THE FUZZY GROUP METHOD OF DATA HANDLING WITH FUZZY INPUTS. Yuriy Zaychenko

Transcription

1 206 5 Knowledge Dalogue - Soluton THE FUZZY GROUP ETHOD OF DATA HANDLING WITH FUZZY INPUTS Yury Zaycheno Abstract: The problem of forecastng models constructng usng expermental data n terms of fuzzness, when nput varables are not nown exactly and determned as ntervals of uncertanty s consdered n ths paper. The fuzzy group method of data handlng s proposed to solve ths problem. The mathematcal model of the problem mentoned above s bult and fuzzy GDH wth fuzzy nputs s elaborated n the paper. The correspondng program whch uses the suggested algorthm was developed. The expermental nvestgatons and comparson of FGDH wth neural nets n problems of stoc prces forecastng were carred out and presented n ths paper. Keywords Group ethod of Data Handlng, fuzzy, economc ndexes, stoc prces, forecastng AC Classfcaton Keywords: H.4.2 [Informaton Systems Applcatons] Types of Systems - Decson support Conference: The paper s selected from XV th Internatonal Conference Knowledge-Dalogue-Soluton KDS , Kyv, Urane, October, Introducton The problem of forecast n fnancal sphere s of great mportance. Fnancal processes have some specfc features whch prevent the applcaton classcal statstc methods and stpulate the development of novel approaches and methods based on artfcal ntellgence. One of such novel methods s fuzzy group method of data handlng (FGDH) developed n [Зайченко, 2003, Zaycheno, 2006]. As s well nown, fuzzy GDH allows to construct fuzzy models and has the followng advantages:. The problem of optmal model fndng s transformed to the problem of lnear programmng, whch s always solvable; 2. There s nterval regresson model bult as the result of method wor out. Fuzzy GDH n ts orgnal form has one lmtaton: the nput varables must be crsp, non-fuzzy whle the model constructed s fuzzy. At the same tme stuatons n fnancal sphere often occur when the nput data are uncertan, gven n the form of uncertanty ntervals. The goal of ths paper s the development and nvestgaton of extended verson of FGDH worng n fuzzy envronment when nput varables are fuzzy and ts comparson wth fuzzy neural networs. athematcal odel of Group ethod of data Handlng wth Fuzzy Inputs The ntal fuzzy GDH consders a lnear nterval regresson model: Y=A 0 Z 0 +A Z + +A n Z n, (.) - where A are fuzzy numbers, whch are descrbed by threes of parameters A = ( A, A, A ), where A nterval center, A upper border of the nterval, A - lower border of the nterval, - and Z are nput varables whch are also fuzzy numbers determned by parameters ( Z, Z, Z) border, Z - center, Z - upper border of fuzzy number., Z - lower

2 Internatonal Boo Seres "Informaton Scence and Computng" 207 Then Y s output fuzzy number, whch parameters are defned as follows (n accordance wth L-R numbers multplyng formulas): Center of nterval: y = A * Z. Devaton n the left part of the membershp functon: y y = ( A *( Z Z ) + ( A A )* Z ). Thus the upper border of the nterval y = ( A *( Z Z ) + Z *( A A ) + A * Z ). For the nterval model to be correct, the real value of nput varable Y should lay n the nterval got by the method worflow. So, the general requrements to estmaton lnear nterval model are followng: to fnd such values of parameters ( A, A, A) of fuzzy coeffcents, whch enable: a) Observed values y lay n estmaton nterval for Y ; b) Total wdth of estmaton nterval be mnmal. Input data for ths tas s Z =[ Z ] - nput tranng sample, and also y nown output values, =,, s a sample sze ( number of observaton ponts). There are two cases of fuzzy values membershp functon nvestgated n ths wor: - Trangular membershp functons - Gaussan membershp functons. FGDH wth Fuzzy Inputs for Trangular embershp Functons Let s consder the specal case of the lnear nterval regresson model: Y = A 0 Z 0 +A Z + +A n Z n, (.2) where A fuzzy number of trangular shape, whch s descrbed by threes of parameters A = ( A, a, A ), where a center of the nterval, A ts upper border, A - ts lower border. Current tas contans the case of symmetrcal membershp functon for parameters A, so they can be descrbed va par of parameters ( a, c ). A = a c, A = a + c, c nterval wdth, c 0, Z also fuzzy numbers of trangular shape, whch are defned by parameters ( Z, Z, Z) center, Z - upper border of fuzzy number. Then Y s a fuzzy number, whose parameters are defned as follows: Center of the nterval: y = a * Z., Z - lower border, Z -

3 208 5 Knowledge Dalogue - Soluton Devaton n the left part of the membershp functon: y y = ( a *( Z Z ) + c Z ). The lower border of the nterval: The upper border of the nterval: y = ( a * Z c Z ). y = ( a * Z + c Z ). For the nterval model to be correct, the real value of nput varable Y should lay n the nterval got by the method It can be descrbed n such a way: ( a * Z c Z ) y, ( a * Z + c Z ) y, =,. Where Z =[ Z ] s nput tranng sample, y nown output values, =,, number of observaton ponts. So, the general requrements to estmaton lnear nterval model are to fnd such values of parameters ( a, c ) of fuzzy coeffcents, whch enable: a) Observed values y lay n estmaton nterval for Y ; b) Total wdth of estmaton nterval s mnmal. These requrements can be redefned as the followng tas of lnear programmng: under constrants: mn ( ( a * Z + c Z ) ( a * Z c Z )) a, c = (.3) ( a * Z c Z ) y, (.4) ( a * Z + c Z ) y, =,. Formalzed problem formulaton n case of trangular membershp functons Let s consder partal descrpton f( x, x ) = A + Ax + Ax + Axx + Ax + Ax. (.5) 2 2 j 0 2 j 3 j 4 5 j Rewrtng t n accordance wth the model (.) needs such substtuton z 0 =, z = x, z = x, z3 = xx j, 2 j z = x, 2 4 z = x. 2 5 j Then math model (.3)-(.4) wll tae the form

4 Internatonal Boo Seres "Informaton Scence and Computng" 209 mn (2 c + a ( x x ) + 2 c x + a ( x x ) + 2c x + a, c = = = = + a ( x ( x x ) + x ( x x )) + 2c x x + 2 a x ( x x ) = = = + 2c x + 2 a x ( x x ) + 2c x ) = = = wth the followng condtons: a0+ ax + a2x + a3( x ( x x ) x ( x x ) + x x ) a4( 2 x ( x x ) + x ) + a5(2 x ( x x ) + x ) c0 c x 2 2 c x c x x c x c x y, a0+ ax + a2x + a3( x ( x x ) + x ( x x ) x x ) + a4(2 x ( x 2 2 x ) x ) + a5(2 x ( x x ) x ) + c0+ c x + c2 x + c3 x x + cx + cx y (.6) (.7) cl 0, l = 0,5. (.8) As we can see, ths s the lnear programmng problem, but there are stll no lmtatons for non-negatvty of varables δ +. a, so we may pass to a dual problem, ntroducng dual varables { δ } and { } Wrte down dual problem: Under constrants: max( y δ y δ ). + = = (.9) δ δ = 0. + = = x δ x δ = ( x x ), (2.0) + = = = x δ x δ = ( x x ). + = = = And several constrants nequaltes correspondng to varables C are not presented here for brevty. It was proved that the dual LP problem s always solvable as well as the prmary LP problem (.6)-(.8). Expermental Investgatons of FGDH and Comparson wth Fuzzy Neural Networs The numerous expermental nvestgatons of the suggested method-fgdh wth fuzzy nputs were carred out at the problem of forecastng stoc prces of shares at Russan stoc maret RTS and comparson of effcency of FGDH and fuzzy neural networs. Some of the obtaned results are presented below. Experment. LUKOIL stoc prce forecastng.. Stoc prce forecastng usng fuzzy group method of data handlng wth trangular F and Gaussan F: а) Forecastng based on prevous data about stoc prces at the perod: 0.2./

5 20 5 Knowledge Dalogue - Soluton The followng results were obtaned whch are presented at the table and Fg. Table. Experment results usng FGDH wth fuzzy data and trangular F Date Real Values Lower Border Center Upper Border Devaton SE = Left border Center Rght border Real values Fg.. Experment 3 results usng FGDH wth fuzzy nput data and trangular F Experment 2. LUKOIL stoc prce forecastng based on prevous data about stoc prces of leadng Russan energetc companes for the same perod: EESR shares of РАО ЕЭС России jont-stoc company, YUKO shares of ЮКОС jont-stoc company, SNGSP prvleged shares of Сургутнефтегаз jont-stoc company, SNGS common shares of Сургутнефтегаз jont-stoc company. The followng results were obtaned: SE =

6 Internatonal Boo Seres "Informaton Scence and Computng" Left border Center Rght border Real values Fg. 2. Experment 2 results usng FGDH wth fuzzy data and Gaussan F Experment 3. Stoc prce forecastng usng fuzzy neural nets wth amdan and Tsuamoto algorthm. 267 everyday ndexes of stoc prces durng perod from to were used for neural net tranng. The followng results were obtaned: Table 2. Experment 3 results usng FNN amdan wth amdan wth Tsuamoto wth Tsuamoto wth Date Real Value Gaussan F Trangular F Gaussan F Trangular F SE As experment 3 results show, forecastng usng FNN wth amdan controller wth Gaussan F was the best place, Tsuamoto controller wth Gaussan F taes the second place. Forecastng usng FGDH wth fuzzy nput data gves best results when usng Gaussan F.

7 22 5 Knowledge Dalogue - Soluton Table 3. Summarzng forecastng results for FGDH and FNN wth Gaussan F: FGDH Wth fuzzy nputs FGDH wth fuzzy nputs amdan controller Tsuamoto Controller (4 nput varables) (prevous values on nput) SD For trangular F: Table 4. Summarzng forecastng results for FGDH and FNN wth trangular F FGDH wth fuzzy nputs FGDH wth fuzzy nputs amdan controller Tsuamoto Controller (4 nput varables) (prevous values on nput) SD As we can see from tables 2 and 3, the best SD are gven by FGDH wth fuzzy nputs, and ths method also allows to buld nterval estmaton of forecasted value. Gaussan F s more accurate than trangular F n FGDH wth fuzzy nputs, and also n FNN. Experment 4. RTS ndex forecastng (openng prce) Forecastng of RTS ndex (openng prce) usng fuzzy neural nets wth amdan and Tsuamoto algorthm. 267 everyday ndexes of stoc prces durng perod from to were used for neural net tranng. The followng fnal results were obtaned: whch are presented at the table 5 ( for Gaussan F) and table 6 ( trangular F) As experment 4 results show, forecastng usng FNN wth amdan controller wth Gaussan F was the best, amdan controller wth trangular F s on the second place. Forecastng usng FGDH wth fuzzy nput data gves best results when usng Gaussan F and usng 5 nput varables: Table 5. Forecastng results of experment 4 for FGDH and FNN wth Gaussan F amdan Controller Tsuamoto Controller FGDH wth fuzzy nputs (5 nput varables ) FGDH wth fuzzy nputs (prevous values on the nput) SD МАРЕ. % Table 6. Forecastng results of experment 4 for FGDH and FNN wth trangular F amdan Controller Tsuamoto Controller FGDH wth fuzzy nputs FGDH wth fuzzy nputs (5 nput varables ) (prevous values of the nput varable) SD МАРЕ. % Experment 5. RTS 2 ndex forecastng (openng prce) Table 7. Forecastng results of experment 5 for FGDH and FNN wth Gaussan F FGDH wth fuzzy nputs FGDH wth fuzzy nputs amdan controller Tsuamoto Controller (4 nput varables) (prevous nput values) SE

8 Internatonal Boo Seres "Informaton Scence and Computng" 23 Table 8. Forecastng results of experment 5 for FGDH and FNN wth trangular F FGDH wth fuzzy nputs FGDH wth fuzzy nputs amdan controller Tsuamoto Controller (4 nput varables) (prevous values on nput) SE As current experment results show, forecastng usng FGDH wth fuzzy nput data usng Gaussan membershp functon was the best, fuzzy amdan controller wth trangular and Gaussan F taes the second place. Best SD are gven by FGDH wth fuzzy nputs, and ths method also allows to buld nterval estmaton of forecasted value. Gaussan F s more accurate than trangular F n FGDH wth fuzzy nputs, and also n FNN. Concluson In ths paper new method of nductve modelng FGDH wth fuzzy nputs s descrbed. Ths method represents the development of fuzzy GDH when nformaton s fuzzy and gven n the form of uncertanty ntervals. The mathematcal model was constructed and correspondng algorthm was elaborated. The expermental results of applcaton of the suggested method n the forecastng of maret ndex and stoc prces and ther comparson wth FNN are presented and dscussed. The man advantages of the suggested method are followng: - It operates wth fuzzy and uncertan nput nformaton and constructs the fuzzy model; - The constructed model has mnmal possble total wdth and n ths sense t s optmal; - For fndng optmal model we solve correspondng lnear programmng problem whch s always solvable. Experments have shown that FNN and FGDH wth fuzzy nputs are effectve methods for forecastng stoc prces and maret ndexes. The best results were obtaned n cases of Gaussan F both for fuzzy neural nets and FGDH. Besdes accurate results, whch are n many cases better than the results of FNN, FGDH also has the followng advantages: possblty to wor wth arbtrary number of fuzzy nput varables. Acnowledgements The paper s publshed wth fnancal support by the project ITHEA XXI of the Insttute of Informaton Theores and Applcatons FOI ITHEA Bulgara and the Assocaton of Developers and Users of Intellgent Systems ADUIS Urane Bblography [Zaycheno Yu, 2006] Zaycheno Yu. The Fuzzy Group ethod of Data Handlng and Its Applcaton for Economcal Processes Forecastng - Scentfc Inqury, - Vol. 7, No., June, p [Зайченко, 200]. Ю.П. Зайченко, І.О. Заєць. Синтез та адаптація нечітких прогнозуючих моделей на основі методу самоорганізації //Наукові вісті НТУУ КПІ, 3, 200.-с [Зайченко, 2003] Ю.П. Зайченко. Нечеткий метод индуктивного моделирования в задачах прогнозирования макроэкономических показателей. // Системні дослідження та інформаційні технології, 3, с Authors' Informaton Yury Zaycheno Professor NTUU Kev Polytechnc Insttute, Insttute of Appled System Analyss, Poltehnchesaya 4, Kev, Urane, phone: , e-mal: baservl@volacable.com