The Optimal Interval Partition and Second-Factor Fuzzy Set B i on the Impacts of Fuzzy Time Series Forecasting

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1 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong The Optmal Interval Partton and Second-Factor Fuzzy Set B on the Impacts of Fuzzy Tme Seres Forecastng CHI-CHEN WANG 1 1 Department of Fnancal Management, Natonal Defense Unversty No 70, Sec 2, Zhongyang N.Rd, Petou Dstrct, Pe-Tou,Tape, TAIWAN 1 ch_chenwang@hotmal.com.tw YUEH-JU LIN 2 YU-REN ZHANG 3 2,3 Department of Accountng, Kanan Unversty No. 1 Kanan Road, Luzhu, Taoyuan, TAIWAN 2 udyln@mal.knu.edu.tw; 3 faunfaun@mal.knu.edu.tw HSIEN-LUN WONG 4* 4 Insttute of Management, MngHsn Unversty of Scence and Technology No 1, Hsnhsng Road, HsnFong, HsnChu, TAIWAN Abstract: - Ths study uses two sets of Tawanese data, the export values as the predcton varalbe and ts foregn exchange spot rates as the auxlary varable, to dscuss two mportant ssues of forecastng effects n the fuzzy tme seres analyss by usng One- and Two-factor models. The frst ssue s the relaton between the optmum number of partton equal ntervals and forecastng error. The second ssue s the settng of fuzzy matrx (B ) n the model to compare ts mpacts on forecastng error when t s statc or dynamc. The above two ssues are nvestgated wth the emprcal results. Frst, the optmum number of partton equal ntervals s to select 14 ntervals for the nformaton to have the smallest forecastng error n all models for one- or twofactor, or dfferent number of wndow bass selected. However, f parttonng the nformaton nto more than 14 equal ntervals, the forecastng error can not be reduced but presents a wavng pattern. Second, when the nformaton perod s longer and f the selectng wndow bass s two, under any number of partton ntervals, the forecastng error s always smaller for the dynamc B than for the statc one. However, when the nformaton perod s shorter and the wndow bass s two or three, only parttonng nto fve or eght equal ntervals, the forecastng error wll also be smaller for the dynamc B. Key-Words: Fuzzy tne seres, Two-factor model, Interval partton, Fuzzy relatonshp matrx, Wndow base, Tawan exports, MSE. 1 Introducton Before fuzzfng tme seres data for fuzzy model predcton, one needs to determne the unverse of dscourse and to partton the set. The selected partton ntervals wll affect the forecastng accuracy of the fuzzy models [6]. Consequently, t s a vtal research topc to explore how the selected number of partton ntervals for the better results of fuzzy tme seres model. Prevous studes have generally parttoned nformaton nto fve or seven ntervals. Song and Chssom [13-16] and Chen [1] have used seven ntervals to fuzzfy tme seres data to predct the number of students enterng the Unversty of Alabama. In addton, usng the same partton technque on the same sample, Hwang et al. [7] proposed another one-factor model. Hsu et al. [4] appled the bvarate Markove model and parttoned fve ntervals to predct prce lmts and tradng volume dfferences of weghted Tawan stock ndex. Chen and Hwang [2] have used two-factor model to fuzzfy seven ntervals n a fuzzy set and engaged n temperature predcton. To avod the operatonal * Correspondng author. Tel: E-mal:alan@mal.must.edu.tw (H.-L. Wong). ISSN: Issue 10, Volume 10, October 2011

2 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong complexty, these studes select only the smple random partton method (fve or seven ntervals) to ther analyss among whch the forecastng error (MSE) s not the smallest. Huarng [] has dscussed the length of ntervals used to segment the data and appled two methods (dstrbuton-based and average-based) to fnd better ntervals and mprove forecastng results. Basng on the base-mappng table to determne the length of the nterval, Huarng [6] dd not explan the causes but only concluded that the more the partton ntervals, the better the predcton accuracy. Huarng [6] has also suggested the multvarate gudance models to partton 16 ntervals on a fuzzfy tme seres data for the predcton of weghted ndex of Tawan futures contracts. L and Chen [9] appled the natural parttonng technque, whch can recursvely partton the unverse of dscourse level by levelng n a natural way for the purpose of replacng the base-mappng table. Both Huarng [6] and L and Chen [9] have decded the length of ntervals based on beleves n experences learned from doman experts. Lee et al. [8] had appled the two-factors hgh-order fuzzy tme seres model on the nformaton of daly cloud densty data and parttoned the unverse of dscourse (U ) nto nne ntervals to predct changes n temperatures. L and Cheng [11], based on the state-transton analyss, had overcome the hurdle of determnng the korder n Chen s model. They quantfed a determnstc maxmum length of subsequence n the fuzzy tme seres whch leaded to a certan state. They suggested that the length of nterval wll reflect the senstvty of the nvested nformaton. The forecastng models must follow consstent prncples under whch the shorter the partton length, the more the ntervals resultng n better forecastng precson. Prevous research ndcates that the more of the partton sets, meanng the shorter the length of ntervals, the hgher the precson resulted. However, to provde every nterval wth a vocabulary and value or descrpton, so called defnng the fuzzy set, s an extremely dffcult or even mpossble task. Due to the ncreasng dffcult tes and complexty durng the process of emprcal calculaton and deducton, the pragmatc applcaton of large number of partton sets s lmted or restrcted. Therefore, t s necessary to compromse between precson and complexty. No other related studes have dscussed comprehensvely to the subect but only consdered dfferences n models and nformaton perods, the resulted forecastng error from usng dfferent ntervals to partton, n addton to ther effects on future studes. The approprateness of partton nformaton nto tradtonal fve or seven ntervals s worth of further verfcaton. Snce Song and Chssom s [13-16] fuzzy tme seres model, there are three man forecastng models been developed, ncludng the two-factor, the heurstc model, and the Markov model. Future studes have based on one of those three to mprove model constructon to ncrease forecastng effects. Yu [18] proposed weghted and refned fuzzy tme seres models for TAIEX forecastng. Wang and Yang [17] ntroduced the applcaton concept of entropy to measure the degrees of fuzzness when a tme-nvarant relaton matrx s derved. Cheng et al [3] developed fuzzy tme seres models for forecastng IT costs applyng two approaches: mnmzng entropy and trapezod fuzzfcaton. Lee and Chen [9] constructed two-factor hgh-order fuzzy logcal relatonshps based on the hstorcal data to ncrease the forecastng accuracy rate. Sngh [12] developed a smple computatonal method for fuzzy forecastng models based on dfferent parameters. L et al [10] used the fuzzy C mean method to cluster fuzzy data for forecastng TAIEX futures and enrollment. All of these papers modfed the formulaton of the model to mnmze the forecastng error (MSE) aganst sample data. They adusted parameters such as fuzzy data partton, the membershp functon, fuzzy relatons and clusterng. These mproved models tred to compare forecastng effects wth ther correspondng models under the set condton of one nformaton perod. However, f the emprcal nformaton perods s prolonged or shorten, whether the forecastng effect of the model s subect to be nfluenced accordngly or f the model stll behaves stably has never been nvestgated n the prevous studes. Hwang et al. [7] proposed a new fuzzy tme seres forecastng method. It follows the heurstc rules of one to three to confrm fuzzy relatonshps and to forecast the enrollment at year t; t must decde the number of years of enrollment nformaton used. The number of years of the enrollment data we used s called the wndow bass whch s set to w years. Usng last year s varaton as the crteron matrx and the varatons of the past years (w) as the operaton matrx to perform the forecastng, called one-factor tme-varant fuzzy tme seres model. Chen and Hwang [2] expanded Hwang s model and named the predcted varable the man-factor whch ntroduced the addtonal nformal as the second-factor to auxlary the predcton of the man-factor. Chens new model s called the two-factor model. In the two-factor model, the determnaton of the second-factor fuzzy set (B ) ISSN: Issue 10, Volume 10, October 2011

3 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong s based on the dynamc relatonshps wth the manfactor. If the relatonshp s negatve, then the second-factor fuzzy set (B ) has an opposte drectonal settng for the man-factor fuzzy set (A ). For example, n Chen et al. model to forecast temperature, the average temperature s the manfactor and the cloud densty s the second-factor, and f parttonng nto seven ntervals, the manfactor fuzzy set (A ) s A 1, A2, A3, A4, A, A6, A7 and A1 1/ u1 0./ u2 0 / u3 0 / u4 0 / u 0 / u6 0 / u7. Based on the negatve relatonshp between the man- and second-factor, the second-factor fuzzy set could be formed accordngly and B 0 / u 0 / u 0./ u 1/ u 1/ u 1/ u 1 u / On the contrary, f the relatonshp s postve, B 1/ u 1/ u 1/ u 0./ u 0 / u 0 / u 0 u / however, there s an mportant assumpton that the way of settng s rrelevant wth the number of partton ntervals. Settng the values of B not movng wth the ncreasng number of partton ntervals s called the statc B. The gnored effect of dfferent partton ntervals should also be consdered nto the model. On the other hand, the values of B movng wth the number of ncreasng partton ntervals are called dynamc B. When the model has added the second-factor as the ncrement nformaton to auxlary the forecastng of manfactor, the dfferences n the settng of B could affect the forecastng error n the two-factor model whch has never been mentoned n prevous lteratures. Therefore, ths study focuses on the settng of the fuzzy set B and compares how the settng nfluences the forecastng error. Ths study ams at the above mentoned two ssues n the fuzzy tme seres forecastng model, usng the one-factor and two-factor model as standards to measure and compare. Under dfferent length of nformaton perods, could these two fuzzy forecastng models mantan ther stabltes n ther forecastng effects? In the meantme, ths study also parttons ntervals nto dfferent numbers to verfy the concept of the shorter the nterval, the smaller the forecastng error. Further deducton on these two fuzzy tme seres models wll be engaged to propose a vald number of partton ntervals n order to reduce forecastng error. Fnally, we wll present an approprate settng method for the second-factor fuzzy set (B ) and dscuss ts effects on the forecastng error, when there are dfferences n the nformaton perods, partton ntervals, and retroactng wndow bass perods. 2 Fuzzy set theory and fuzzy tme seres Defnng the unverse of dscourse and the ntervals. Let U be the unverse of dscourse n whch U u u,...,. As the problem doman, U can 1, 2 u t be defned properly. After the length of ntervals s determned, U can be parttoned nto equal length ntervals u u, u,..., u. The mdponts of these ntervals are 1, 2 3 t m 1, m2, m3,..., mt respectvely. 2.2 Defnng the fuzzy sets A and fuzzfyng the data Each fuzzy set A s assgned to a lngustc term, and can be defned by the ntervals u, u, u,..., u }, A ( u1) / u1 f A ( u2 ) / u 2... { t A f f ( u ) / u, where f s the membershp functon of fuzzy A set f A. u k s an element of the fuzzy set A, and f u ) s the degree of A, : U 0,1 belongngness of 1 k n. A ( k u k to A A, ( ) 0,1 A u k t f and 2.3 Fuzzy tme seres Suppose that Y( t)( t...,0,1,2,...) s a subset of R. Let Y (t) be the unverse of dscourse defned by the fuzzy set f (t). If F(t) conssts of f ( t)( 1,2,...), then F(t) s defned as a fuzzy tme seres on Y ( t)( t...,0,1,2,...). Futhermore, we can also see that F (t) s a functon of tme t,.e., the values of F (t) can be dfferent at dfferent tmes. Accordng to Song and Chssom (1993), f F (t) s caused by F ( t 1) only, then ths relatonshp s represented by F( t 1) F( t). Let F (t) be a fuzzy tme seres. If for any tme t, F( t 1) F( t) and F (t) only have fnte elements, then F (t) s called a tme-nvarant fuzzy tmes. Otherwse, t s called a tme-varant fuzzy tme seres. 2.4 One-factors tme-varant fuzzy tme seres model One-factor crteron vector C (t) and operaton matrx O w (t) One-factor crteron vector s defned as followng: C t f t [ C, C,, C, 0 C 1 (1) ] m Operaton matrx s defned as followng: t ISSN: Issue 10, Volume 10, October 2011

4 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong f t 2 O11 O12 O1 m w f t 3 O21 O22 O2m O t, f t w O ( w1)1 O( w1) m 0 O 1, 1 w 1, 1 m (2) f s the fuzzfed varaton between tme ( t 1) and ( t 2 ) of the factor F t, m the number of nterval n the unverse of dscourse, and w the wndow bass. Where t One-factor fuzzy relatonshp matrx R (t) R( t) O w t C t O11 C1 O12 C2 O1 m Cm R11 R12 R1 m O21 C1 O22 C2 O2m Cm R21 R22 R2m O( w1)1 C1 O( w1)2 C2 O( w1) m Cm R( w1)1 R( w1)2 R( w1) m 0 O 1, 1 w 1, 1 m, 0 C 1 (3) Where R O C, s the multplcaton operator. From the matrx model (3), the fuzzfed varaton f (t) can be descrbed as followng: f ( t) =, max [ max ( R ) ( ) ] 11, R21,..., R( w 1)1, max R12, R22,..., R( w 1)2,... ( R, R,..., R ) ] 1m 2m ( w 1) m (4) The model algorthm of One-factor fuzzy tme seres can be presented as follows: Step 1. Compute the varatons of the enrollments between any two contnuous years. Step 2. Partton the unverse of dscourse and the length of ntervals. Step 3. Defne fuzzy sets on unverse of dscourse U. Step 4. Fuzzfy the values of hstorcal data. Step. Choose a sutable wndow bass w, and calculate the output from operaton matrx O w (t) and crteron matrx C (t). Step 6. Fuzzfy the fuzzy forecasted varatons derved n Step. Step 7. Calculate the forecasted enrollments. 2. Two-factors tme-varant fuzzy tme seres model 2..1 Two-factor crteron vector C (t), S(t) and C S operaton matrx O w (t) t f t 1 [ C1, C2,, Cm ] t gt [ S, S,, S ], 0 C 1 (), 0 S 1 (6) m Operaton matrx s defned as followng: f t 2 O11 O12 O1 m w f t 3 O21 O22 O2m O t, f t w O ( w1)1 O( w1) m 0 O 1, 1 w 1, 1 m (7) f s the fuzzfed varaton between tme ( t 1 ) and ( t 2 ) of the frstfactor F t, s t 1 the fuzzfed varaton at tme ( t 1) of the second-factor S t, m the number of nterval n the unverse of dscourse, and w the wndow bass. Where t Two-factor fuzzy relatonshp matrx R (t) R( t) O w t St Ct O11 S1 C1 O12 S 2 C2 O1 m S m Cm R11 R12 R1 m O21 S1 C1 O22 S 2 C2 O2m S m Cm R21 R22 R2m O( w1)1 S1 C1 O( w1)2 S 2 C2 O( w1) m S m Cm R( w1)1 R( w1)2 R( w1) m 0 O 1, 1 w 1, 1 m, 0 S 1, 0 C 1 (8) Where R O S C, s the multplcaton operator. From the matrx model (8), the fuzzfed varaton f (t) can be descrbed as followng: Where R O S C, s the multplcaton operator. From the matrx model (8), the fuzzfed varaton f (t) can be descrbed as followng: f ( t) = max R, R,..., R, max R, R,..., R, max [ ( ) ( ) ] ( w 1) ( w 1)2,... ( R, R,..., R ) ] 1m 2m ( w 1) m (9) The model algorthm of Two-factor fuzzy tme seres can be presented as follows: Step 1. Decde the frst factor and calculate ts varatons. Step 2. Decde unverse of dscourse and the length of ntervals. S t as Step 3. Determne the second factor presented n step (1) and (2). Step 4. Decde the wndow bass for fuzzy relatonshp matrx. Step. Compute the fuzzfed varaton and defuzzfy. ISSN: Issue 10, Volume 10, October 2011

5 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong Ths study uses mean square error (MSE) to measure forecastng accuracy. The MSE value can be represented by: MSE n 1 n (0) (0) 2 (10) k 1 ( x ( k) x ( k)) where (0) X ( K) s the actual value; X K) the predcted value; n denotes the number of data. The smaller MSE value s, the closer s predcted value of the model to the hstorcal data, meanng the hgh predcton capablty of the model. 3 Data nformaton and results 3.1 Data nformaton The data are obtaned from AREMOS economc database, whch ncludes the amount of Tawan exports and the foregn exchange spot rate from January 199 to March 2002, wth a total of 87 data ponts. Accordng to the nternatonal fnance theory, amount of exports and foregn exchange spot rate have a close relatonshp. Therefore, n our analyss, the amount of exports s the predcted varable and the foregn exchange spot rate s the auxlary varable as the ncrement nformaton for forecastng. The correlaton between the two varables s calculated to understand ther movng drecton. The correlaton s found to be (p-value=0.0001), meanng that there s a sgnfcant postve relaton between the amount of exports and foregn exchange spot rate. The Mean Square Error (MSE) s the forecastng error to evaluate model s forecastng accuracy. Due to the possble nfluence of nformaton perod on forecastng power, the length for nformaton perod s controlled for n ths study and further separated nto three sample sets, ncludng a short perod of a total 27 data ponts (January 2000 to March 2002), a mddle term of a total 1 data ponts (January 1998 to March 2002) and a long perod of a total 87 data ponts (January 199 to March 2002). Under dfferent length of nformaton perods, how pror nformaton, nterval partton, ncremenet nformaton and the settng of secondfactor fuzzy set affect model forecastng power are dscussed. 3.2 Emprcal result Ths study ncludes the wndow bass wth a range of two to seven ( w 2 ~ w 7 ) and selects fve to 16 equal partton ntervals to dscuss the varaton (0) ( of forecastng error n the two-factor model when under ether one- or two-factor condton The comparson of forecastng error for dfferent partton ntervals In the search of the best partton ntervals for a smaller forecastng error model to verfy the exstence of the most approprate length of ntervals, ths study adopts a generally used range of fve to 16 equal partton ntervals n our three separate nformaton data sets. Through the applcaton of the fuzzy tme seres two-factor model, comparson results for changes n the forecastng error (MSE) are summarzed n the followng. Based on table 3-1, 3-2 & 3-3, fgure 3-1, 3-2 and 3-3 are presented to compare the partton ntervals wth the frequency accumulated for four smaller forecastng errors (MSE) obtaned from the three separate data nformaton sets. The frequency n each fgure represents the accumulated occurrence of the four smallest forecastng errors for each partton ntervals, from one- and two-varable, and under the wndow bass of two to seven. These four smallest forecastng errors are regarded as the most approprate equal partton ntervals wth relatvely smaller forecastng errors. Ths study dscovers that under the two-factor model, the selecton of 14 equal partton ntervals has the smallest forecastng error, no matter the nformaton perods n the three data set, one- or two-varable type, or the wndow bass. After the partton ntervals has reached to 14, forecastng errors show a wavng condton and even addng more partton ntervals wll not further reduce the forecastng error The settng of second-factor fuzzy B on model forecastng error In order to nvestgate whether the settng of secondfactor fuzzy B would nfluence model forecastng error, ths study sets B as statc or dynamc. The method of settng s n the appendx 1 & 2. Under three dfferent (short or long) nformaton perods ncludng January 199 to March 2002, January 1998 to March 2002, and January 2000 to March 2002 as well as dfferent retroactng wndow bass perods ncludng the range of w 2 ~ w 7, changes n forecastng errors are compared. The emprcal results are ncluded n table ~ 3-4-6, 3--1 ~ 3--6 and ~ ISSN: Issue 10, Volume 10, October 2011

6 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong partton nterval Amount Table 3-1 Forecastng error (MSE) comparson for one-and two varables under two factor model (Tawan export amounts, January 199~March 2002, n mllon NT dollars) model Two-factor model type One varable Two varables wndow w=2 w=3 w=4 w= w=6 w=7 w=2 w=3 w=4 w= w=6 w=7 ntervals *** **** ** * *** * **** ** *** * **** ** **** ** **** *** * **** ** **** *** * **** ** **** *** * **** ** **** *** * **** ** **** *** * **** ** **** *** * **** ** **** *** * **** ** **** *** * **** ** **** *** * ****: the smallest forecastng error (MSE), ***: the second smallest, **: the thrd smallest, *: fourth smallest MSE compared horzontally. partton nterval Amount Table 3-2 Forecastng error (MSE) comparson for one-and two varables under two factor model (Tawan export amounts, January 1998~March 2002, n mllon NT dollars) model Two-factor model type One varable Two varables wndow w=2 w=3 w=4 w= w=6 w=7 w=2 w=3 w=4 w= w=6 w=7 ntervals *** **** ** * *** **** ** * *** **** ** * *** **** ** * *** **** ** * *** **** ** * *** * **** *** ** *** **** ** * *** **** ** * *** * **** * *** * **** ** *** * **** ** ****: the smallest forecastng error (MSE), ***: the second smallest, **: the thrd smallest, *: fourth smallest MSE compared horzontally. Amount Table 3-3 Forecastng error (MSE) comparson for one-and two varable under two factor model (Tawan export amounts, January 2000~March 2002, n mllon NT dollars) model Two-factor model partton type One varable Two varables nterval wndo w=2 w=3 w=4 w= w=6 w=7 w=2 w=3 w=4 w= w=6 w=7 w * * *** **** ** ntervals *** * **** ** *** * **** ** *** * **** ** *** * **** ** **** ** *** * **** ** *** * **** ** *** * **** ** ** * **** *** ** ** **** ** *** * **** *** ** * **** *** ** * **** ** *** * ****: the smallest forecastng error (MSE), ***: the second smallest, **: the thrd smallest, *: fourth smallest MSE compared horzontally. ISSN: Issue 10, Volume 10, October 2011

7 frequency frequenc frequency WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong two-factor model ntervals Fgure 3-1 Frequency of the best partton ntervals (Tawan export amounts, ~ ) two-factor model ntervals Fgure 3-2 Frequency of the best partton ntervals (Tawan export amounts, ~ ) two-factor model ntervals Fgure 3-3 Frequency of the best partton ntervals (Tawan export amounts, ~ ) (1) For the nformaton perods from January 199 to March 2002 When the wndow bass s two ( w 2 ) and partton ntervals are fve and seven, the rato of forecastng errors for the dynamc B smaller than the statc B s (2/12). However, when the wndow bass s three to seven ( w 3 ~ w 7 ), except for the partton ntervals of sx and 16, the rato s (10/12). (2) For the nformaton perods from January 1998 to March 2002 When the wndow bass s two ( w 2 ) and partton ntervals are fve, seven, eght, 13, 14, 1, and 16, the rato of forecastng errors for the dynamc B smaller than the statc B s 0.83 (7/12). On the other hand, when the wndow bass s four to seven ( w 4 ~ w 7 ), except for the partton ntervals of sx, the rato s (11/12). Table3-4-1 Forecastng error (MSE) comparson for wndow bass at two under two factor model (Tawan export amounts, January 199~March 2002, w =2, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * MSE = * * * * * * * * * * * ISSN: Issue 10, Volume 10, October 2011

8 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong Table3-4-2 Forecastng error (MSE) comparson for wndow bass at three under two factor model (Tawan export amounts, January 199~March 2002, w =3, n mllon NT dollars) fuzzy set B ntervals statc B x10 9 dynamc B MSE > * * * * * * * * * * MSE = * * Table3-4-3 Forecastng error (MSE) comparson for wndow bass at four under two factor model (Tawan export amounts, January 199~March 2002, w =4, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * * * * * * * * MSE = * fuzzy set B ntervals Table Forecastng error (MSE) comparson for wndow bass at fve under two factor model (Tawan export amounts, January 199~March 2002, w =, n mllon NT dollars) statc B dynamc B MSE > * * * * * * * * * * MSE = * fuzzy set B ntervals Table 3-4- Forecastng error (MSE) comparson for wndow bass at sx under two factor model (Tawan export amounts, January 199~March 2002, w =6, n mllon NT dollars) statc B dynamc B MSE > * * * * * * * * * * MSE = * fuzzy set B ntervals Table Forecastng error (MSE) comparson for wndow bass at seven under two factor model (Tawan export amounts, January 199~March 2002, w =7, n mllon NT dollars) statc B dynamc B MSE > * * * * * * * * * * MSE = * (3) For the nformaton perods from January 1998 to March 2002 When the wndow bass s two and three ( w 2 ~ w 3 ) and partton ntervals are fve, seven, eght, and nne, the rato of forecastng errors for the dynamc B smaller than the statc B s 0.2 (3/12). Moreover, when the wndow bass s four and fve ( w 4 ~ w ) and partton ntervals are fve, eght, and nne, the rato s (4/12). Fnally, when the wndow bass s four and fve ( w 6 ~ w 7 ) and partton ntervals are fve, and eght, the rato s (2/12). From the above emprcal results, for those rather longer nformaton perods, such as the sample data sets of January 199 ~ March 2002 and January 1998 ~ March, most fuzzy sets have the smaller forecastng error n the dynamc B than n the statc B.However, wth the ncreasng retroactng wndow bass, the dfferences n forecastng errors between the dynamc and statc B gradually lessen ISSN: Issue 10, Volume 10, October 2011

9 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong or become equal, and even reverse to the opposte that the dynamc B has a larger forecastng error than the statc B. These results reveal that the length of nformaton perods and the settng of fuzzy set B have an absolute nfluencng effect on forecastng errors. Partcularly, when selectng the wndow bass at greater than two under a longer nformaton perod, for any equal partton ntervals, the dynamc B wll always has a smaller forecastng error than the statc B. On the contrary, when the nformaton perod s shorter, the retroactng wndow bass selected should be at two to three ( w 2 ~ w 3 ) and the partton ntervals should be at fve or eght. Then, the dynamc B wll have a smaller forecastng error than the statc B. The emprcal results are summarzed n the followng tables for the three data set perods. Table3--1 Forecastng error (MSE) comparson for wndow bass at two under two factor model (Tawan export amounts, January 1998~March 2002, w =2, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * * * * * MSE = * Table 3--2 Forecastng error (MSE) comparson for wndow bass at three under two factor model (Tawan export amounts, January 1998~March 2002, w =3, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * * * * * * * MSE = * * Table3--3 Forecastng error (MSE) comparson for wndow bass at four under two factor model (Tawan export amounts, January 1998~March 2002, w =4, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * * * * * * * * * MSE = * Table3--4 Forecastng error (MSE) comparson for wndow bass at fve under two factor model (Tawan export amounts, January 1998~March 2002, w =, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * * * * * * * * * MSE = * ISSN: Issue 10, Volume 10, October 2011

10 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong Table 3-- Forecastng error (MSE) comparson for wndow bass at sx under two factor model (Tawan export amounts, January 1998~March 2002, w =6, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * * * * * * * * * MSE = * Table 3--6 Forecastng error (MSE) comparson for wndow bass at seven under two factor model (Tawan export amounts, January 1998~March 2002, w =7, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * * * * * * * * * MSE = * Table Forecastng error (MSE) comparson for wndow bass at two under two factor model (Tawan export amounts, January 2000~March 2002, w =2, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * * MSE = * * * Table3-6-2 Forecastng error (MSE) comparson for wndow bass at three under two factor model (Tawan export amounts, January 2000~March 2002, w =3, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * * MSE = * * * Table3-6-3 Forecastng error (MSE) comparson for wndow bass at four under two factor model (Tawan export amounts, January 2000~March 2002, w =4, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * MSE = * * * * Table Forecastng error (MSE) comparson for wndow bass at fve under two factor model (Tawan export amounts, January 2000~March 2002, w =, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * * MSE = * * * * ISSN: Issue 10, Volume 10, October 2011

11 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong Table 3-6- Forecastng error (MSE) comparson for wndow bass at sx under two factor model (Tawan export amounts, January 2000~March 2002, w =6, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * MSE = * * * * * * Table Forecastng error (MSE) comparson for wndow bass at seven under two factor model (Tawan export amounts, January 2000~March 2002, w =7, n mllon NT dollars) fuzzy set B ntervals statc B dynamc B MSE > * * MSE = * * * * * * : Statc B s set as appendx 1. : Dynamc B s set as appendx 2. 4 Conclusons Ths paper proposes the fuzzy two-factor forecastng model to dscuss equal partton ntervals under dfferent ranges to locate forecastng models wth achevng the smallest forecastng error. In the three nformaton data sets, dsregardng the length of nformaton perod, one- or two-varable type, or the number of wndow bass, the smallest forecastng error occurs when the 14 equal partton ntervals s selected. After that, addtonal ntervals wll cause the forecastng error becomng a wavng pattern that further reducton n forecastng error s no longer possble. These results ndcate that the statement of the lengther the ntervals (hgher partton ntervals), the smaller the forecastng error s ndeed naccurate. In addton, randomly selectng the partton ntervals at fve or seven equally as the prevous lterature wll produce rather larger forecastng errors n the two-factor model. Therefore, n applcaton, ether consderng the vocabulary defnton of fuzzy nterval or controllng the amount of forecastng error, selectng 14 equal partton ntervals could reach the smallest forecastng error n the two-factor forecastng model. From the emprcal evdences, among the two nformaton data sets, January 199 to March 2002 and January 1998 to March 2002, representng the longer nformaton perod, almost all of the fuzzy sets wth the dynamc B have rather smaller forecastng error than the statc B, no matter the number of wndow bass. However, for the data set from January 2000 to March 2002 whch s a shorter perod relatvely, dfferences n forecastng errors between the dynamc and statc B lessen, become equal, or even reversed to the opposte that dynamc B has a larger forecastng error than the statc B. That reveals the length of nformaton perod and the settng n the fuzzy set B have absolute nfluence effects on forecastng error. Therefore, under the two-factor emprcal research model, a pre-test focusng on the fuzzy set B should be done frst to consder all possble condtons on the fuzzy set B. Then, the research would be comprehensve and thoroughly and then the goal of reducng forecastng errors could be accomplshed. References [1]Chen, S. (1996). Forecastng enrolments based on fuzzy tme seres, Fuzzy Sets and Systems, 18, [2]Chen, S.and Hwang, J. (2000). Temperature predcton usng fuzzy tme seres, IEEE Transactons on System, Man and Cybernetc,-Part B: Cybernetcs 30 (2) [3]Cheng, C.H., Chang, J.R., and Yeh, C., (2006). Entropy-based and trapezod fuzzfcaton-based fuzzy tme seres approaches for forecastng t proect cost, Technologcal Forecastng and Socal Change 73 () ISSN: Issue 10, Volume 10, October 2011

12 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong [4]Hsu, Y.Y, Tse, S.M, and Wu, B. (2003). A new approach of bvarate fuzzy tme seres analyss to the forecastng of a stock ndex, Internatonal Journal of Uncertanty Fuzzness Knowledge Based System 11(6) []Huarng, K. (2001a). Heurstc models of fuzzy tme seres for forecastng, Fuzzy Sets and Systems [6]Huarng, K. (2001b). Effectve lengths of ntervals to mprove forecastng n fuzzy tme seres, Fuzzy Sets and Systems [7]Hwang, J. Chen, S. and Lee, C. (1998). Handlng forecastng problems usng fuzzy tme seres, Fuzzy Sets and Systems [8]Lee, L., Wang, L., and Chen, S. (2006). Handlng forecastng problems based on two-factor hgh-order fuzzy tme seres, IEEE Transactons on fuzzy System 14(3) [9]L, S.T, and Chen, Y.C. (2007). Determnstc fuzzy tme seres model for forecastng enrolments, Computers and Mathematcs wth Applcatons [10]L, C.H., Huang, W.C., Kuo, B.C, and Hung, C.C., (2008). A novel fuzzy weght C-means method for mage classfcaton, Internatonal Journal of Fuzzy Systems 10(3) [11]L, S.T. and Chen, Y. P. (2004). Natural parttonng based forecastng model for fuzzy tmeseres, Budapest, Hungary [12]Sngh, S.R. (2007). A smple method of forecastng based on fuzzy tme seres, Appled Mathematcs and Computaton 186(1) [13]Song, Q. and Chssom, B. S. (1993a). Forecastng enrolments wth fuzzy tme seres-part I, Fuzzy Sets and Systems [14]Song, Q. and Chssom, B. S. (1993b). Fuzzy tme seres and ts models, Fuzzy Sets and Systems [1]Song, Q.and Chssom, B. S. (1993b) Forecastng enrolments wth fuzzy tme seres Part II, Fuzzy Sets and Systems [16]Song, Q. and R.P. Leland. (1996). Adaptve learnng defuzzfcaton technques and applcatons, Fuzzy Sets and Systems 81(3) [17] Wang, H.F. and Yang, J C., (200). Fuzzy relaton analyss n fuzzy seres model, Computers and Mathematcs wth Applcatons [18]Yu, H.K. (200). Refned fuzzy tme seres model for forecastng, Physcal A ISSN: Issue 10, Volume 10, October 2011

13 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong Appendx 1 Multvarate fuzzy tme seres two-factor model: settng of statc B ntervals ntervals ntervals ntervals ntervals ntervals ntervals ntervals ntervals ntervals ISSN: Issue 10, Volume 10, October 2011

14 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong 1 ntervals ntervals ISSN: Issue 10, Volume 10, October 2011

15 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong Appendx 2 Multvarate fuzzy tme seres two-factor model: settng of dynamc B ntervals ntervals ntervals ntervals ntervals ntervals ntervals ntervals ntervals ntervals ISSN: Issue 10, Volume 10, October 2011

16 Ch-Chen Wang, Yueh-Ju Ln, Yu-Ren Zhang, Hsen-Lun Wong 1 nterva 閱 s ntervals ISSN: Issue 10, Volume 10, October 2011