Designing Fuzzy Time Series Model Using Generalized Wang s Method and Its application to Forecasting Interest Rate of Bank Indonesia Certificate

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1 The Frst Internatonal Senar on Scence and Technology, Islac Unversty of Indonesa, 4-5 January 009. Desgnng Fuzzy Te Seres odel Usng Generalzed Wang s ethod and Its applcaton to Forecastng Interest Rate of Bank Indonesa Certfcate Agus aan Abad, Subanar, 3 Wdodo, 4 Sasubar Saleh Departent of atheatcs Educaton, Faculty of atheatcs and Natural Scences, Yogyakarta State Unversty, Indonesa Karangalang Yogyakarta 558,3 Departent of atheatcs, Faculty of atheatcs and Natural Scences, Gadah ada Unversty, Indonesa Sekp Utara, Bulaksuur Yogyakarta Departent of Econocs, Faculty of Econocs and Busness, Gadah ada Unversty, Indonesa Jl. Huanora, Bulaksuur Yogyakarta 558 Eal: aanabad@yal.co, subanar@yahoo.co, 3 wdodo_ath@yahoo.co, 4 huas@paue.ug.ac.d Abstract Fuzzy te seres s a dynac process wth lngustc values as ts observatons. odellng fuzzy te seres developed by soe researchers used the dscrete ebershp functons and table lookup schee (Wang s ethod) fro tranng data. The Wang s ethod s a sple ethod that can be used to overcoe the conflctng rule by deternng each rule degree. The weakness of fuzzy te seres odel based on the ethod s that the fuzzy relatons ay not be coplete so the fuzzy relatons can not cover all values n the doan. Ths paper presents generalzaton of the Wang s ethod usng the contnuous ebershp functon based on fuzzy te seres data. Furtherore, ths ethod s appled to forecast nterest rate of Bank Indonesa Certfcate (BIC) based on one-factor two-order fuzzy te seres. The predcton of nterest rate of BIC usng the proposed ethod has a hgher accuracy than that usng the Wang s ethod. Keywords: fuzzy relaton, fuzzy te seres, generalzed Wang s ethod, nterest rate of BIC.. Introducton Fuzzy te seres s a dynac process wth lngustc values as ts observatons. In recently, fuzzy te seres odel was developed by soe researchers. Song and Chsso developed fuzzy te seres by fuzzy relatonal equaton usng adan s ethod [0]. In ths odelng, deternng the fuzzy relaton need large coputaton. Then, Song and Chsso constructed frst order fuzzy te seres for te nvarant and te varant cases

2 The Frst Internatonal Senar on Scence and Technology, Islac Unversty of Indonesa, 4-5 January 009. [], []. Ths odel need coplexty coputaton for fuzzy relatonal equaton. Furtherore, to overcoe the weakness of the odel, Chen desgned fuzzy te seres odel by clusterng of fuzzy relatons [4]. Hwang constructed fuzzy te seres odel to forecast the enrollent n Alabaa Unversty [8]. Fuzzy te seres odel based on heurstc odel gave ore accuracy than ts odel desgned by soe prevous researchers [7]. Then, forecastng for enrollent n Alabaa Unversty based on hgh order fuzzy te seres resulted ore accuracy predcton [5]. Frst order fuzzy te seres odel was also developed by Sah and Degtarev [9] and Chen and Hsu [6]. Abad [] constructed fuzzy te seres odel usng table lookup schee (Wang s ethod) to forecast nterest rate of Bank Indonesa certfcate (BIC) and the result gave hgh accuracy. Then, forecastng nflaton rate usng sngular value decoposton ethod have a hgher accuracy than that usng Wang s ethod [], [3]. The weakness of the constructng fuzzy relatons based on the Wang s ethod s that the fuzzy relatons ay not be coplete so the fuzzy relatons can not cover all values n the doan. In ths paper, we wll desgn fuzzy te seres odel usng generalzed Wang s ethod to prove the predcton accuracy. Then, ts result s used to forecast nterest rate of BIC. The proposed ethod has a hgher predcton accuracy than the Wang s ethod n applcaton to forecastng nterest rate of BIC. The rest of ths paper s organzed as follows. In secton, we present the Wang s ethod to construct fuzzy odel. In secton 3, we brefly revew the defntons of fuzzy te seres and ts propertes. In secton 4, we present a generalzaton of Wang s ethod to construct fuzzy te seres odel based on tranng data. In secton 5, we apply the proposed ethod to forecastng nterest rate of BIC. We also copare the proposed ethod wth the Wang s ethod n the forecastng nterest rate of BIC. Fnally, soe conclusons are dscussed n secton 6.. Wang s ethod for desgnng fuzzy rules In ths secton, we wll ntroduce the Wang s ethod to construct fuzzy rules [3]. Suppose that we are gven the followng N nput-output data: ( x, x,..., x ; y ) p p np p, p =,,3,..., N where x [ α, β ] R and y [ α, β ] R, =,,, n. Desgnng fuzzy odel usng p p y y Wang s ethod s gven by the followng steps: Step. Defne fuzzy sets to cover the nput and output doans. For each space [ α, β ], =,,, n, defne N fuzzy sets A, =,,, N whch are coplete n [ α, β ]. Slarly, defne N fuzzy sets B, =,,, N y y whch are noral and coplete n [ α, β ]. y y Step. Generate one rule fro one nput-output par. For each nput-output par ( x, x,..., x ; y ), deterne ebershp value of x p p np p p, =,,, n n fuzzy sets A, =,,, N and ebershp value of y n fuzzy sets B, =,, p, N y. Then, for each nput varable x p, =,,, n, deterne the fuzzy set n whch x p * has the largest ebershp value. In other word, deterne A such that µ * ( x ) µ ( x ), A p p A l* =,,..., N. Slarly, deterne B such that µ l* ( y ) µ l( y ), l =,,..., N. Fnally, we B p B p y construct a fuzzy IF-THEN rule: IF x s A and x s A and... and x s A, THEN y s B * * * l * n n Step 3. Copute degree of each rule desgned n step.

3 The Frst Internatonal Senar on Scence and Technology, Islac Unversty of Indonesa, 4-5 January 009. Fro step, one rule s generated by one nput-output par. If the nuber of nput-output data s large, then t s possble that there are the conflctng rules. Two rules becoe conflctng rules f the rules have sae IF parts but dfferent THEN parts. To resolve ths proble, we assgn a degree to each rule desgned n step. The degree of rule s defned as follows: suppose that the rules n Step s constructed by the nput-output par ( x, x,..., x ; y ), then ts degree s defned as p p np p n D ( rule ) = µ ( x ) µ ( y ) = * l* A p B Step 4. Construct the fuzzy rule base. The rule base conssts of the followng three sets of rules: () The rules desgned n Step that do not conflct wth any other rules; () The rule fro a conflctng group that has the axu degree; (3) Lngustc rules fro huan experts. Step 5. Construct the fuzzy odel usng the fuzzy rule base. We can use any fuzzfer, nference engne and defuzzfer and cobne wth the fuzzy rule base to desgn fuzzy odel. If the nuber of tranng data s N and the nuber of all possble cobnatons of the n fuzzy sets defned for the nput varables s N, then the nuber of fuzzy rules generated by = n Wang s ethod ay be uch less than both N and N. Then, the fuzzy rule base generated by ths ethod ay not be coplete so that the fuzzy rules can not cover all values n the nput spaces. 3. Fuzzy te seres In ths secton, we ntroduce the followng defntons and propertes of fuzzy te seres referred fro Song and Chsso [0]. Defnton. Let Yt () R, t =..., 0,,,...,, be the unverse of dscourse on whch fuzzy sets f () t ( =,, 3,...) are defned and Fts () the collecton of f () t, =,, 3,...,then Ft () s called fuzzy te seres on Yt, () t =..., 0,,, 3,... In the Defnton, Ft () can be consdered as a lngustc varable and f () t as the possble lngustc values of Ft (). The value of Ft () can be dfferent dependng on te t so Ft () s functon of te t. The followng procedure gves how to construct fuzzy te seres odel based on fuzzy relatonal equaton. Defnton. Let I and J be ndces sets for Ft ( ) and Ft () respectvely. If for any f() t Ft (), J, there exsts f( t ) Ft ( ), I such that there exsts a fuzzy relaton R tt and f( t) = f( t ) R( tt, ), Rtt (, ) = R( tt, ) where s unon operator, then (, ) Rtt (, ) s called fuzzy relaton between Ft () and Ft ( ). Ths fuzzy relaton can be wrtten as Ft () = Ft ( ) Rtt (, ). () where s ax-n coposton. In the equaton (), we ust copute all values of fuzzy relatons R( tt, ) to deterne value of Ft (). Based on above defntons, concept for frst order and -order of fuzzy te seres can be defned. Defnton 3. If Ft () s caused by Ft ( ) only or by Ft ( ) or Ft ( ) or or Ft ( ), then the fuzzy relatonal equaton, = p

4 The Frst Internatonal Senar on Scence and Technology, Islac Unversty of Indonesa, 4-5 January 009. Ft ( ) = Ft ( ) R( tt, ) or Ft ( ) = ( Ft ( ) Ft ( )... Ft ( )) R0 ( tt, ) () s called frst order odel of Ft (). Defnton 4. If Ft () s caused by Ft ( ), Ft ( ),... and Ft ( ) sultaneously, then the fuzzy relatonal equaton Ft ( ) = ( Ft ( ) Ft ( )... Ft ( )) Ra ( tt, ) (3) s called -order odel of Ft (). Fro equatons () and (3), the fuzzy relatons Rtt (, ), R(, tt ), R(, tt ) are portant a o factors to desgn fuzzy te seres odel. Furtherore for the frst order odel of Ft (), for any f() t Ft (), J, there exsts f( t ) Ft ( ), I such that there exsts fuzzy relatons R( tt, ) and f( t) = f( t ) R( tt, ). Ths s equvalent to f f ( t ), then f () t, and then we have the fuzzy relaton R( tt, ) = f ( t ) f () t. Because of Rtt (, ) = (, ) R tt, then Rtt (, ) = aks {n( f ( t), f ( t ))}. (4), For the relaton R(, tt ) of the frst order odel, we get o R(, tt ) = aks{ aks{n( f ( t k), f ( t))} }. (5) o p k k, Based on -order odel of Ft (), we have R(, tt ) = aks{ n ( f ( t ) f ( t )... f ( t ) f ( t) )} (6) a p,,,..., k Fro equatons (4), (5) and (6), we can copute the fuzzy relatons usng ax-n coposton. Defnton 5. If for t t, Rt (, t ) = Rt (, t ) or Ra( t, t ) = Ra( t, t ) or R ( o t, ) t = R ( o t, ) t, then Ft () s called te-nvarant fuzzy te seres. Otherwse t s called tevarant fuzzy te seres. Te-nvarant fuzzy te seres odels are ndependent of te t, those ply that n applcatons, the te-nvarant fuzzy te seres odels are spler than the te-varant fuzzy te seres odels. Therefore t s necessary to derve propertes of te-nvarant fuzzy te seres odels. Theore. If Ft () s fuzzy te seres and for any t, Ft () has only fnte eleents f () t, =,, 3,..., n, and Ft () = Ft ( ), then Ft () s a te-nvarant fuzzy te seres. Theore. If Ft () s a te-nvarant fuzzy te seres, then Rtt (, ) =... f ( t ) f ( t) f ( t ) f ( t )... f ( t ) f ( t + )... 0 where s a postve nteger and each par of fuzzy sets s dfferent. Based on the Theore, we should not calculate fuzzy relatons for all possble pars. We need only to use one possble par of the eleent of Ft () and Ft ( ) wth all possble t s. Ths ples that to construct te-nvarant fuzzy te seres odel, we need only one observaton for every t and we set fuzzy relatons for every par of observatons n the dfferent of te t. Then unon of the fuzzy relatons results a fuzzy relaton for the odel. Theore s very useful because we soete have only one observaton n every te t. Let F () tbe fuzzy te seres on Yt. () If F() t s caused by ( F( t ), F ( t )), ( F( t ), F ( t )),..., ( F( t n), F ( t n)), then the fuzzy logcal relatonshp s presented by ( F( t n), F ( t n)),..., ( F ( t ), F ( t )), ( F( t ), F ( t )) F() t and t s called two-factor n-order fuzzy te seres forecastng odel, where F (), t F () t are called the an factor and the secondary factor fuzzy te seres respectvely. If a fuzzy logcal relatonshp s presented as,

5 The Frst Internatonal Senar on Scence and Technology, Islac Unversty of Indonesa, 4-5 January 009. ( F( t n), F ( t n),..., F ( t n)),..., ( F ( t ), F ( t ),..., F ( t )), ( F( t ), F ( t ),..., F ( t )) F() t (7) then the fuzzy logcal relatonshp s called -factor n-order fuzzy te seres forecastng odel, where F() t are called the an factor fuzzy te seres and F ( t ),..., F ( t ) are called the secondary factor fuzzy te seres. f t = f t R tt s equvalent to the fuzzy rule IF f ( t ), THEN () Because ( ) ( ) (, ) f t, and the fuzzy relaton R( tt, ) = f ( t ) f () t, then we can vew the fuzzy rule as the fuzzy relaton and vce versa. 4. Desgnng fuzzy te seres odel usng generalzed Wang s ethod Lke n odelng tradtonal te seres data, we use tranng data to set up the relatonshp aong data values at dfferent tes. In fuzzy te seres, the relatonshp s dfferent fro that n tradtonal te seres. In fuzzy te seres, we explot the past experence knowledge nto the odel. The experence knowledge has for IF THEN. Ths for s called fuzzy rules. Furtherore, an step to odelng fuzzy te seres data s to dentfy the tranng data usng fuzzy rules. Let A ( t ),..., A ( t ) be N, k N, k fuzzy sets wth contnuous ebershp functon that are noral and coplete n fuzzy te seres F ( t ), =,, 3,, n, k =,,,, then the k fuzzy rule: R : IF ( x ( t n) s A ( t n) and...and x ( t n) s A ( t n)) and,, and ( x ( t ) s A ( t ) and...and x ( t ) s A ( t )), THEN x,, ()s t A,() t (8) s equvalent to the fuzzy logcal relatonshp (7) and vce versa. So (8) can be vewed as n fuzzy relaton n U V where U = U... U R, V R wth n µ A( x( t n),..., x( t ),..., x( t n),..., x( t )) = µ µ,, µ µ A A A,, A ( x ( t n))... ( x ( t ))... ( x ( t n)... ( t ), where A = A ( )... ( )... ( )... ( ), t n A, t A, t n A, t Let F( t ), F ( t ),..., F ( t ) F( t) be -factor one-order fuzzy te seres forecastng odel. Then F ( t ), F ( t ),..., F ( t ) F ( t) can be vewed as fuzzy te seres forecastng odel wth nputs and one output. In ths paper, we wll desgn -factor one-order te nvarant fuzzy te seres odel usng generalzed Wang s ethod. But ths ethod can be generalzed to -factor n-order fuzzy te seres odel. Suppose we are gven the followng N tranng data: ( x ( t ), x ( t ),..., x ( t ); x ( t)), p p p p p =,,3,..., N. We wll ntroduce a ethod to construct fuzzy logcal relatonshps fro tranng data presented as follows: Step. Defne the unverse of dscourse for an factor and secondary factor. Let U = [ α, β ] R be unverse of dscourse for an factor, x p( t ), x p( t) [ α, β ] and V = [ α, β ] R, =,3,...,, be unverse of dscourse for secondary factors, x ( ) [, ] p t α β. Step. Defne fuzzy sets on the unverses of dscourse. Let A ( t ),..., A ( t ) be N, k N, k,k fuzzy sets n fuzzy te seres F ( t ) that are contnuous, noral and coplete n [ α, β ] R, k k k k =,,3,..., =0,. Step 3. Deterne all possble antecedents of canddates of fuzzy logcal relatonshps. Based on the Step, there are N antecedents of canddates of fuzzy logcal relatonshps. k,. k= The antecedent has for: A ( t ), A ( t ),..., A ( t ), =,,..., N,,,

6 The Frst Internatonal Senar on Scence and Technology, Islac Unversty of Indonesa, 4-5 January 009. Step 4. Deterne consequence of each canddate of fuzzy logcal relatonshp. For each antecedent A ( t ), A ( t ),..., A ( t ), we choose A (), * t as the consequence,,, of the antecedent f there exsts tranng data ( x *( t ), x *( t ),..., x *( t ); x *( t)) such that µ ( x ( t )) µ ( x ( t ))... µ ( x ( t )) µ ( x ( t)) A * * * *, p A, p A, p A *, p µ µ µ µ A, p A, p A, p A p, p p p p ( x ( t )) ( x ( t ))... ( x ( t )) ( x ( t)) for all tranng data ( x ( t ), x ( t ),..., x ( t ); x ( t)) p p p p Fro ths step we have the followng = N collectons of fuzzy logcal relatonshps, k desgned fro tranng data: R l l l l l : ( A ( t ), A ( t ),..., A ( t )) A * ( t), l =,, 3,,. (9),,,, k= Step 5. Deterne the ebershp functon for each fuzzy logcal relatonshp resulted n the Step 4. If we vew each fuzzy logcal relatonshp as fuzzy relaton n U V wth U = U... U R, V R, then the ebershp functon for the fuzzy logcal relatonshp (9) s defned by µ ( x ( t ), x ( t ),..., x ( t ); x ( t)) R l p p p p = µ ( x ( t )) µ ( x ( t ))... µ ( x ( t )) µ l ( x ( t )) A ( ) ( ) ( ),,, A * () t p, Step 6. For gven nput fuzzy set A ( t ) n nput space U, copute the output fuzzy set A () tn l output space V for each fuzzy logcal relatonshp (9) as µ ( x( t)) = sup( µ ( xt ( )) ( ( ); ( )))) µ l xt x t where xt ( ) = ( x( t ),..., x ( t )). Al A R xu Step 7. Copute fuzzy set A () t as the cobnaton of fuzzy sets A (), t A (), t A (),...,() t A t by 3 µ ( x ( t)) = ax( µ ( x ( t),..., µ ( x ( t))) A () t A () t A () t l= = µ A µ l l= R xu ax (sup( ( xt ( )) ( xt ( ); x( t))) = µ A µ µ A, ( ) l f f t f l= A xu, f =. ax (sup( ( xt ( )) ( x ( t )) ( x( t)))) Step 8. Calculate the forecastng outputs. Based on the Step 7, f we are gven nput fuzzy set A ( t ), then the ebershp functon of the forecastng output A () ts µ ( x ( t)) A = µ () t A µ µ A, ( ) l f f t f l= A xu, f = ax (sup( ( xt ( )) ( x ( t )) ( x( t)))). (0) Step 9. Defuzzfy the output of the odel. If the goal of output of the odel s fuzzy set, then stop n the Step 8. We use ths step f we want the real output. For exaple, f gven the nput * ( x ( t ) x ( t )) fuzzy set A ( t ) wth Gaussan ebershp functon µ ( ( )) exp( ) A xt ( t = ), = a then the forecastng real output usng the Step 8 and center average defuzzfer s ( x ( t ) x ( t )) exp( ) * y = = a + σ, ( ) = ( ( ),..., ( )) = * ( x ( t ) x ( t )) exp( ) = = a + σ, A t. x t f x t x t where y s center of the fuzzy set,() Fro Step 4, the set of fuzzy logcal relatonshps (9) constructed by ths ethod contans fuzzy relatons desgned by the Wang s ethod. Therefore the proposed ethod s generalzaton of the Wang s ethod. ()

7 The Frst Internatonal Senar on Scence and Technology, Islac Unversty of Indonesa, 4-5 January Applcaton of the proposed ethod In ths secton, we apply the proposed ethod to forecast nterest rate of BIC based on one-factor two-order fuzzy te seres odel. The data are taken fro January 999 to February 003. The data fro January 999 to Deceber 00 are used to tranng and the data fro January 00 to February 003 are used to testng. We apply the procedure n Secton 4 to predct nterest rate of BIC of k th onth usng data of (k-) th and (k-) th onths. We use [0, 40] as unverse of dscourse of two nputs and one output and we defne seven fuzzy sets A, A,..., A wth Gaussan ebershp functon on each unverse of dscourse of 7 nput and output. Then, we apply the Step 4 of the proposed ethod to yeld forty nne fuzzy logcal relatonshps. The fuzzy logcal relatonshps generated by the proposed ethod are shown n Table I. Table I. Fuzzy logcal relatonshp groups for nterest rate of BIC usng generalzed Wang s ethod Nuber ( x ( t ), x ( t ) ) xt () Nuber ( x ( t ), x ( t ) ) xt () Nuber ( x ( t ), x ( t ) ) xt () (A, A) A 7 (A3, A3) A3 33 (A5, A5) A3 (A, A) A 8 (A3, A4) A3 34 (A5, A6) A7 3 (A, A3) A 9 (A3, A5) A 35 (A5, A7) A7 4 (A, A4) A 0 (A3, A6) A 36 (A6, A) A 5 (A, A5) A3 (A3, A7) A7 37 (A6, A) A 6 (A, A6) A3 (A4, A) A 38 (A6, A3) A 7 (A, A7) A3 3 (A4, A) A 39 (A6, A4) A3 8 (A, A) A 4 (A4, A3) A 40 (A6, A5) A3 9 (A, A) A 5 (A4, A4) A 4 (A6, A6) A5 0 (A, A3) A3 6 (A4, A5) A 4 (A6, A7) A6 (A, A4) A3 7 (A4, A6) A7 43 (A7, A) A (A, A5) A3 8 (A4, A7) A7 44 (A7, A) A 3 (A, A6) A3 9 (A5, A) A 45 (A7, A3) A3 4 (A, A7) A7 30 (A5, A) A 46 (A7, A4) A3 5 (A3, A) A 3 (A5, A3) A 47 (A7, A5) A3 6 (A3, A) A 3 (A5, A4) A 48 (A7, A6) A5 49 (A7, A7) A6 Based on the Table II, the average forecastng errors of nterest rate of BIC usng the Wang s ethod and the proposed ethod are % and.7698%, respectvely. So we can conclude that forecastng nterest rate of BIC usng the proposed ethod results ore accuracy than that usng the Wang s ethod. Table II. Coparson of average forecastng errors of nterest rate of BIC fro the dfferent ethods ethod Nuber of fuzzy relatons SE of tranng data SE of testng data Average forecastng errors (%) Wang s ethod Generalzed Wang s ethod The coparson of predcton and true values of nterest rate of BIC usng the Wang s ethod and the proposed ethod s shown n Fgure.

8 The Frst Internatonal Senar on Scence and Technology, Islac Unversty of Indonesa, 4-5 January 009. (a) (b) 6. Conclusons Fgure. Predcton and true values of nterest rate of BIC usng (a) Wang s ethod, (b) generalzed Wang s ethod In ths paper, we have presented a generalzaton of Wang s ethod to construct fuzzy te seres odel based on the tranng data. We appled the proposed ethod to forecast the nterest rate of BIC. The result s that forecastng nterest rate of BIC usng the proposed ethod has a hgher accuracy than that usng the Wang s ethod. It s portant to deterne the optal nuber of fuzzy logcal relatonshps to get effcent coputatons and to prove predcton accuracy. The precson of forecastng depends also to takng factors as nput varables. In the future works, we wll construct the optal nuber of fuzzy logcal relatonshps and select the portant varables to prove predcton accuracy. References [] Abad, A., Subanar, Wdodo & Saleh, S Forecastng nterest rate of Bank Indonesa Certfcate based on unvarate fuzzy te seres. Internatonal Conference on atheatcs and Its applcatons SEAS. Yogyakarta: Gadah ada Unversty, Indonesa. [] Abad, A., Subanar, Wdodo & Saleh, S.. 008a. Constructng coplete fuzzy rules of fuzzy odel usng sngular value decoposton. Proceedng of Internatonal Conference on atheatcs, Statstcs and Applcatons (ICSA). Banda Aceh: Syah Kuala Unversty, Indonesa. [3] Abad, A., Subanar, Wdodo & Saleh, S.. 008b. Desgnng fuzzy te seres odel and ts applcaton to forecastng nflaton rate. 7 Th World Congress n Probablty and Statstcs. Sngapore: Natonal Unversty of Sngapore. [4] Chen, S Forecastng enrollents based on fuzzy te seres. Fuzzy Sets and Systes. 8: [5] Chen, S Forecastng enrollents based on hgh-order fuzzy te seres. Cybernetcs and Systes Journal. 33: -6. [6] Chen, S.. & Hsu, C.C A new ethod to forecastng enrollents usng fuzzy te seres. Internatonal Journal of Appled Scences and Engneerng. (3):

9 The Frst Internatonal Senar on Scence and Technology, Islac Unversty of Indonesa, 4-5 January 009. [7] Huarng, K Heurstc odels of fuzzy te seres for forecastng. Fuzzy Sets and Systes. 3: [8] Hwang, J.R., Chen, S.. & Lee, C.H Handlng forecastng probles usng fuzzy te seres. Fuzzy Sets and Systes. 00: 7-8. [9] Sah,. & Degtarev, K.Y Forecastng enrollents odel based on frst-order fuzzy te seres. Transacton on Engneerng, Coputng and Technology VI. Enforatka VI: [0] Song, Q. & Chsso, B.S.. 993a. Forecastng enrollents wth fuzzy te seres, part I. Fuzzy Sets and Systes. 54: -9. [] Song, Q. & Chsso, B.S.. 993b. Fuzzy te seres and ts odels. Fuzzy Sets and Systes. 54: [] Song, Q. & Chsso, B.S Forecastng enrollents wth fuzzy te seres, part II. Fuzzy Sets and Systes. 6: -8. [3] Wang LX A course n fuzzy systes and control. Upper Saddle Rver: Prentce-Hall, Inc.