.,0.75,.) (33885,3.,3.) 993 (5860.4,0.4,0.75) (4866.7,0.5,0.75) ( ,4,3.6) 994 ( ,.,0.) (5445.,0.9,) ( ,4.7,4) 995 (67534.,0.9,0.5) (464.8,0.8,0.75) ( ,5,4.9) 996 ( ,0.6,0.9) (474.4,.,) ( ,5.,5.6) 997 (9768.7,0.5,0.45) (3948.4,.4,.75) (50956.,6.,5.9) 998 ( ,0.75,0.5) (494.6,,0.5) (5955.8,6.5,6) 999 (30494.,0.,0.75) (576,0.5,0.5) ( ,7.,6.7) 000 (30069,,.5) (463.9,0.9,0.5) ( ,7.5,7) 00 (330565,,.5) (4780.9,0.9,0.5) ( ,8,8.) 00 (355554,0.75,0.87) (569,,.5) ( ,8.,8.5) 003 (379837,,) (603.8,0.8,0.75) ( ,8.5,8.6) 004 (39834,,) (6653.5,0.5,0.75) ( ,9.4,9.5) 005 (49705,,.5) (637,,0.5) ( ,0,0.) 006 (445790,0.75,0.5) (650.3,0.5,0.5) (047637,.3,) Table 3: Inputs and output data for labor supply estate n exaple 6.. Year X X 988 (808.5,,) (90506,9,) ( ,0.,0.) 989 (89597.,9,7) (964.,8.8,9.8) (38.69,0..3,0.0) 990 (8538.7,4,9) (06.4,6.6,8.9) (38.999,0.34,0.08) 99 ( ,5.6,7.4) (778.4,5.6,6.9) ( ,0.7,0.) 99 (54059.,.,3.5) (859.,7.9,5) ( ,0.,0.) 993 (5860.4,.6,7) (4866.7,4,6.3) (37.999,0.,0.4) 994 ( ,.5,.75) (5445.,6.9,8.9) ( ,0.8,0.7) 995 (67534.,.8,.) (464.8,,0) (36.4,0.3,0.3) 996 ( ,.4,) (474.4,4.6,8.9) (35.3,0.35,0.37) 997 (97687,0,9) (3948.4,5.6,8.6) (35.7,0.39,0.4) 998 ( ,6.4,8.) (494.6,9,8.5) (36.,0.4,0.43) 999 (30494.,9.8,6.9) (576,,7.5) ( ,0.45,0.48) 000 (30069,,) (4638.5,8.9,8.8) (37.,0.49,0.5) 00 (330565,,5) (4780.9,9.5,7.5) ( ,0.54,0.56) 00 (355554,,5) (569,,) ( ,0.59,0.58) 003 (379837,6,) (603.8,9.5,8.5) ( ,0.6,0.6) 004 (39834,0,8) (6653.5,,) ( ,0.63,0.66) 005 (49705,.5,.5) (637,9.8,6.7) (39.,0.68,0.68) 006 (445790,0.5,) (650.3,0,) ( ,0.7,0.7) Y Y ny = n(.4734,0.03,0.034) (0.9,0.0087,0.0063) nx ( 0.78,0.0075,0.0056) nx (0.886,0.004,0.009) ny. ny = n(0.907,0.003,0.0085) (0.0788, , ) nx (38) (39) ( 0.385,0.00,0.006) nx (0.863,0.0073,0.0084) ny. lnyt Eq.(38) s labor deand and Eq.(39) s labor supply, te coeffcent 0.9 eans tat, oldng lnx and, one percent ncrease of GDP s predcted to ncrease t by 0.3 percent. te oter coeffcents y ave a slar nterpretaton. Te functon for n current researc s a sple Cab-Daglas for. We nclude yt n equaton, because t s explans all anoter varables tat are pact on y t. Te key pont n ts artcle s tat newly 83

19 estaton by OS neverteless all of econoetrc coents s stll, suc as statonery te seres and Ceters parbus assupton. More dfference of ts artcle s anoter type of OS estaton by fuzzy ndependents and dependent varables. Tus te coeffcents are fuzzy and tey are not exactly. 7 Suary and Conclusons: Solvng fuzzy lnear regresson (FR) and fuzzy nonlnear regresson (FNR) by usng unversal approxators (A), tat s, FNN s presented n ts paper. Te proble forulaton of te proposed AM s qute stragtforward. To obtan te "Best-approxated" soluton of FRs and FNRs, te adjustable paraeters of FNN are systeatcally adjusted by usng te learnng algort.n ts paper we used Iran labor arket data and estated labor supply and deand.te coeffcents are are not exact.becuase te data of labor arket are not exact,ten tese coeffcents are better tan tradtonal OS estators. In ts paper, we derved a learnng algort of fuzzy wegts of two-layer feedforward fuzzy neural networks wose nput-output relatons were defned by extenson prncple. Te effectveness of te derved learnng algort was deonstrated by coputer sulaton of nuercal exaples. Coputer sulaton n ts paper was perfored for two-layer feedforward neural networks usng te back-propagaton-type learnng algort. If we use oter learnng algorts, we ay ave dfferent sulaton results. For exaple, soe global learnng algorts suc as genetc algorts ay tran fuzzy connecton wegts uc better tan te back-propagaton-type learnng algort. REFERENCES Abbasbandy, S., M. Otad, 006. Nuercal soluton of fuzzy polynoals by fuzzy neural network, Appl. Mat. Coput., 8: Abbasbandy, S., M. Otad, M. Mosle, 008. Nuercal soluton of a syste of fuzzy polynoals by fuzzy neural network, Infor. Sc., 78: Alefeld, G., J. Herzberger, 983. Introducton to Interval Coputatons, Acadec Press, New York. Buckley, J.J., E. Esla, 997. Neural net solutons to fuzzy probles: Te quadratc equaton, Fuzzy Sets and Sestes, 86: Cang, P.T., 997. Fuzzy seasonalty forecastng, Fuzzy Sets Syst., 90: -0. Cen, T., M.J.J. Wang, 999. Forecastng etods usng fuzzy concepts, Fuzzy Sets Syst., 05: Conway, K.S., T.J. Knesner, 99. How fragle are ale labor supply functon estates, Eprcal Econocs, 7: Cowell, A.F., 004. Mcroeconocs prncples and analyss, Stcerd and departent of econocs, ondon scool of econocs. Dubos, D., H. Prade, 978. Operatons on fuzzy nubers, J. Systes Sc., 9: Feurng,T.H., W.M. ppe, 995. Fuzzy neural networks are unversal approxators, IFSA World Congress, Sao Paulo, Brasl, : Jorgenson, W.D., R.J. Goettle, S.M. Ho, T.D. Slesnck, J.P. Wlcoxen, 008..S. abor supply and deand n te long run. Journal of polcy odelng, 30: Hayas, Y., J.J. Buckley, E. Czogala, 993. Fuzzy neural network wt fuzzy sgnals and wegts, Internat. J. Intellgent Systes, 8: Heckan, J.J., T.E. Macurdy, 980. A lfe-cycle odel of feale labor supply, revew of econoc studes, 47: Henderson, J.M., R.E. Quandt, 985. Mcroeconoc teory a ateatcal approac, Trd edton, MCGRAW-HI. Isbuc, H., K. Kwon, H. Tanaka, 995. A learnng algort of fuzzy neural networks wt trangular fuzzy wegts, Fuzzy Sets and Systes, 7: Isbuc, H., M. N, 00. Nuercal analyss of te learnng of fuzzfed neural networks fro fuzzy f-ten rules, Fuzzy Sets and Systes, 0: Kaleva, O., 987. Fuzzy dfferental equatons, Fuzzy Sets and Systes, 4: Kao, C., C.. Cyu, 003. east-squares estates n fuzzy regresson analyss, European Journal of Operatonal Researc, 48: Kaufann, A., M.M. Gupta, 985. Introducton Fuzzy Artetc, Van Nostrand Renold, New York. a,y.j., S.I. Cang, 994. A fuzzy approac to ultperson optzaton: an off-lne qualty engneerng proble, Fuzzy Sets and Syst., 63:

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