A Fuzzy Time Series Analysis Approach by Using Differential Evolution Algorithm Based on the Number of Recurrences of Fuzzy Relations

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1 Amercan Journal of Intellgent Systems 2013, 3(2): DOI: /.as A Fuzzy Tme Seres Analyss Approach by Usng Dfferental Evoluton Algorthm Based on the Number of Recurrences of Fuzzy Relatons Ere n Bas 1, Vedde Rezan Uslu 2, Ufuk Yolcu 1, Erol Egroglu 2,* 1 Department of Statstcs, Gresun Unversty, Gresun, 28000, Turkey 2 Department of Statstcs, Unversty of OndokuzMays, Samsun, 55139, Turkey Abstract Fuzzy tme seres approaches are used when the observatons of tradtonal tme seres approaches contan uncertanty. Besdes, fuzzy tme seres approaches do not need the assumptons vald for tradtonal tme seres approaches. Generally, fuzzy tme seres methods consst of three stages. These are fuzzfcaton, determnaton of fuzzy relatons and defuzzfcaton stages, respectvely. All these stages of fuzzy tme seres are very mportant on the forecastng performance of the model. There are many studes that contrbute for each stage n the lterature. In ths study, we contrbuted the fuzzfcaton and defuzzfcaton stages. In fuzzfcaton stage, we used Dfferental Evaluaton Algorthm to avod subectve udgments for determnng the nterval lengths and also as known; the forecastng performance may be mproved f the fuzzy relatons must be occurred, properly. From ths pont of vew, we take nto account the recurrence numbers of fuzzy relatons n defuzzfcaton stage. Then, our proposed method has been appled to the real data sets whch are often used n other studes n lterature. The results are compared to the ones obtaned from other technques. Thus t s concluded that the results present superor forecasts performance. Keywords Fuzzy Tme Seres, Recurrence Numbers, Forecastng, Dfferental Evaluaton Algorthm. 1. Introducton Fuzzy tme seres procedures do not requre the assumptons such as the large sample and that the model s true. Recent studes are about fuzzy tme seres procedures snce they do not requre the strct assumptons and generally provde remarkable forecastng performances. The fuzzy set was frstly ntroduced by Zadeh[1] and ths concept has found many applcaton areas snce then. Fuzzy tme seres were ntroduced frstly n the studes of[2, 3, 4]. These proposed fuzzy tme seres technques n lterature generally consst of the three stages; these are fuzzfcaton, determnng of fuzzy relatons and defuzzfcaton. In the lterature, the decomposton of unverse of dscourse was mostly used n the fuzzfcaton stage and ntervals of t was determned arbtrarly the studes of[2, 3, 4, 5,6]. In addton, Huarng[7] has put forward the mportance of the nterval length on the forecastng performance and proposed two new technques based on the mean and the dstrbuton n order to fnd ntervals. Egroglu et al.[8, 9] have suggested formng the problem of fndng ntervals as an optmzaton * Correspondng author: erole1977@yahoo.com (Erol Egroglu) Publshed onlne at Copyrght 2013 Scentfc & Academc Publshng. All Rghts Reserved problem. Chen and Chung[10] and Lee et al.[11] used genetc algorthm to fnd the nterval lengths, and also Fu et al.[12], Huang et al.[13], Kuo et al.[14, 15], Davar et al.[16], Park et al.[17] and Hsu et al.[18] used the partcle swarm optmzaton. Besdes these studes, Cheng et al.[19] and L et al.[20] used fuzzy C-means clusterng method n ther studes and also Egroglu et al.[21] used Gustafson-Kessel fuzzy clusterng method n ths stage. In the stageof determnng of fuzzy relatons, whle Song and Chssom[2, 3, 4] used matr operatons, Chen[5] and some others were used the fuzzy logc relatons group table and also artfcal neural networks were used n the studes of[7, 22, 23, 24, 25, 26] for determnng the fuzzy relatons. The other studes n ths stage were proposed n the studes of[23, 27, 28, 29]. In defuzzfcaton stage, studes n the lterature mostly used the centrod method. Chen[5], Huarng[7]. Cheng et al.[30], Aladag et al.[28] preferred to use adaptve epectaton method n the defuzzfcaton process. The determnng of the fuzzy relatons s very mportant for the model structure. Besdes, the forecastng performance may be mproved f the fuzzy relatons defned well. Fro m ths pont of vew, we used Dfferental Evaluaton Algorthm (DEA) n fuzzfcaton stage to avod subectve decsons and also we amed to obtan more realstc

2 76 Eren Bas et al.: A Fuzzy Tme Seres Analyss Approach by Usng Dfferental Evoluton Algorthm Based on the Number of Recurrences of Fuzzy Relatons forecasts by usng fuzzy relatons recurrence numbers. Because, usng recurrence numbers of fuzzy relatons s mportant as well as fuzzy relatons occur or not. In ths study, a fuzzy tme seres approach that uses DEA n fuzzfcaton stage and takes nto account the recurrence numbers of fuzzy relatons was proposed and also the proposed method was supported by the applcatons and t s superor forecastng performance was shown. The rest part of the paper can be outlned as below: The fundamental defntons of fuzzy tme seres has been gven n Secton 2. Short nformaton about DEAhas been gven n Secton 3. In Secton 4, the proposed method has been ntroduced. In Secton 5, the results from the applcaton of the proposed method to three real lfe data sets have presented. In secton 6, dscussonshave been presented and fnally n secton 7, conclusons havebeen presented. 2. Fuzzy Tme Seres The defnton of fuzzy tme seres was frstly ntroduced by Song and Chssom[2, 3, 4]. In contrast to conventonal tme seres methods, varous theoretcal assumptons do not need to be checked n fuzzy tme seres approaches. The most mportant advantage of the fuzzy tme seres approaches s to be able to work wth a very small set of data. Let U be the unverse of dscourse,where U = { u1, u2,, u n }. A fu zzy set, A of U can be defned as, A = µ A ( u1)/ u1+ µ A ( u2)/ u2 + + µ A ( un)/ u n (1) Where µ A s the membershp functon of the fuzzy set A and µ A ; [0,1] U. In addton to, µ A ( u ), = 1,2,, n denotes s a generc element of fuzzy set A ; µ A ( u ), s the degree of belongngness of u to A ; µ A ( u ) [0,1]. Defnton 1 Let Y ( t)( t =,0,1,2, ), a subset of real numbers, be the unverse of dscourse by whch fuzzy sets f (t) are defned. If F (t) s a collecton of f 1( t), f2 ( t), then F (t) s called a fuzzy tme seres defned on Y(t). Defnton 2 Fuzzy tme seres relatonshps assume that F(t) s caused only by F( t 1), then the relatonshp can be epressed as: F ( t) = F( t 1) * R( t, t 1), whch s the fuzzy relatonshp between F(t) and F( t 1), where * represents as an operator. To sum up, let F ( t 1) = A and F ( t) = A. The fuzzy logcal relatonshp between F(t) and F( t 1), can be denoted as A A where A (current state) refers to the left-hand sde and A (net state) refers to the rght-hand sde of the fuzzy logcal relatonshp. Furthermore, these fuzzy logcal relatonshps can be grouped to establsh dfferent fuzzy relatonshp. 3. Dfferental Evaluaton Algorthm DEA was proposed by Prce and Storn[31]. DEA s a heurstc algorthm based on the populaton. It has some operators such as mutaton and crossover operatons. These operators are used to create new generatons. At the end of the process, canddate solutons are found by usng some mathematcal operatons and these solutons are compared wth the current solutons n the populaton. The best soluton s transferred to the new generaton accordng to the evaluaton functon and the best chromosome s taken as the optmal soluton. DEA and ts operators dscussed n more detal n the secton of proposed method and also those who want more nformaton can look the study of Prce and Storn[31]. 4. Proposed Method As t s well known that all stages of the fuzzy tme seres approaches nfluences very much ntensvely on the forecastng performance of the appled model and researchers have used dfferent technques to make a contrbuton to each stages. In recent years, artfcal ntellgence algorthms have been used n fuzzfcaton stage by the researchers and also the researchers have generally preferred to use the centrod method n the last stage so far and also the stage of determnng of the fuzzy relatons s very mportant as well as the other stages of fuzzy tme seres approaches. Besdes, the forecastng performance may be mproved f the fuzzy relatons defned well. From ths pont of vew; we have notced that none of them have consdered how many tmes a fuzzy relaton has occurred. To overcome ths problem, the weghts can be gven for each fuzzy relaton accordng to ther recurrences. Then, these weghts can be used n the defuzzfcaton stage. Because usng recurrence numbers of fuzzy relatons s mportant as well as fuzzy relatons occur or not and the forecastng performance may be mproved wth the help of usng these recurrence numbers In ths study, a fuzzy tme seres method that uses DEA n fuzzfcaton stage and takes nto consderaton of recurrence numbers of fuzzy relatons n defuzzfcaton stage to obtan more realstc forecasts has been proposed. The man advantages of proposed method are as follows: The nterval lengths are determned by avodng subectve decsons because of usng DEA. A more fleble soluton process s provded by obtanng f nterval length nstead of constant length nterval. The most basc element n the structure of the model s fuzzy relatons. To obtan forecasts as well as the fuzzy relatons, usng ther recurrence numbers cause to obtan

3 Amercan Journal of Intellgent Systems 2013, 3(2): more realstc forecasts. The search space has been dfferentated by usng DEA and new generatons are obtaned. Brefly, we can summarze our proposed method as below. Algorthm. Step 1 D mn and D ma are the mnmum and mamum values of tme seres, respectvely and the unverse of dscourse s defned n Equaton 2. U = [ Dlower = Dmn D1, Dupper = Dma + D2 ] (2) D1and D 2 are the numbers determned arbtrarly. If genes are shown wth ( = 1,2,,( m 1)) a chromosome structure of DEA can be shown as follows m 1 F gure 1. A chromosome st ruct ure of DEA Accordng to the values of these genes, the parts of unverse of dscourse,.e., sub-ntervals are shown as follows: u1 = [ Dlower, 1], u2 = [ 1, 2], um = [ m 1, Dupper ] (3) Step 2 The generaton of ntal populaton. n k,, s to show the n. gene of k. chromosome of. generaton n, k, = 0 = Dlower + randn[0,1]* ( Dupper Dlower ) (4) The genes produced by chromosomes are sorted by ascendng order. Step 3 For each chromosome n the populaton, the root of the mean squared error (RMSE) selected as the evaluaton functon s calculated by applyng the steps from 3.1 to 3.5. Step 3.1 mm ntervals based on the values of genes n chromosomes are used to form fuzzy sets as below. A = a u + a / u + + a / u, 1,2, m (5) a 1 / m m = here, k s the membershp degrees and t s shown n equaton 6. 1, k = a k = 0.5, k = 1, + 1 0, otherwse The observatons of crsp tme seres are converted to the fuzzy sets n whch the nterval n whch the correspondng observaton s ncluded, has got the hghest degrees of membershp value. Step 3.2 Obtanthe FLRs and FLRG tables. For eample, when we observe the relaton such as F ( t 1) = A and F ( t) = A for any tme t ths fuzzy logc relaton s represented by A A. In the whole seres f we get the relaton F ( t 1) = A and F ( t) = Ak (6) for any tme t then we epress the fuzzy logc relaton as. Also we save the number of how many tmes A A A k the fuzzy logc relaton such as the weghts w. A A s occurred, nto Step 3.3 Obtan the fuzzy forecasts. The fuzzy forecasts are obtaned wth respect to the fuzzy logc relaton table. If F ( t 1) = A and there s a relaton A A such as n the fuzzy logc relaton table then the fuzzy forecast wll be A. If F ( t 1) = A and there s a A A A relaton such as k n the fuzzy logc relaton table then the fuzzy forecast wll be A, Ak. If F ( t 1) = A and A Empty n the fuzzy logc relaton table then the fuzzy forecast wll be A. Step 3.4 Defuzzfy the fuzzy forecasts. The weghts ww obtaned from the fuzzy logc relaton table are used n the defuzzfcaton stage. For eamp le; If F ( t 1) = A and there ests the relaton wll be A A n the table then the defuzzfed forecast m whch s the mdpont of u whch s the subnterval of the fuzzy set A wth the largest membershp degree. That s, we don t regard how many tmes that relaton s repeated n the table. If F ( t 1) = A and there ests the relaton A A, A k and ww s the number of how many tmes the relaton AA AA s repeated n the whole tme seres and w s the number of how many tmes the relaton k A s repeated then the defuzzfed forecast s A k calculated as below. w m + w m k k ˆ t = (7) w + wk If F ( t 1) = A and there ests the relaton A Empty n the fuzzy logc relaton table then the defuzzfed forecast wll be m whch s the mdpont of the subnterval u whch s the fuzzy set A wth the largest membershp degree. Step 3.5 Let t be the orgnal tme seres and ˆ t be ts defuzzfed forecasts wth the nobservatons. RMSE s calculated by the equaton 8. 1 RMSE = T = t 1 Step 4 utaton and Crossover operatons. T ( t ˆt) (8)

4 78 Eren Bas et al.: A Fuzzy Tme Seres Analyss Approach by Usng Dfferental Evoluton Algorthm Based on the Number of Recurrences of Fuzzy Relatons Mutaton and Crossover operatons are appled to for each chromosome(chr) n the ntal populaton, respectvely. Step 4.1 The applcaton of Mutaton operaton For applyng mutaton operaton n DEA, frstly we choose four chromosomes. The frst one of these four chromosomes s called as current chromosome and the remanng three chromosomes are selected randomly ecept current chromosome. The frst two of these selected three chromosomes are subtracted each other and t s called as the dfference vector.then, dfference vector s multply by F and a new chromosome s obtaned (In general the parameter F gets values between 0 and 2. We take ths F value as 0.8 whch s general value n the lterature n ths study). Ths new chromosome s called as the weghted dfference vector. The weghted dfference vector s summed wth the thrd chromosome and mutaton s completed. Ths new created chromosome s called as the total vector. Chromosome 7 Chromosome Dfference vector Weghted Dfference vector *F= Weghted Dfference Chromosome 1 Total vector vector F gure 2. An eample of mutaton operaton Table 1. An eample of a random populaton Chr Chr Chr Chr Chr Chr Chr Thus, the chromosome to be used n the crossover operaton s created wth the help of mutaton operaton. To better understand mutaton operaton, let s look at the Fgure 2 and take a populaton wth 7 chromosomes and the unverse of dscourse s U = [13000,20000] as gven n Table 1. and assume that the Chromosome 2 s the chromosome to be mutated and Chromosome 6, Chromosome 7, Chromosome 1 are the chromosomes selected randomly ecept Chromosome 2. Step 4.2 The applcaton of Crossover. To apply Crossover operaton, the total vector obtaned from at the end of the mutaton operaton s compared wth current chromosome and nomnee chromosome s obtaned. Whle nomnee chromosome s obtaned, each gene of total vector and current chromosome s evaluated one by one. Frst at all, a crossover rate (cor) s determned. Then, a random number s generated between 0 and 1 wth the help of unform dstrbuton. If ths random number s smaller than the crossover rate, the gene s taken from total vector. If t s not, the gene s taken from currentchromosome and nomnee chromosome s generated and the ftness value of nomnee chromosome s calculated. Then, let s determne the (cor) rate. For eample let s take t 0.10 than generate random numbers for each gene, respectvely. (0.01, 0.08, 0.15, 0.20, 0.16) and obtan the nomnee chromosome. An eample of crossover has been gven n Fgure 3. Total vector Chromosome 2 Nomnee chromosome F gure 3. An eample of Crossover Step 5 The comparson of ftness values Nomnee chromosome and current chromosome are compared n terms of ftness values. The chromosome whch has the smaller RMSE value used as evaluaton functon s transferred to new generaton. For eample let s assume that the RMSE value of Nomnee chromosome s sma ller than chromosome 2 then, nomnee chromosome s transferred to the new generaton lke shown n Table 2. Table 2. An eample of creaton of a new generaton Chr Chr Chr Chr Chr Chr Chr

5 Amercan Journal of Intellgent Systems 2013, 3(2): All these operatons are appled to each chromosome n ntal populaton ndvdually. Step 6 Steps 3-5 are repeated as much as a predetermned number of teratons. 5. Applcaton In order to show the performance of the proposed method ths s appled to three dfferent tme seres data. The results are compared wth the results from the methods whch are already n fuzzy tme seres lterature wth regards to RMSE and Mean Absolute Percent Error (MAPE) crtera. n 1 t ˆ t MAPE = *100 n (9) t= 1 t For each tme seres data DEA parameters are defned as follows. cn s epermented as from 10 to 100 wth ncrement 10 cor s epermented respectvely 0.1 to 1 wth ncrement 0.1 m s epermented as from 5 to 20 wth the ncrement 1. For all possble case, DEA s eecuted 100 tmes n MATLAB. At the end of the process, we obtaned 1600 dfferent solutons. Then the parameters (m, cn, cor) wth the smallest RMSE value were taken as the best soluton among these solutons Enrollment Data The performance of proposed method s evaluated separately for both test and tranng sets for enrollment data between the years 1971 and Frstly all data s used for tranng set lke almost all studes n the lterature and then the last three observatons of enrollment data s taken as test set and t s compared wth the other studes n the lterature. It s clearly seen that our proposed method has superor forecastng performance. As a frst eperment, the proposed method s solved for tranng data. We conclude that the best result s obtaned n the case where m=19, cn=80, cor=0.9. Table 3 presents the all results, whch nclude forecasts and the RMSE and MAPEvalues, obtaned from the proposed method and the other methods proposed n lterature, comparatvely. These results from both are belongng to the best case. Addtonally, a comparatve presentaton of enrollments forecasts n terms of RMSE values for some other methods s gven Table 4. As a second eperment, the proposed method s solved for test data. We conclude that the best result s obtaned n the case where m=16, cn=60, cor=0.3. Table 5 presents the all results, whch nclude RMSE values, obtaned from the proposed method and the other methods proposed n lterature, comparatvely Tafe Data In the second case, the proposed method was appled to TAIFEX data whose observatons are between and The last 16 observatons are used for test set, respectvely. We conclude that the best result s obtaned n the case where m=12, cn=70, cor=0.9. Table 6 presents the all results, whch nclude forecasts and the RMSE and MAPE values, obtaned from the proposed method and the other methods proposed n lterature, comparatvely. These results from both are belongng to the best case. Table 3. A comparatve presentaton of enrollments forecasts for tranng data [32] [33] [34] (MEP A) [34] (T FA) [35] [36] [7] The Proposed Method Table 4. A comparatve of enrollments forecasts for tranng set n terms of RMSE [3] [5] [27] [30] [23] [9] [8] [21] [37] Proposed Met hod

6 80 Eren Bas et al.: A Fuzzy Tme Seres Analyss Approach by Usng Dfferental Evoluton Algorthm Based on the Number of Recurrences of Fuzzy Relatons TAIFEX Test Dat a Table 5. The comparson of the results for test set data Meth ods RMSE [2] [4] [27] [5] [7] [38] [9] [19] [21] Propose d me thod Table 6. The comparson of the results for test data [39] [11] [23] [18] [40] Propose d Method RMSE MAPE Car Road Accdent n Belgum Data In the thrd case, the proposed method was mplemented on the tme seres data of klled n car road accdent n Belgum. We conclude that the best result s obtaned n the case where m=20, cn=80, cor=0.2. Table 7 presents the all results, whch nclude forecasts and the RMSE and MAPE values, obtaned from the proposed method and the other methods proposed n lterature, comparatvely. These results from both are belongng to the best case. As seen n Table 7, the method we propose gves the best result wth respect to the forecastng performance. 6. Dscussons Usng recurrence numbers of fuzzy relatons s mportant as well as fuzzy relatons occur or not.and also the forecastng performance may be mproved wth the help of usng these recurrence numbers of fuzzy relatons Besdes, more relable results may be obtaned by gvng weghts as the number of repettons of recurrent fuzzy relatons. Besdes, the contrbuton of fuzzy relatons recurrent more s more than the fuzzy relatons recurrent less. We also consdered ths contrbuton and gave more weghts to the fuzzy relatons recurrent more. Table 7. A comparatve presentaton of klled forecasts and the actual observatons by varous methods Year Act ual [11] [41] [42] Propose d RMSE MAPE Conclusons Fuzzy tme seres forecastng methods have attracted much attenton n recent years. Dfferent approaches have been proposed for the each stage of fuzzy tme seres approaches. The stage of determnng of fuzzy relaton s very mportant for fuzzy tme seres approachesbecause t affects defuzzfcaton stage. Revealng the effects of fuzzy relatons, properly may mprove the forecastng performance. From ths pont of vew, we used DEA n fuzzfcaton stage to avod subectve decsons and also we consder the recurrence numbers of the repeated relatons n fuzzy logc relaton table whch s necessary n the defuzzfcaton stage n ths study. By usng recurrence numbers of fuzzy relatons the forecastng performance was mproved. The proposed method was supported by the real data sets analyss and ts superor forecastng performance was shown. We epect that n future studes researchers can concentrate on a dfferent optmzaton technque for fndng the weghts used n the defuzzfcaton stage and may use dfferent artfcal ntellgence technques n fuzzfcaton

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