T-Norm of Yager Class of Subsethood Defuzzification: Improving Enrolment Forecast in Fuzzy Time Series

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1 Gadng Busness and Management Journal Vol. 10 No. 2, 1-11, 2006 T-Norm of Yager Class of Subsethood Defuzzfcaton: Imrovng Enrolment Forecast n Fuzzy Tme Seres Nazrah Raml Faculty of Informaton Technology and Quanttatve Scences Unverst Teknolog MARA (UTM), Malaysa Emal: nazrahr@ahang.utm.edu.my Abu Osman Md. Ta Faculty of Scence and Technology Kolej Unverst Sans dan Teknolog Mengabang Telot, Kuala Terengganu Emal: abuosman@kustem.edu.my ABSTRACT Fuzzy tme seres has been used to model observatons that contan multle values. Ths aer rooses the t-norm of Yager class of subsethood defuzzfcaton to forecast unversty enrolments based on fuzzy tme seres and the data of hstorcal enrolments hch are adoted from Song and Chssom (1994). The roosed method aled seven and ten nterval th equal length and the ma-roduct and ma-mn as the comoston oerator n the fuzzy relatons () t = F( t 1) R( t, t 1) F. The result shos that the t-norm of Yager class of subsethood defuzzfcaton models th (10, ma-roduct) s the best forecastng method n terms of accuracy. The roosed method has also mroved the forecastng results by revous researchers. Keyords: forecastng enrolments, t-norm of Yager Class, subsethood defuzzfcaton, ma-mn comoston, ma-roduct comoston ISSN Unverst Teknolog MARA (UTM), Malaysa 1

2 Gadng Busness and Management Journal Introducton The fuzzy set theory as orgnally develoed to handle roblems nvolvng human lngustc terms, for eamle, hot, arm, cool and cold for temerature and short, medum and tall for heght. In economc forecastng, the classcal tme seres method cannot deal th forecastng roblems n hch the values of tme seres are lngustc terms reresented by fuzzy sets. In ve to ths, Song and Chssom (1993) roosed a secal dynamc rocess called fuzzy tme seres to overcome the draback of the classcal tme seres methods. In ths alcaton of fuzzy tme seres, Song and Chssom (1993; 1994) used to models to forecast students enrolments n the Unversty of Alabama by fuzzfyng the hstorcal data. Snce then, several other studes have been carred out. Hoever, there are stll many crtcal ssues ertanng to Song and Chssom s methods. Undoubtedly, the determnaton of defuzzfcaton rncles s one of these ssues. In the early days, defuzzfcaton roblem never had the chance to be formally defned or analysed (Olvera 1995). Noadays, many dfferent studes on t have been develoed. Defuzzfcaton s the rocess to select an arorate crs value based on a fuzzy set n such a ay that the selected crs value may reresent the fuzzy set. Ths ste s necessary n most alcatons of fuzzy systems. For eamle, the outut of fuzzy controllers must be n the defuzzfed form snce mechancal, electrcal, neumatc and other actuators can only accet and use these determnstc sgnals. Song and Chssom (1993) combned to defuzzfcaton rncles: the mean of mama (MOM) and the centre of area (COA) n ther research. In 1994, Song and Chssom aled a three-layer back roagaton neural netork to convert the outut of the varant model. Insred by Song and Chssom s aroach, Song and Leland (1996) roosed the concet of the otmal defuzzfcaton mang and used Song and Chssom s tme varant fuzzy tme seres model to forecast the unversty enrolments. An adatve learnng of the otmal defuzzfcaton mang s develoed to fnd the otmal arameters. Besdes, research fndngs of Nazrah and Abu Osman (2000) also ndcated that the number of ntervals and the defuzzfcaton technques could nfluence the forecastng results. In ther study, the unverse dscourse s dvded nto seven and ten ntervals th equal length. Fuzzy ntervals th defuzzfcaton requrement (reasonable but not necessary (RNN)) s 2

3 T-Norm of Yager Class of Subsethood Defuzzfcaton aled to obtan the result occurred at the mamum value of the membersh functon. In other related study, Tsaur et al. (2005) roosed the concet of entroy to measure the degree of fuzzness and modfed Song and Chssom s tme nvarant fuzzy tme seres model to forecast the unversty enrolments. The fuzzy outut s defuzffed by usng combned rncle as n Song and Chssom s (1993). Nazrah and Abu Osman (2006) roosed the concet of subsethood defuzzfcaton th algebrac roduct t-norm μ ( ). to forecast the unversty enrolments. They aled mamn and ma-roduct comoston oerator n the fuzzy relaton F(t) = F(t 1) R(t, t 1). Ths aer attemts to aroach the forecastng ssue by roosng the t-norm of Yager class, t ( ) 1 = ( μ( ), ) 1 mn 1, ( 1 μ( )) + ( 1 ) (0, ) n the subsethood defuzzfcaton ndcated n the study by Nazrah and Abu Osman (2006). The roosed method aled seven and ten nterval th equal length and the ma-roduct and ma-mn as the comoston oerator n the fuzzy relatons, F() t = F( t 1) R( t, t 1) The analyss shos that the t-norm of Yager class of subsethood defuzzfcaton th (10, ma-roduct) s the best forecastng method n terms of accuracy. The roosed method ll also mrove the forecastng results by other revous researchers. Prelmnares In 1965, Lotf Zadeh roosed the dea of fuzzy set for dealng th the vagueness tye of uncertanty (Wang 1997). A fuzzy set A defned on the unverse X s charactersed by a membersh functon such that μ A : X [0, 1]. The nearer the value of μ() s unty, the hgher the grade of membersh of n A. Intersecton of Fuzzy Subsets: T-Norms (Trangular Norms) Trangular norms (brefly t-norms) are an ndsensable tool for the nterretaton of the conjuncton n fuzzy logcs and subsequently, for the ntersecton of fuzzy sets. A t-norm s a bnary oeraton t on the unt 3

4 Gadng Busness and Management Journal nterval [0, 1] hch s commutatve, assocatve, monotone and has 1 as the neutral element, that s, t s a functon t : [0, 1] [0, 1] [0, 1] such that for all a, b, c [0, 1]: Aom t1: t(a, b) = t(b, a), Aom t2: t[t(a, b),c] = t[a, t(b, c)], Aom t3: t(a, b) t(a, c) for b c, Aom t4: t(a, 1) = a. Subsethood Defuzzfcaton The subsethood defuzzfcaton (SD) s based on sgma-count measurement and mean value theorem (Olvera 1995). The SD concdes th the unfyng structure of fuzzy set theory, that s, arallel th accordance to the Kosko s subsethood theorem (Kosko 1992). Wthout the loss of generalty, the unverse dscourse s normalsed to X = [0,1]. For a gven fuzzy set A, the averagng defuzzfcaton n ^ M ( A ) terms of set measures s gven by here the sgmacount measure of a set denoted as: M = ( ) + + μ n ( ) n n = μ ( ) / M ( ) A A ( X ) = μ 1, the mrror set of suort A s eressed as A and ( ) of fuzzy sets A and A μ. The ntersecton Μ denoted as A Μ A, s defned as μ defuzzfcaton, the SD method can be rtten as: A Μ ( ) = μ A( ) t A here t s a trangular norm. From the averagng SD = n =1 ( ( ) ) n μ =1 A μ A =1 t ( ) here for = 1, 2, n, ( ) μ s the A degree of the membersh of the -th element n the suort of A. 4

5 T-Norm of Yager Class of Subsethood Defuzzfcaton Enrolment Forecastng Song and Chssom (1994) develoed a tme-varant fuzzy tme seres model to forecast students enrolment n the Unversty of Alabama. A three-layer neural netork as aled to defuzzfy the outut of the fuzzy tme seres model. In 2006, Nazrah and Abu Osman roosed the subsethood defuzzfcaton th t-norm of algebrac roduct to aroach ssues on students enrolment forecastng at the Unversty of Alabama. The follong secton ll sho that the subsethood defuzzfcaton th t-norm of Yager class yelds better than the forecast result by other researchers mentoned before. The enrolment forecastng usng the fuzzy tme seres (Song and Chssom 1994) th the t-norm of Yager class subsethood defuzzfcaton s descrbed ste by ste as follos. Seven and ten equal ntervals are used along th the ma-mn and ma-roduct comoston oerator. Ste 1: Defne the unverse of dscourse U thn the hstorcal data. Let U = [13000, 2000] and for seven and ten equal ntervals, the length s 1000 and 700 resectvely. Table 1: The Fuzzfed Hstorcal Enrolments (Ten Intervals) Year Actual enrolment Fuzzfed enrolment A 1 A 1 A 2 A 3 A 5 A 6 A 6 A 5 A 5 A 6 A 8 A 9 A 10 A 10 A 9 5

6 Gadng Busness and Management Journal Ste 2: Ten or seven (based on the number of equal nterval) lngustc value must be determned to defne fuzzy sets on the unverse U. The A ( = 1, 2, 3, ) are the ossble lngustc values of enrolment. For ten ntervals, each A s defned by the ntervals u 1, u 2, u 3,, u 10 as follos: A 1 = 1/u /u 2 + 0/u 3 + 0/u /u 9 + 0/u 10 A 2 = 0.5/u 1 + 1/u /u 3 + 0/u /u 9 + 0/u 10 A 3 = 0/u /u 2 + 1/u /u /u 9 + 0/u A 9 = 0/u 1 + 0/u 2 + 0/u 3 + 0/u /u /u 8 + 1/u /u 10 A 10 = 0/u 1 + 0/u 2 + 0/u 3 + 0/u /u 7 + 0/u /u 9 + 1/u 10 Ste 3: Choose a model bass = 4 and at a gven tme t, calculate the fuzzy relaton T T T R ( t, t 1) = f ( t 2) f ( t 1) f ( t 3) f ( t 2) f ( t ) f ( t + 1) and fuzzy forecasted F() t = F( t 1) R( t, t 1) here s the ma-mn and ma-roduct comoston. Ste 4 : Interret the forecast oututs. By usng the SD th t-norm of ( ) 1 Yager class, t ( ( ) ) = ( ( )) + ( ) μ, 1 mn 1, 1 μ 1, the forecast enrolment can be rtten as SD = 1 ( ( 1 μ( ) + ( 1 ) ) n 1 = 1 mn 1, n = 1 μ ( ) Alyng the above rncles, the redcted enrolments are tabulated n Table 2 and shon n Fgures 1 and 2. Results The follong table shos the forecastng results of four dfferent ars of nterval and comoston oerator. 6

7 T-Norm of Yager Class of Subsethood Defuzzfcaton Table 2: Forecast Enrolments Usng T-Norm of Yager Class of Subsethood Defuzzfcaton Year (10, ma- (10, ma- (7, ma- (7, ma- Actual mn) roduct ) mn) roduct) enrolments (, ) (1.41, 45) (1.52, 10) (1.55, 1) (1.55, 1) Number of Students Actual Ma-mn Ma-roduct Year Fgure 1: Forecast Enrolments and Actual Enrolments (Ten Intervals) 7

8 Gadng Busness and Management Journal Number of Students Year Actual Ma-mn Ma-roduct Fgure 2: Forecast Enrolments and Actual Enrolments (Seven Intervals) Dscussons In ths study, the t-norm of Yager class of subsethood defuzzfcaton s resented to forecast unversty students enrolments based on fuzzy tme seres. The hstorcal data are adoted from Song and Chssom (1994). The forecast enrolment n Ste 4 s calculated for (0, ) and 1 50 usng Mathcad softare. It s qute dffcult to fnd th the best forecastng model snce there s no obvous attern beteen the forecastng error and. Hoever, after numbers of effort n tral and error, fnally the value of th the best forecastng enrolments (Table 2) s found. The best forecast enrolments (th the smallest forecastng errors) for (10, ma-roduct) and (10, ma-mn) occurred at ( = 1.51, = 45) and (1.52, 10) resectvely. Hoever, for (7, ma-roduct) and (7, ma-mn), the best forecastng occurred at the same ont (1.55, 1). The accuracy of ths forecastng method as roosed n the model s measured by usng four crtera namely mean absolute devaton (MAD), Table 3: Comarson of Performance Indcator for the Proosed Methods MAD MSE RMSE MAPE (10, ma-roduct) (10, ma-mn) (7, ma-roduct) (7, ma-mn)

9 T-Norm of Yager Class of Subsethood Defuzzfcaton mean square error (MSE), root mean square error (RMSE) and mean absolute ercentage error (MAPE). Among the forecast enrolments th ten ntervals, the ma-roduct comoston oerator has the best accuracy snce ts MSE, RMSE, MAD and MAPE are the loest. For the seven ntervals th equal length, the ma-roduct comoston oerator s also the best forecastng model snce all the forecastng errors are the smallest. Hoever, the dfference n errors beteen (7, ma-roduct) and (7, ma-mn) s very small and 16 out of 19 (84.2%) forecast enrolments sho the same results. Ths ndcates that there s no dfference n the forecastng results beteen (7, ma-roduct) and (7, ma-mn) RMSE (10,ma-roduct) (10, ma-mn) (7, ma-roduct) (7, ma-mn) Proosed Methods Fgure 3: Comarson of RMSEs for the Proosed Methods Based on the results n Table 3 and Fgure 3, t s obvous that the t- norm of Yager class of the subsethood defuzzfcaton th (10, maroduct) s ranked frst folloed by (10, ma-mn), (7, ma-roduct) and (7, ma-mn). Table 4: Comarson of RMSEs th Varous Models Song & Song Nazrah & Yu Nazrah & Proosed Chssom Leland Abu Osman (2005) Abu Osman method (1994) (1996) (2000) (2006) (10, maroduct) RMSE

10 Gadng Busness and Management Journal 1200 RMSE (10, maroduct) Song & Chssom Song & Leland Nazrah & Abu Osman Table 4 comares the RMSE of dfferent forecastng models th the roosed method (10, ma-roduct). From Table 4 and Fgure 4, t can be seen that the root mean square error (RMSE) for the roosed method (10, ma-roduct) s and ths s better than n Song and Chssom (1994), (Song & Leland 1996), (Nazrah & Abu Osman 2000), (Yu 2005) and (Nazrah & Abu Osman 2006). It shos that the roosed method th (10, ma-roduct) outerforms the models by Song and Chssom (1994), Song and Leland (1996), Nazrah and Abu Osman (2000), Yu (2005) and Nazrah and Abu Osman (2006). Yu Forecastng Models Nazrah & Abu Osman Fgure 4: Comarson of RMSEs th Varous Models Proosed method Concluson Ths study rooses the t-norm of Yager class of subsethood defuzzfcaton to forecast unversty students enrolments based on fuzzy tme seres and the data of hstorcal enrolments ere adoted from Song and Chssom (1994). The model th (10, ma-roduct) s the best forecastng method comared to (10, ma-mn), (7, ma-roduct) and (7, ma-mn). The (10, ma-roduct) models has also mroved the forecastng results by Song and Chssom (1994), Song and Leland (1996), Nazrah and Abu Osman (2000), Yu (2005) and Nazrah and Abu Osman (2006). The dffculty of fndng the relatonsh beteen and the average forecastng error s one of the roblems n ths study. In order to overcome ths shortcomng, t s suggested genetc algorthms to be aled n the future studes (Day 1995, as cted n Hang 1998). 10

11 T-Norm of Yager Class of Subsethood Defuzzfcaton References Hang, J.R., Chen, S.M. & Lee, C.H. (1998). Handlng forecastng roblems usng fuzzy tme seres. Fuzzy Sets and System, 100, Kosko, S. (1992). Neural Netorks and Fuzzy Systems. Ne Jersey: Prentce Hall. Nazrah, R. & Abu Osman, M.T. (2000). Ramalan kaburkemasukan elajar suatu alternatf. Prosdng Smosum Kebangsaan Sans Matematk Ke-8, Kolej Unverst Sans dan Teknolog, Terengganu. Nazrah, R. & Abu Osman, M.T. (2006). Forecastng enrollments usng fuzzy tme seres th subsethood defuzzfcaton (algebrac roduct t- norm). Proceedngs of the Natonal Semnar on Scence, Technology and Socal Scences 2006, 2, Kuantan, Pahang. Olvera, J.V. (1995). A set-theoretcal defuzzfcaton method. Fuzzy Sets and System, 76, Song, Q. & Chssom, B.S. (1993a). Fuzzy tme seres and ts model. Fuzzy Sets and System, 54, 1-8. Song, Q. & Chssom, B.S. (1993b). Forecastng th fuzzy tme seres Part I. Fuzzy Sets and System, 54, Song, Q. & Chssom, B.S. (1994). Forecastng th fuzzy tme seres Part II. Fuzzy Sets and System, 62, 1-8. Song, Q & Leland, R.P. (1996). Adatve learnng defuzzfcaton technques and alcatons. Fuzzy Sets and System, 81, Tsaur, R.C., Yang, J.C.O. & Wang, H.F. (2005). Fuzzy relaton analyss n fuzzy tme seres model. Comuters and Mathematcs th Alcatons, 49, Wang, L.X. (1997). A course n fuzzy systems and control. Ne Jersey: Prentce Hall. Yu, H.K. (2005). A refned fuzzy tme seres model for forecastng. Physca A, 346,