An Algorithm Forecasting Time Series Using Wavelet

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1 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 04 ISSN (Prnt): ISSN (Onlne): An Algorthm Forecastng Tme Seres Usng Wavelet Kas Ismal Ibraheem,Eman Bacheer Abdelahad Department of Computer, Unversty of Mosul, Mosul, Iraq Department of Mathematc, Unversty of Mosul, Mosul, Iraq Abstract In ths paper we used the technque of wavelets wth fuzzy logc to forecast enrollment of Alabama unversty from 97 to 994 where data were taken analyzed by usng wavelets, then logc, we used the mean square error (MSE) to compare the forecastng results wth prevous dferent forecastng methods. The results were acceptable compared wth the results of prevous research. - Introducton Tme seres forecastng are wdely used n many areas, Such as economcs, nventory, systems, statstcs, etc. Forecastng s one of the mportant actvtes n busness Enrollment, fnance, etc. that helps n decson makng. The Classcal tme seres methods can not deal wth forecastng problems n whch the values of tme seres are lngustc terms represented by usng wavelet fuzzy logc. Wavelets turned out to be very useful when appled to many Problems ncludng analyss synthess of tme seres n both tme scale [3]. Foundatons of wavelet based analyss method were lad n the begnnng of the 0 th century. Back then, n the year 909 Hungaran mathematcan Alfred haar ntroduced hs two-state functon n appendx to hs doctoral thess publshed later on [] lately a very fast development of wavelet-based data mnng [8] technques may be observed. Fuzzy set theory s frst presented by Zadeh (95) for treatment of uncertan envronment nseveral felds. Partcularly fuzzy logc desgns are well accepted establshed for electronc devces later fuzzy sets found a broad applcaton potental on varous studyfelds [5]. Song Chssom [5] ntroduced a theory for fuzzy tme Seres appled fuzzy tme seres methods [], [7] that modeled the enrollments of the unversty of Alabama, n recent years a number of technques have been proposed for forecastng based on fuzzy set theory methods. Chen presented a method to forecast the enrollments of the unversty of Alabama based on fuzzy tme seres []. In [8] Huang extended Chen s work presented n [] used smpled calculatons wth the addton of heurstc rules to forecast the enrollments. The rest of ths paper s organzed as follows. In secton () we brefly revew wavelet transform, n secton (3) we deal wth Defntons of the fuzzy tme seres. In secton (4) we use the theory of wavelet transform Copyrght (c) 04 Internatonal Journal of Computer Scence Issues. All Rghts Reserved.

2 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 04 ISSN (Prnt): ISSN (Onlne): fuzzy tme seres to propose a new method to forecast the enrollment of the unversty of Alabama. In secton (5) we compare the forecastng result of the forecastng result of the proposed method wth the exstng methods, the conclusons are dscussed n ths secton. -Wavelet Transform Accordng to Fourer theory, a sgnal can be expressed as the sum of a seres of snes cosnes. Ths sum s also called a Fourer expanson (see Eq. ()).However, a serous drawback of the Fourer transform s that t only has frequency resoluton no tme resoluton. Therefore, we can denty all the frequences present n a sgnal, but we do not know when they are present. The wavelet theory s proposed [] F w f f t e wt dt t cos wt sn wtdt... Mathematcally, a wavelet can be defned as a functon wth a zero average: If s close to 0, t may be seen that only n an nterval (-T,T) Correspondng to ths values t are dferent than 0. Outsde of ths nterval they must equal 0 nterval(-t,t) s small compared to an nterval, on whch a wale functon s determned condton () mples that t has some postve values, t also has to have some negatve ones. If Haar functon, whch s a two-state functon of real varable (fg. ) t 0 Would be transformed nto H t 0 t 0 0 t otherwse t 0 0 t otherwse Then the resultng functon satsfes condton () (3) called Haar basc wavelet functon (fg. ) [] H t dt 0 t dt Where be a real functon of real varable t. Condton () means that for any from an nterval (0,) there s an nterval (-T,T) such that [] T T t dt Fg Haar funcaton Copyrght (c) 04 Internatonal Journal of Computer Scence Issues. All Rghts Reserved.

3 order IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 04 ISSN (Prnt): ISSN (Onlne): Defnton (3) Let R ( t, t ) be a frst-order model of F (t).if for any t, R ( t, t ) R( t, t ), then F (t) s called a tme-nvarant fuzzy tme seres. Otherwse, t s called a tme-varant fuzzy tme seres.[ 4 ] Fg. Moded functon-the frst basc wavelet functon 3-Fuzzy tme seres defnton In ths secton brefly summarzes basc fuzzy tme seres. Defnton () Assume t R ( t...0,,,...) Y to be a unverse of dscourse defned by the fuzzy set f ( t). F( t) consstng of t,,,... f, s defned as a fuzzy tme seres on Y t.at that t F can be understood as a lngustc varable, where t,,,... f are possble lngustc values of Defnton () F t. [4] If there exsts a fuzzy relatonshp R( t, t ) such that F( t) F( t ) R( t, t) where symbol ( ) s an operator, then F (t) s sad to be caused by F ( t ). The exstng relatonshp between F (t) F ( t ) can be denoted by the expresson F( t ) F( t). [4] Defnton (4) the[[frst [ Suppose F( t ) A [ F( t) A, unvarate fuzzy logcal relatonshp can be defned as where A A A, A are called the left-h sde (LHS) rght-h sde (RHS) of the fuzzy logcal relatonshp respectvely.[5] 4- Anew method proposed In ths secton, we present a new method to forecast the enrollments of the unversty of Alabama by usng wavelets transform fuzzy. Ths method s descrbed n three parttons. Partton one. The orgnal data were taken from the unversty of Alabama, they were applyng the Haar wavelet by usng algorthm A whch s descrbed below we get the followng data whch s explaned n table () Algorthm -A- Procedure Stard Decomposton k (c: array [...,... ] of reals) for row to do k Decomposton( c [ row,... ] ) end for for col to k do Copyrght (c) 04 Internatonal Journal of Computer Scence Issues. All Rghts Reserved.

4 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 04 ISSN (Prnt): ISSN (Onlne): Decomposton ( c[..., col ] ) end for end procedure Table -- The orgnal data wavelet Year enrollment wavelet Partton two. After that the data whch we obtaned from the prevous partton were appled n fuzzy logc algorthm B whch s descrbed below, we get these data n table -3- Algorthm B- - Defne the unverse of dscourse U = a. b where a s the mnmum value lttle less than t of the wavelet data obtaned by partton one b s the maxmum value or lttle more than t of the wavelet data obtaned by partton one. - Dvde U = a. b nto several equal-length ntervals. u,..., postve nteger number., u un where n s the 3- Defne ench fuzzy set based on the re dvded ntervals. 4- Determne fuzzy logcal relatonshps where s fuzzed enrollment wavelet of the year n s the fuzzed enrollments wavelet of the year n,table () shows fuzzy logcal relatonshp. Table -- fuzzy logcal relatonshps Dvde each nterval derved n step () nto four subntervals of equal length where the 0.5 pont 0.75 pont of each nterval are used as the upward downward forecastng ponts of the forecastng. We use the followng rules to determne whether the trend of the forecastng goes up, down or mddle. -If, (( W n Wn ) ( Wn Wn3 )) 0 the trend of the forecastng wll go up we use rule () to compute the forecast value, Copyrght (c) 04 Internatonal Journal of Computer Scence Issues. All Rghts Reserved.

5 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 04 ISSN (Prnt): ISSN (Onlne): If, (( Wn Wn ) ( Wn Wn3 )) 0 the trend of the forecastng wll go down we use rule () to compute the forecast value Rule () Let ( Wn Wn ) ( Wn Wn3 ) W * W n ) ( - If or ( Wn W / ) W then the trend of the forecastng of the nterval wll be upward (0.75 pont). W / W n ) ( - If or ( Wn W / ) then the trend of the forecastng of ths nterval wll be downward (0.5 pont), nether of these cases, then the trend of the forecastng of ths nterval wll be mddle. Rule () Let ( Wn Wn ) ( Wn Wn3 ) W / W n ) ( - or ( Wn W / ) W then the trend of the forecastng of ths nterval wll be downward(0.5 pont) W W n ) ( - If or ( Wn W ) then the trend of the forecastng of ths nterval wll be upward (0.75 pont) If nether of these cases, then the trend of the forecastng of ths nterval wll be mddle. [3] Partton three. After that the data whch taken from the second step were be back by usng algorthm nverse wavelet C. Algorthm C- Procedure Stard Redecom Poston k (c:array[..,.. ] of reals) for col to k do Red composton(c[.., col ]) end for for row to Red composton(c[ end for end procedure. do row.. k, ]) Table-3- Actual enrollment forecastng enrollment by Year fuzzy wavelet of the Unversty of Alabama Enrollment Fuzzy Set Forecast by Fuzzy Wavelet e Copyrght (c) 04 Internatonal Journal of Computer Scence Issues. All Rghts Reserved.

6 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 04 ISSN (Prnt): ISSN (Onlne): Table -4- A comparson of the forecastng results of dferent forecastng methods Year MSE Enrollment Song[] Song[7] Chen[] Hwang[8] Huarng[7] Chen[] Jlan[0] Proposed method Copyrght (c) 04 Internatonal Journal of Computer Scence Issues. All Rghts Reserved.

7 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 04 ISSN (Prnt): ISSN (Onlne): Concluson : In ths paper we proposed a new algorthm to forecast enrollment of Alabama unversty from 97 to 994, where we used wavelet wth logc appled ths method to predct these data. Table -3- contans regstraton data real values forecasted by usng the method descrbed prevously. Table -4- ncluded a comparson of predctons of the proposed method wth prevous methods. Researchers formerly used mean square error (MSE) to compare the forecastng results of dferent methods. Results obtaned usng ths method were good compared wth the methods used prevously ; except for three values where error rate appeared to be of farly large compared wth other values. Each value appeared n the range of three groups nto whch these data are. Refrans [] Chen, S.N. "Forecastng enrollments based on fuzzy tme seres", Set System, Vol. 8, 99, pp [] Chen, S.M. "Forecastng enrollments based on hgh-order fuzzy tmeseres". Cybernetcs Systems: An Internatonal Journal, Vol. 33, 00, pp. -. [3] Chen, S.M. Hsu, C.C."A new method to forecast enrollmentsusng fuzzy tme seres". Internatonal Journal of appled scence engneerng, Vol. 3, No., 004, pp [4] Chen, S. Chung, N. " Forecastng enrollments of students by usng fuzzy tme seres genetc algorthms", nformaton management scences, Vol. 7, No. 3, 00, pp. -7. [5] Duru, O. Yoshda, S. "Fuzzy tme seres methods for shppngfreght markets". Lsbon, portual, vol., no. 5, 00, pp. -3. [] Harr, A. "Zur theore der orthogonalen funkon systeme", Math.Ann. Vol. LI. 90, pp [7] Huarng, K. "Heurstc models of fuzzy tme seres for forecastng",fuzzy sets systems, 00, 3: pp [8] Hwang, J.R., Chen, S.M. Lee, C.H. "Hlng forecastng proble- ms usng fuzzy tme seres". Fuzzy sets systems, Vol. 00,pp [9] Jasm, H.T., Ibraheem, K.I. Salm, A.G.J. "A hybrd algorthms toforecast enrollment based on genetc algorthms fuzzy tmeseres", Iat Frst Onlne Publcaton, 03. [0] Jlan, T.A., Burney, S.M.A. Ardl, C. "Fuzzy metrc approachfor fuzzy tme seres forecastng based on frequency densty based parttonng proceedngs of world academy of scence, engneerng technology". Vol. 3, 007, pp [] Kozlowsk, B."Tme seres denosng wth wavelet transform", Journal of telecommuncaton nformaton technology, Vol. 3, 005, pp [] Norgaard, C. "Fnancal tme seres forecastng usng mprovedwavelet neural network", Pedersen, 009. [3] Percval, D.B. Walden, A.T. "Wavelet methods for tme seres analyss", Cambrdge: Cambrdge unversty press, 000. [4] Sah, M. Degtarev, K.Y. "Forecastng enrollment model based on frst-order fuzzy tme seres" Proceedngs of world academy of scence, engneerng technology. Vol., 005, pp Copyrght (c) 04 Internatonal Journal of Computer Scence Issues. All Rghts Reserved.

8 IJCSI Internatonal Journal of Computer Scence Issues, Vol., Issue, No, January 04 ISSN (Prnt): ISSN (Onlne): [5] Song, Q. Chssom, B.S. "Fuzzy tme seres ts models", Fuzzy set systems.vol. 54,993, pp [] Song, Q. Chssom, B.S. "Forecastng enrollments wth fuzzy tme seres ", Part, fuzzy sets systems,vol. 54, 993, pp. -9. [7]Song, Q. Chssom, B.S., " Forecastng enrollments wth fuzzytme seres", Part, fuzzy sets systems. Vol., 994, pp. -8. [8] Zhu, S., L, Q. Oghara, M., "Survey on wavelet applcatons n data mnng", Vol. 4, no., 003, pp Frst Author: Kas Ismal Ibraheem receved the Msc n Appled Mathemtcs Phd degree n Electrcal Engneerng from Gadah Mada Unversty Indonesa, n , respectvely. He s currently a head of department of Computer Scences faculty of educaton, Mosul Unversty, Iraq. Hs research nterests nclude artcal ntellgence,wavelet, numercal analyss, etc. Second Author: Eman Bacheer Abdelahad receved her Bsc degree n Mathematcs, Department of Mathmatcs, College of Educaton, Mosul Unversty, Iraq, 985. Her Msc Degree n Tme Seres from Department of Mathmatcs, College of Scences, 990. Her research nterests nclude forecastng Tme Seres by usng wavelet Fuzzy Tme Seres.She s author of research artcles. Copyrght (c) 04 Internatonal Journal of Computer Scence Issues. All Rghts Reserved.