A Hybrid Forecast of Exchange Rate based on Discrete Grey-Markov and Grey Neural Network Model

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1 A Hybrd Forecast of Exchange Rate based on Dscrete Grey-Marov and Grey eural etwor Model Gol Km a, R Su Yun b ( a Center of atural Scence, Unversty of Scences, Pyongyang, DPR Korea, E-mal: golm4@yahoocom b Foregn Economc General Bureau, Pyongyang, DPR Korea Abstract: We propose a hybrd forecast model based on dscrete grey-fuzzy Marov and grey neural networ model and show that our hybrd model can mprove much more the performance of forecast than tradtonal grey-marov model and neural networ models Our smulaton results are shown that our hybrd forecast method wth the combnatonal weght based on optmal grey relaton degree method s better than the hybrd model wth combnatonal weght based mnmzaton of error-squared crteron Keywords: Fnancal forecastng, Hybrd forecast, Dscrete Grey-Fuzzy Marov,Grey eural etwor Introducton Investors have been tryng to fnd a way to predct exchange rates and stoc prce accurately, but haven t been obtaned the good resultsrecently, artfcal ntellgence technques le artfcal neural networs (A and genetc algorthms (GA, and wavelet transform have been appled to ths area However, the above-mentoned concern method showed that A had some lmtatons n learnng the patterns because foregn exchange rates data has tremendous nose and complex dmensonalty Moreover, the sheer quantty of exchange rates data sometmes nterferes wth the learnng of patterns Hybrd forecast s a well-establshed and well-tested approach for mprovng the forecastng accuracy Therefore, the mportance of hybrd forecast methods has steadly ncreased and t acts stll on tme seres forecastng Hybrd forecast system usng rough sets and artfcal neural networs has been proposed by MDurara, K Meena (0 and hybrd forecast method by ARIMA and neural networ model Has been researched by Reza Asar Moghadam, Mehrnaz Keshmrpour (0 Aobng Sun, Yubo Tan, Dexan Zhang (008 have researched the hybrd forecast model based on BP neural networ Hybrd Forecast Model of ARIMA and eural etwor has been gven by Šterba Ján, Hl ovs Katarína (00 and staced heterogeneous neural networs for tme seres forecastng has been suggested by Florn Leon, Mha Hora Zahara(00 Tare Aboueldahab, Mahumod Fahreldn (0 have studed the forecast of stoc maret ndces usng hybrd genetc algorthm/ partcle swarm optmzaton wth perturbaton term Improvement method n forecastng accuracy usng the hybrd model of ARFIMA and feed forward neural networ has been proposed by Cagdas Haan Aladag, Erol Egroglu, Cem Kadlar (0 Forecast of future stoc close prce usng based on hybrd A Model of functonal ln fuzzy logc neural model has been researched by Kumaran Kumar J, Kalas (0 Forecastng of foregn exchange rate by grey- chaotc forecast hybrd has been researched by Km Gol (008 and forecastng of stoc prce and ndex of ol by grey- fractal forecast hybrd model has been studed by Km Gol (009 Zhu Jang-lang (006 has studed the forecastng of the electrc power load based on the grey theory and BP neural networ and Su Bo (006 has consdered the comparson and research of gan producton forecastng wth methods of GM (, grey system and BP Method of the wavelet neural networ n the forecast of stoc maret has been suggested by,yao Hong-Xng, Shong Zhao-han, Chen Hong-xang (00 and Genetc algorthms approach to feature

2 dscretzaton n artfcal neural networs for the forecast of stoc prce has been researched by Kyong-ac Km, Ingoo Han (000 Ths paper deals wth the development of an mproved forecast model usng grey-marov chan models and the grey-neural networ and ts applcaton to fnancal tme seres forecastng Our method ncludes dscrete grey- fuzzy weght Marov chan model, grey-neural networ model, The hybrd model wth combnatonal weght based mnmzaton of error-squared crteron, hybrd model wth combnatonal weght based on optmal grey relaton degree method We demonstrate the advantage of our method compared wth the tradtonal grey-marov or neural networ methods through forecastng smulatons Dscrete grey- Fuzzy weght Marov model Grey forecastng model Grey forecastng model (GM has three basc operaton such as ; (і accumulated generaton, (ⅱ nverse accumulated generaton, and (ⅲ grey modelng The grey forecastng model uses the operatons of accumulated generaton to buld dfferental equatons Intrnscally speang, t has the characterstcs of requrng less data The GM (, grey model, e, a - varable frst-order grey model, s summarzed as follows Step : Selecton of the orgnal sequence; x = ( x (, x (, x (,, x ( (5 where x ( s the tme seres data at tme Step : Accumulated generatng; Based on the tme sequence x, a new sequence operaton (AGO, where ( ( ( ( ( x = ( x (, x (, x (,, x ( and s = ( x s gven by the accumulated generatng x ( ( derved as follows: ( x = x ( (6 Step 3: Mang the dfferental equaton; The frst-order dfferental equaton holds true: ( dx ( + ax = u (7 dt Step 4: Obtanng the forecast model; We have by (7 ( u a u xˆ ( + = ( x ( e + (8 a a ( ( xɵ ( + = xɵ ( + xɵ (9 where [, ] ( T T T a u = B B B y ( xˆ ( ( ( 05( x ( + x ( ( ( 05( x ( + x (3 B= ( ( ( 05( x ( n + x ( y = ( x (, x (3, x (,, x ( T ( + s the predcted value of x ( ( + at tme +, The authors used the posteror test to evaluate the accuracy of the grey forecastng The forecastng error wear defned as q = x xˆ, =,,,, the mean and the

3 standard devaton of the forecastng error s q and S The mean and the standard devaton of ntal tme seres are x = m=, S = ( x m / = The posteror rato C s derved by dvdng S by S The lower C s, the better the model s The posteror rato can ndcate the change rate the forecastng error Probablty of small s defned as P= prob{ q q S }, where =,3,, P Ths shows the probablty that the relatve bas of the forecastng error s lower than P s commonly requred to be larger than 095 The pars of the forecastng ndcators P and C can characterze four grades of forecastng accuracy, as shown Table It s called that above-mentoned grey forecast model s tradtonal grey model on homogenous dscrete grey model Let x = ( x (, x (, x (,, x ( be an orgnal sequence and = ( (, (, (,, ( be accumulated generatng sequence by AGO ( ( ( ( ( x x x x x Where x ( ( s gven such as: = ( x x ( = ([] Then, we defne non-homogenous dscrete grey model (DGM such as ; ( ( ( xˆ ( + = βxˆ + βxˆ + β3+ β 4 ( xˆ ( = ξ ( where ˆx ( fttng value of orgnal s sequence and β, β, β 3, β 4 are parameters of system By least square method, the parameters of system β= [ β, β, β3, β 4] are gven by β = [ β, β, β 3, β 4 ] = ( B B T ( ( x ( x ( x ( ( ( Here, = x ( x ( B x (3 Y = ( ( x ( n x ( n n x ( n The algorthm of non-homogenous dscrete grey model (DGM s gven such as; Step ; Fnd the parameters of systemβ= [ β, β, β3, β 4] Step ; Put ˆ ( x ( = ξ and fnd the smulaton values xˆ (, =,3,, n ˆ x ( 0 s the functon ofξ Step 3; CalculateQ= Table The Grades of Forecastng Accuracy Forecastng ndex Grade P C Good >095 <035 Qualfed >08 <05 Just >07 <065 Unqualfed n = B T Y [ xˆ x ] Q s the functon ofξ 3

4 dq Step 4; Put = 0 and calculate the value ξ whch Q have to tae mnmum dξ ˆ Step 5; Calculate the value x ( correspondng to ξ obtaned from step 4 Step 6; calculate the forecast value ˆ ( ( operator (IAGO That s, xˆ = xˆ ( + xˆ x ( 0, =,3,, n by nverse accumulated generatng 3 Fuzzy weght Marov model We assume that Xˆ t s the fttng curve whch s obtaned through forecastng for the tme seres Y t by non homogenous dscrete grey model By consderng the actual meanng of the orgnal sequence, we have generated a new tme seres Z ˆ t = ( Yt X t / Yt ( t=,3,, whch the fttng curve Xˆ t taes as reference Here Yt s one step of tme delay value In accordance wth the dstrbuton of the random sequence Z t, t s dvded by peces of state and s taen ts partton value That s, e = [ m0, m ], e = [ m, m ],,, e = [ m, m ], e = [ m, m ] Here m m A one-step transton probablty P s assocated wth each possble transton from stat e to stat e, and P can be estmated usng P = M / M (, =,,, m M means the numbers of dvded peces whose resduals are stat e, and M s number of transton from e to e that have occurred by passed one step These P values can be presented as a transton matrx R Then, we accept the statstcal quantty such as; P P0 = M / M, χ = M log P = = = = Then, when s comparatvely large, the statstcal quantty χ s accordng to χ -dstrbuton wth the degree of freedom ( When gvng the confdence degree α, f χ > χ α ( m, then we confrm that the random sequence Zt have Marov s property, unless, the sequence haven t Marov s property Suppose U s the range whch random varable of Marov chan taes the value We construct the fuzzy state set S, S,, Sl If for arbtrary u U the condton l m= µ ( u = S m s satsfed, and then µ S ( u s called the membershp degree of the fuzzy state S m M for numercal valueu Suppose that µ S ( Z ( t µ S Z t+ s the fuzzy state transton coeffcent from the state S to S when tme s turned from t tot+ Then, a = µ ( Z µ ( Z S t S t+ t= s called fuzzy transton frequency number from the state S to the state S When the state Z t belongs to S wth degree of member µ S ( Z t and belongs to S wth membershp degree µ ( Z, the transton order s only expressed by the product between S membershp degreesµ S ( Zt µ S ( Zt+ t 4 0

5 Owng to the fuzzy transton probablty from the state S to the state S s denoted by a P =, ( (, =,,, M a =, the tme seres tme seres whch we are gong to establsh s gven such as; Yˆ ˆ t = X t + S ( Zt = = µ ( m + m PYt, t =,,, Here, Xˆ t s the predcted value obtaned by usng the dscrete grey model The forecastng model obtaned by above methods s called the dscrete grey -fuzzy weght Marov model We denote ths model by DGM-FMarov 3 The forecast of JPY/USD exchange rate based on the dscrete grey -fuzzy weght Marov model We tae JPY/USD exchange rate of total 78 barter perod from September 994 to July 997 The data of ths paper s taen from G Shnarst, P Wamso (997 The concrete data s omtted We assume that ŷ s the forecasted value of the orgnal data obtaned by DGM model n tme The curve y ˆ = xˆ + deflects the total change tendency of the orgnal data Based on the real phase of the sample data and consderng the real meanng, let dvde the state 6 th tem, that s, = [, ] = [ y + a x, y + b x ] ( =,,,6 where a = ( a, a,, am = ( 009, 005, 00, 0, 00, 005, b = ( b, b,, bm = ( 005, 00, 0, 00, 005, 009 :[ y 009 x, y 005 x ], :[ y 005 x, y 00 x ] 3 :[ y 00 x, y ], 4 :[ y, y + 00 x ] :[ y + 00 x, y x ], :[ y x, y x ] where 5 6 ŷ s the predct value of the foregn exchange rate obtaned by DGM model n tme The state shows the degree of devaton of the ntal data devated from the curve ŷ Where denote that the dsadvantage s putted wthn ~9% range and dsadvantage s putted wthn ~5% range And 3 and 4 denote that the proft and dsadvantage are putted wthn % range and 5 denote that the proft s putted wthn ~5% range and 6 denote that the proft s putted wthn 5% range The states are,, 3, 4, 5, 6 The numbers of sample pont lay at each state are respectvely n = 73, n = 4, n3 = 9, n4 = 9, n5 = 44, n6 = 7 The -step state matrx composed by each state s gven by ( n 6 6 =

6 The last state transton of the ntal sample s uncertan Therefore, when calculatng the state transton matrx, last data have to exclude From the formula (5, we can obtan M 0 [066, 05, 0043, 00683, 0583, 0554] T = From the formula (6, we can obtan the state transton matrx by ( M 6 6 = Calculatng the statstcal quantty, we have m m M χ = n log 8089 M = = 0 Tang the degree of confdence α = 00 and nspectng the tables, then we have χ00(5 = 443, χ χ00(5 Therefore, the -step transton matrx ( M 6 6 then we confrm that the sequence s concdence to Marov s property, unless, the sequence s not concdence to Marov s property Then, based on ths transton matrx, we can predct the foregn exchange rates whch would be putted on the next -perod or -perod Therefore, the forecast value of the exchange rates of the 79 th perod s gven by ɵ ɵ ɵ ɵ x ɵ ɵ 79 = [ y78+ (00x x78 p65+ (005x x78 p66] = 749 From the step-by-step formula the forecast value of the stoc prce of the perod whch would be putted on afterward the state 79 th perod s gven by ɵ ɵ x+ = y + x ˆ [05 p66 ( p65 ( + 00 p63( p6 ( + 05 p6( + ], 79 Accuracy was the most mportant crteron, followed by the cost savngs generated from mproved decsons In partcular, executon ssues such as ease of nterpretaton and ease of use were also hghly rated In ths study, there are three crtera used to evaluate forecastng models The frst evaluate crtera s mean square error, MSE: MSE= ( xˆ x / (7 = where xˆ ( s the predcted value at tme, x ( s the actual value at tme, and s the number of forecasts The second crteron s mean absolute error, MAE: (8 MAE = e / xˆ x / = = The thrd crteron s mean absolute percent error, MAPE: ˆ MAPE= q 00(% 00(% The fourth crteron s Thel Coeffcent, µ : x x = = x (9 6

7 µ x xˆ [ ( ( ( ]/ = xˆ [ ( ( ]/ If x = xˆ (, e, forecast value s concdng wth actual value perfectly, we haveµ = 0 Therefore, f µ s approached to 0, then forecast method become fner (0 4 Grey eural etwor The Grey GM (, model was establshed forecast model by usng new data accumulated generator The accumulated generator data can mae wea the randomness a certan degree and can easly fnd the data change rule The grey GM (, model has the advantage whch demands the small sample data The neural networ has the capacty of self-learnng, nonlnear mappng, and parallel dstrbuton processng, etc These tow forecast method have already used n foregn exchange rate predctng The system of foregn exchange rate forecast s a complcated system whch have the great randomness and there are many ndex nfluenced n that system Combnng grey system dea and neural networ, we can construct the grey neural networ (G and suffcently demonstrate the advantage of the model by usng the modelng method of grey system wth the small data and the characters of nonlnear mappng of neural networ Thus, we can more enhance forecast accuracy Here we dscuss the method of combng the GM model and neural networ model There are three nds of forecastng model structure That s, Parallel grey neural networ (PG, seres grey neural networ (SG, and nlad grey neural networ (IG 4 Parallel grey neural networ (PG PG uses grey model and neural networ to predct separately, then uses neural networ to combne the predctng results Fg shows the prncple scheme of PG Intal data GM(, model eural networ model Hybrd Forecast model Forecast results Fg Parallel grey neural networ model PG model s essental s the hybrd model The am of mang the PG model s to decrees the randomness of data and avods data lac used by sngle model, and enhances the forecast accuracy degree by usng data offered from varous nds methods synthetcally On the ground of hybrd forecast prncple, there are varous hybrd methods that s, arthmetc mean hybrd method, geometrc mean hybrd method, harmonc mean hybrd method, etc Those hybrd formulas are gven by xɵ = wɵ ɵ x + w x, =,,, ɵ ɵ ɵ ( ( w w x = x + x, =,,, ɵ x =, =,,, w / xɵ ɵ + w / x 7

8 Where s total forecast data number, x ɵ ( and x ɵ are the forecast value by GM(, model and neural networ model, respectvely, and w and w are weghts of two nd of forecast method x ɵ s actual value ow, we consder the concepton of effectve degree The concepton of effectve degree has some ratonalty because t reflects the effcent of the methods Its dea s as follow; We put x xˆ x ( wɵ ɵ x + w x A = = x x Then, A s the accuracy seres of hybrd forecast We denote the mean value E and mean square devaton of A( respectvely by / E= A, σ = ( ( A E = = We defne effectve degree of hybrd forecast method by S = E( σ If S s more great, then forecast accuracy s more rased and forecast error s more safety It says that the model s more effectve To fnd the weghts w, w, we can use optmzaton method by nf S( w, w S = E( σ { w, w } But, ths method s very complcated ow, we would use a smple method by usng of physcal meanng Let A and A are the seres of forecast accuracy by grey GM (, model method and neural networ model method respectvely, that s, x xˆ A =, =,; =,,, x Let S are the effectve degree by GM (, Model and neural networ model respectvely Then, we can defned weghts w and w by S w = ( =, S = 4 Seres grey neural networ (SG Seres grey neural networ employ grey model to predct, then use neural networ to combne the predctng results Establshed forecast model GM(, wth each other data for already gven the same seres, then the ganed forecast results are dfferent each other To obtan forecast results approached to actual value, for varous grey forecast results we can combne ts results by usng neural networ Ths s mmedately SG model Fg shows the prncple scheme of SG The nput nerve element number of neural networ s the number of each other GM model and the number of output nerve element number of neural networ s sngle element The hdden nerve element number s confrmed by test methods The learnng of neural networ s progressed by error bac-propagaton method PG and SG are all essental hybrd forecast models Theoretcally, we can prove that the forecast results of hybrd model are advantage than sngle GM model or neural networ model 8

9 In SG model, we fnd the combned weghts of several nds grey model by usng the nonlnear matchng ablty of neural networ The combnaton of SG s nonlnear, the other sde, the combnaton of PG s lnear GM model Intal data GM model eural etwor Model Forecast results GM model m Fg Seres grey neural networ model 43 Inlad grey neural networ (IG IG model s bult by addng a grey layer before neural nput layer and a whte layer after neural output layer Fg 3 shows the prncple scheme of IG Intal data Grey layer eural networ whte layer Forecast results The acton of grey layer s to weaen the randomness of ntal data Generally, the role of grey layer s to mae tranng sample of neural networ wth new data ganed from -AGO for ntal data In that case, the approxmaton n nonlnear exctaton functon of neural networ s easy and studyng tme of networ s very short, and convergence process s quc and forecast accuracy s rased because accumulated operator data have monotone ncreasng tendency 5 The hybrd forecast method 5 The determnaton of combnatonal weght based mnmzaton of error-squared crteron In the actual problem, we can predct for forecastng problem by usng the varous forecast methods Generally speang, the forecast result forecasted by the dstnctve forecast method s dfferent each other To obtan the more mproved forecast result, t s necessary to apply the varous nds of hybrd forecast methods Let x ( =,,, are an actual observaton value and xˆ ( =,,, p are the predct value by the ' th method Let Is the forecast error From ths notaton, we have w s the weght of the ' th method and p x = w ˆ x + e = = ( ( ( ( (,,, ow, let wˆ s the calculated value of the weght and can obtan p = = xɵ w ɵ x 9 e x xˆ ( = ( ( xˆ s the hybrd forecast value Then we To fnd the optmal weght w, we can use varous optmzaton methods For example, fnd optmal weght * Fg3 Inlad grey neural networ model * w for optmzaton problem;

10 p * (( ˆ = = w = = J[ w ] mn J[ w ] ( x w x p = w =, w 0, =,,, p To obtan the soluton of optmzaton problem (, we can apply varous optmzaton methods, for example, quadratc programmng, genetc algorthm, neural networ method, etc For nstance, xˆ and xˆ are the predct value by two methods for the same phenomena Then, ɵ ɵ ɵ ɵ ɵ x = w x + w x = ρ x + ( ρ x s called the hybrd forecast model for two models xˆ and xˆ ( Where 0 ρ ow, we put e = e, e = e, from the hybrd model, we have e= ρe + ( ρ e The square devaton of e s gven by D( e = ρ D( e + ( ρ D( e + ρ( ρcov( e, e dd( e Where cov( e, e s covarance of e and e To obtan least value of D( e, we put 0 dρ = Then, we can obtan the least value of D( e by * D( e cov( e, e ρ = D( e = D( e + D( e cov( e, e * That s, whenρ = ρ, D( e s acheved to the least value In the actual calculaton, we can approvable that e and e are the mutual ndependence random quantty Therefore, we have cov( e, e = 0 Accordngly, we have * D( e ρ = ( D( e = D( e + D( e The fnal forecast model s gven by ɵ * ɵ * ɵ x = ρ x + ( ρ x (3 5 The determnaton of combnatonal weght based on optmal grey relaton degree method Let { y( =,,, } be orgnal value of tme seres, yˆ ( =,,, m, =,,, be predcted value of the th forecast method n tme ( =,,, m, t=,,, We put γ 0 = mn mn e ( t + ρ max max e m t m t t= e ( t + ρ max max e ( t m t Then, γ s called the grey relaton degree of the predcted value sequence 0 Here ρ (0, s called the dentfcaton coeffcent Generally we tae ρ =05 e ( t = y( t yˆ ( t s the predcted error of tme for th forecast method Let ŷ(t = Here w, m = m =, ( t w yˆ ( t be the predcted value of y(t by the hybrd forecast method w, w s the weght coeffcent of m nd of forecast method, whch satsfes m w =, w 0 ( =,,, m The combnatonal weght based on the optmal grey relaton degree s determned such as; ( 0

11 maxγ ( W = m t t= mn mn e ( t + ρ max max e m = m t w e ( t + ρ max max e m t m w = s t = w 0, =,,, m By solvng ths optmal problem, we can determne the combnatonal weght T W= {w, w,, w based on the optmal grey relaton degree m} 6 The hybrd forecast of exchange rates by DGM-FMarov model and Grey - eural etwor ow, let xˆ s the predct value obtaned by DGM-FMarov model and xˆ ( s the forecasted value obtaned by nlad grey neural networ model We have to obtan the calculatng results by DGM-FMarov model and grey neural networ model for exchange rates The structure of neural networ s taen 4 4 BP networ The transfer functon of nput layer and hdden layer are sgmod type and transfer functon of output s lnear functon In G forecast, we have appled IG mode We have progressed -AGO smoothng handlng by addng a grey layer before nput layer of BP neural networ and have progressed -IAGO reducton handlng for whte layer after neural output layer We have calculated accordng to formula (, (3 by unbased grey-marov- grey neural networ hybrd forecast value for foregn exchange rates The mean absolute percent error MAPE of DGM-FMarov model s gven by xˆ x MAPE = 00% 365(% x = Thel coeffcent of DGM-FMarov model s gven by µ= The mean absolute percent error MAPE by hybrd model of DGM-FMarov model - grey neural networ on the bass of the determnaton of combnatonal weght based mnmzaton of errorsquared crteron s gven by xˆ x MAPE = 00% 7(% x = Then, Thel coeffcent s gven by µ = The mean absolute percent error MAPE by hybrd model of DGM-FMarov model - grey neural networ on the bass of the determnaton of combnatonal weght based on optmal grey relaton degree method s gven by xˆ x MAPE = 00% 85(% x = Then, Thel coeffcent s gven by µ = 0 00 ( t ( t

12 7 Concluson As nown from the hybrd forecast model, ts model can use the nformaton of the orgnal data suffcently and absorb the characters and the advantages of two models, and avod the lmtaton of sngle model The forecast accuracy of hybrd model by DGM-FMarov model - grey neural networ s more hgher than the DGM-FMarov model Especally, The forecast accuracy of hybrd model by DGM-FMarov - grey neural networ model on the bass of the determnaton of combnatonal weght based on optmal grey relaton degree method s more hgher than the DGM-FMarov-grey neural networ model on the bass of the determnaton of combnatonal weght based mnmzaton of error-squared crteron It s shown that ths model has hgh forecast accuracy for forecast of exchange rates wth tendency and fluctuaton In ths paper we have researched an hybrd forecast model by usng DGM-FMarov model and grey neural networ forecast model and have exhbted that ths method s effcent to the exchange rates forecast In the hybrd forecast model, we can use some forecast models, fore example; not only forecast not only DGM-FMarov forecast, grey-regresson forecast, but also grey- chaotc forecast and greyfractal forecast ([3, 4], ect References [] M Durara, K Meena (0, A Hybrd Forecast System Usng Rough Sets and Artfcal eural etwors, Internatonal Journal of Innovatve Technology & Creatve Engneerng,(7 6-3 [] Reza Asar Moghadam, Mehrnaz Keshmrpour (0, Hybrd ARIMA and eural etwor Model for Measurement Estmaton n Energy-Effcent Wreless Sensor etwors, Communcatons n Computer and Informaton Scence, 53, [3] Aobng Sun, Yubo Tan, Dexan Zhang(008, Hybrd Forecast Model Based on BP eural etwor for Lung Cancer, Proceedngs of 008 IEEE Internatonal Symposum on IT n Medcne and Educaton, [4] Cagdas Haan Aladag, Erol Egroglu, Cem Kadlar(0, Improvement n Forecastng Accuracy Usng the Hybrd Model of ARFIMA and Feed Forward eural etwor, Amercan Journal of Intellgent Systems, (: -7 [5] Kumaran Kumar J, Kalas (0 A Forecast of Future Stoc Close Prce usng Proposed Hybrd A Model of Functonal Ln Fuzzy Logc eural Model, IAES Internatonal Journal of Artfcal Intellgence,(, 5~30 [6] Tare Aboueldahab, Mahumod Fahreldn(0, Forecast of Stoc Maret Indces usng Hybrd Genetc Algorthm/ Partcle Swarm Optmzaton wth Perturbaton Term, ICSI 0: Internatonal Conference on swarm ntellgence Cergy, France, -0 [7] Florn Leon, Mha Hora Zahara (00, Staced Heterogeneous eural etwors for Tme Seres Forecastng, Mathematcal Problems n Engneerng, 0, -9 [8] Šterba Ján, Hl ovs Katarína (00, The Implementaton of Hybrd ARIMA-eural etwor Forecast Model for Aggregate Water Consumpton Forecast, Aplmat Journal of Appled Mathematcs, 3(3, 3-30 [9] Zhu Jang-lang (006, The forecastng of the electrc power load based on the grey theory and BP neural networ, Electrc Machnes and Control, 0(4, [0] Su Bo (006, Comparson and research of gan producton forecastng wth methods of GM (, grey system and BP, Journal of Chna Agrcultural Unversty, (4, 99-04, [] Yao Hong-Xng, Shong Zhao-han, Chen Hong-xang(00, Method of the wavelet neural networ n the forecast of stoc maret, Systems Engneerng-Theory & Practce, 6, 35-38,

13 [] Kyong-ac Km, Ingoo Han(000, Genetc algorthms approach to feature dscretzaton n artfcal neural networs for the forecast of stoc prce, Expert Systems wth Applcatons, 9, 5-3, [3] Km Gol (008, Forecastng of foregn exchange rate by grey- chaotc forecast hybrd model, Journal of Unversty of Scence, 4, [4] Km Gol (009, Research of forecastng of stoc prce and ndex of ol by grey- fractal forecast hybrd model, Informaton Scence,, 8-35 [5] G Shnarst, P Wamso (997, Exceptng effect of sample n admsson fluctuaton foregn exchange rate model, Internatonal fnance papers, 30, 6-3