ESTIMATION OF SMOOTHING CONSTANT WITH OPTIMAL PARAMETERS OF WEIGHT IN THE MEDICAL CASE OF A TUBE AND A CATHETER

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1 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// ESTIMATION OF SMOOTHING CONSTANT WITH OPTIMAL PARAMETERS OF WEIGHT IN THE MEDICAL CASE OF A TUBE AND A CATHETER Daisuke Takeyasu, Kazuhiro Takeyasu. The Open Universiy of Japan, - Wakaa, Mihama-Disric, Chia Ciy, , Japan. College of Business Adminisraion, Tokoha Universiy,5 Oouchi, Fuji Ciy, Shizuoka, , Japan ake@ homail.co.jp akeyasu@fj.okoha-u.ac.jp ABSTRACT In indusries, how o improve forecasing accuracy such as sales, shipping is an imporan issue. There are many researches made on his. In his paper, a hyrid mehod is inroduced and plural mehods are compared. Focusing ha he equaion of exponenial smoohing mehod(esm) is equivalen o (,) order ARMA model equaion, a new mehod of esimaion of smoohing consan in exponenial smoohing mehod is proposed efore y us which saisfies minimum variance of forecasing error. Generally, smoohing consan is seleced arirarily. Bu in his paper, we uilize aove saed heoreical soluion. Firsly, we make esimaion of ARMA model parameer and hen esimae smoohing consans. Thus heoreical soluion is derived in a simple way and i may e uilized in various fields. A mere applicaion of ESM does no make good forecasing accuracy for he ime series which has non-linear rend and/or rend y monh. A new mehod o cope wih his issue is required. In his paper, comining he rend removing mehod wih his mehod, we aim o improve forecasing accuracy. An approach o his mehod is execued in he following mehod. Trend removing y he cominaion of linear and nd order non-linear funcion and rd order non-linear funcion is carried ou o he sum oal medical daa of producion and impors of A Tue and A Caheer for hree cases (A Tue and A Caheer, The ue for serilized respiraory organs and a caheer, and The ue for serilized lood vessels and a caheer). The weighs for hese funcions are se 0.5 for wo paerns a firs and hen varied y 0.0 incremen for hree paerns and opimal weighs are searched. For he comparison, monhly rend is removed afer ha. Theoreical soluion of smoohing consan of ESM is calculaed for oh of he monhly rend removing daa and he non monhly rend removing daa. Then forecasing is execued on hese daa. The. new mehod shows ha i is useful for he ime series ha has various rend characerisics and has raher srong seasonal rend. The effeciveness of his mehod should e examined in various cases. Keywords: minimum variance, exponenial smoohing mehod, forecasing, rend, ue, caheer. INTRODUCTION Time series analysis is ofen used in such hemes as sales forecasing, sock marke price forecasing ec. Sales forecasing is ineviale for Supply Chain Managemen. Bu in fac, i is no well uilized in indusries. I is ecause here are so many irregular incidens herefore i ecomes hard o make sales forecasing. A mere applicaion of mehod does no ear good resul. The ig reason is ha sales daa or producion daa are no saionary ime series, while linear model requires he ime series as a saionary one. In order o improve forecasing accuracy, we have devised rend removal mehods as well as searching opimal parameers and oained good resuls. We creaed a new mehod and applied i o various ime series and examined he effeciveness of he mehod. Applied daa are sales daa, producion daa, shipping daa, sock marke price daa, fligh passenger daa ec. Many mehods for ime series analysis have een presened such as Auoregressive model (AR Model), Auoregressive Moving Average Model (ARMA Model) and Exponenial Smoohing Mehod (ESM) []-[4]. Among hese, ESM is said o e a pracical simple mehod. For his mehod, various improving mehod such as adding compensaing iem for ime lag, coping wih he ime series wih rend [5], uilizing Kalman Filer [6], Bayes Forecasing [7], adapive ESM [8], exponenially weighed Moving Averages wih irregular updaing periods [9], making averages of forecass using plural mehod [0] are presened. For example, Maeda [6] calculaed smoohing consan in relaionship wih S/N raio under he assumpion ha he oservaion noise was added o he sysem. Bu he had o calculae under supposed noise ecause he could no grasp

2 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// oservaion noise. I can e said ha i does no pursue opimum soluion from he very daa hemselves which should e derived y hose esimaion. Ishii [] poined ou ha he opimal smoohing consan was he soluion of infinie order equaion, u he didn show analyical soluion. Based on hese facs, we proposed a new mehod of esimaion of smoohing consan in ESM efore []. Focusing ha he equaion of ESM is equivalen o (,) order ARMA model equaion, a new mehod of esimaion of smoohing consan in ESM was derived. Furhermore, comining he rend removal mehod, forecasing accuracy was improved, where shipping daa, sock marke price daa ec. were examined [] -[9]. In his paper, uilizing aove saed mehod, a revised forecasing mehod is proposed. A mere applicaion of ESM does no make good forecasing accuracy for he ime series which has non-linear rend and/or rend y monh. A new mehod o cope wih his issue is required. Therefore, uilizing aove saed mehod, a revised forecasing mehod is proposed in his paper o improve forecasing accuracy. In making forecas such as producion daa, rend removing mehod is devised. Trend removing y he cominaion of linear and nd order non-linear funcion and rd order non-linear funcion is execued o he sum oal medical daa of producion and impors of A Tue and A Caheer for hree cases (A Tue and A Caheer, The ue for serilized respiraory organs and a caheer, and The ue for serilized lood vessels and a caheer). The weighs for hese funcions are se 0.5 for wo paerns a firs and hen varied y 0.0 incremen for hree paerns and opimal weighs are searched. For he comparison, monhly rend is removed afer ha. Theoreical soluion of smoohing consan of ESM is calculaed for oh of he monhly rend removing daa and he non monhly rend removing daa. Then forecasing is execued on hese daa. This is a revised forecasing mehod. Variance of forecasing error of his newly proposed mehod is assumed o e less han hose of previously proposed mehod. The new mehod shows ha i is useful especially for he ime series ha has sale characerisics and has raher srong seasonal rend and also he case ha has non-linear rend. The res of he paper is organized as follows. In secion, ESM is saed y ARMA model and esimaion mehod of smoohing consan is derived using ARMA model idenificaion. The cominaion of linear and non-linear funcion is inroduced for rend removing in secion. The Monhly Raio is referred in secion 4. Forecasing is execued in secion 5, and esimaion accuracy is examined..description OF ESM USING ARMA MO DEL [] following equaion. Here, ˆ xˆ x xˆ x xˆ xˆ x : forecasing a x : realized value a : smoohing consan 0 () is re-saed as l0 l x l () xˆ () By he way, we consider he following (,) order ARMA mo del. x x e e () Generally, p, q order ARMA model is saed as x p q aix i e i j e j j (4) Here, x : Sample process of Saionary Ergodic Gaussian P rocess e x,,, N, :Gaussian Whie Noise wih 0 mean variance e MA process in (4) is supposed o saisfy converiiliy condii on. Uilizing he relaion ha E e e, e, 0 we ge he following equaion from (). xˆ x e (5) Operaing his scheme on +, we finally ge xˆ xˆ xˆ e x xˆ (6) If we se, he aove equaion is he same wih (), i.e., equaion of ESM is equivalen o (,) order ARMA model, or is said o e (0,,) order ARIMA model ecause s order AR parameer is [][]. Focusing ha he equaion of exponenial smoohing mehod(esm) is equivalen o (,) order ARMA model equaion, a new mehod of esimaion of smoohing consan in exponenial smoohing mehod is derived (See Appendix in deail). Finally we ge: In ESM, forecasing a ime + is saed in he

3 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// 4 4 (7) Thus we can oain a heoreical soluion y a simple way. Here mus saisfy 0 (8) in order o saisfy 0. Focusing on he idea ha he equaion of ESM is equivalen o (,) order ARMA model equaion, we can esimae smoohing consan afer esimaing ARMA model parameer. I can e esimaed only y calculaing 0h and s order auocorrelaion funcion..trend REMOVAL METHOD[] As ESM is a one of a linear model, forecasing accuracy for he ime series wih non-linear rend is no necessarily good. How o remove rend for he ime series wih non-linear rend is a ig issue in improving forecasing accuracy. In his paper, we devise o remove his non-linear rend y uilizing non-linear funcion. As rend removal mehod, we descrie he cominaion of linear and non-linear funcion. [] Linear funcion We se as a linear funcion. [] Non-linear funcion We se y a x (9) y a (0) x x c y a x x c x () d as a nd and a rd order non-linear funcion. [] The cominaion of linear and non-linear funcion We se a x a x x y () c a x β a x x c x y () β d y γ a x γ a x x c γ a x x c x d (4) as he cominaion of linear and nd order non-linear and rd order non-linear funcion. Here,, β β, γ ( γ γ ). Comparaive discussion concerning (), () and (4) are descried in secion 5. 4.MONTHLY RATIO[] For example, if here is he monhly daa of L years as saed ellow: i,, L j,, x ij Where, x ij R in which j means monh and i means year and x ij is a shipping daa of i-h year, j-h monh. Then, monhly raio x~ j j,, is calculaed as follows. L xij ~ L i x j L (5) xij L i j Monhly rend is removed y dividing he daa y (5). Numerical examples oh of monhly rend removal case and non-removal case are discussed in 5. 5.FORECASTING THE SHIPPING DATA OF MANUFACTURER 5. Analysis Procedure Sum oal daa of A Tue and A Caheer for hree cases (A Tue and A Caheer, The ue for serilized respiraory organs and a caheer, and The ue for serilized lood vessels and a caheer) from January 007 o Decemer 009 are analyzed. These daa are oained from he Annual Repor of Saisical Invesigaion on Saisical-Survey-on-Trends-in-Pharmaceuical-Producion y Minisry of Healh, Laour and Welfare in Japan. Firs of all, graphical chars of hese ime series daa are exhiied in Fig.,,.

4 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// Consan wih minimum variance of forecasing error is esimaed. Then sep forecas is execued. Thus, daa is shifed o nd o 5h and he forecas for 6h daa is execued consecuively, which finally reaches forecas of 6h daa. To examine he accuracy of forecasing, variance of forecasing error is calculaed for he daa of 5h o 6h daa. Final forecasing daa is oained y muliplying monhly raio and rend. Forecasing error is expressed as: xˆ x (6) i i i Fig. : Sum oal daa of A Tue and A Caheer N (7) N i Variance of forecasing error is calculaed y: i N i (8) N i 5. Trend Removing Trend is removed y dividing original daa y (),(),(4). The paerns of rend removal are exhiied in Tale. Fig. : Sum oal daa of he ue for serilized respirao ry organs, and a caheer Tale : The paerns of rend removal Paern, are se 0.5 in he equaion () Paern, are se 0.5 in he equaion () Paern is shifed y 0.0 incremen in () Paern4 is shifed y 0.0 incremen in () Paern5 γ and γ are shifed y 0.0 incremen in (4) Fig. : Sum oal daa of he ue for serilized lood vessels, and a caheer Analysis procedure is as follows. There are 6 monhly daa for each case. We use 4 daa( o 4) and remove rend y he mehod saed in. Then we calculae monhly raio y he mehod saed in 4. Afer removing monhly rend, he mehod saed in is applied and Exponenial Smoohing In paern and, he weigh of,,, are se 0.5 in he equaion (),(). In paern, he weigh of is shifed y 0.0 incremen in () which saisfy he range In paern4, he weigh of is shifed in he same way which saisfy he range In paern5, he weigh of and are shifed y 0.0 incremen in (4) which saisfy he range 0. 00, The es soluion is seleced which minimizes he variance of forecasing error. Esimaion resuls of coefficien of (9), (0) and () are exhiied in Tale. Esimaion resuls of weighs of (), () and (4) are exhiied in Tale. 4

5 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// Tale : Coefficien of (9),(0) and () A Tue and A Cahee r The ue for serilize d respira ory organs, and a caheer The ue for serilize d lood vessels, and a caheer s nd rd a a c a c d Tale Weighs of (), () and (4) A Tue and A Caheer The ue for serilized respiraory organs, and a caheer The ue for serilized lood vessels, and a caheer Monhly raio Paern Paern Paern Paern4 Paern5 Used No used Used No used Used No used Graphical char of rend is exhiied in Fig. 4, 5, 6 for he cases ha monhly raio is used. 5

6 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// Fig. 4: Trend of A Tue and A Caheer Fig. 5: Trend of The ue for serilized respiraory organs, and a caheer 6

7 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// Fig. 6: Trend of The ue for serilized lood vessels, and a caheer 5. Removing rend of monhly raio Afer removing rend, monhly raio is calculaed y he mehod saed in 4. Calculaion resul for s o 4h daa is exhiied in Tale 4 hrough 8. Tale 4: Monhly raio (Paern) Monh A Tue and A Caheer The ue for serilized respiraory organs, and a caheer The ue for serilized lood vessels, and a caheer Tale 5: Monhly raio (Paern) Monh A Tue and A Caheer The ue for serilized respiraory organs, and a caheer The ue for serilized lood vessels, and a caheer

8 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// Tale 6: Monhly raio (Paern) Monh A Tue and A Caheer The ue for serilized respiraory organs, and a caheer The ue for serilized lood vessels, and a caheer Tale 7: Monhly raio (Paern4) Monh A Tue and A Caheer The ue for serilized respiraory organs, and a caheer The ue for serilized lood vessels, and a caheer Tale 8: Monhly raio (Paern5) Monh A Tue and A Caheer The ue for serilized respiraory organs, and a caheer The ue for serilized lood vessels, and a caheer Esimaion of Smoohing Consan wih Minimum Variance of Forecasing Error Afer removing monhly rend, Smoohing Consan wih minimum variance of forecasing error is esimaed uilizing (7). There are cases ha we canno oain a heoreical soluion ecause hey do no saisfy he condiion of (A-9). In hose cases, Smoohing Consan wih minimum variance of forecasing error is derived y shifing variale from 0.0 o 0.99 wih 0.0 inerval. Calculaion resul for s o 4h daa is exhiied in Tale 9. A Tue and A Caheer Tale 9: Esimaed Smoohing Consan wih Minimum Variance Monhly raio Paern Paern Used No used The ue for serilized respiraory organs, and a Used

9 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// caheer No used The ue for serilized lood vessels, and a caheer Used No used A Tue and A Caheer The ue for serilized respiraory organs, and a caheer The ue for serilized lood vessels, and a caheer Monhly raio Paern Paern4 Paern5 Used No used Used No used Used No used Forecasing and Variance of Forecasing Error Uilizing smoohing consan esimaed in he previous secion, forecasing is execued for he daa of 5h o 6h daa. Final forecasing daa is oained y muliplying monhly raio and rend. Variance of forecasing error is calculaed y (8). Forecasing resuls are exhiied in Fig. 7, 8, 9 for he cases ha monhly raio is used. Fig. 7: Forecasing Resuls of A Tue and A Caheer 9

10 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// Fig. 8: Forecasing Resuls of The ue for serilized respiraory organs, and a caheer Fig. 9: Forecasing Resuls of The ue for serilized lood vessels, and a caheer Variance of forecasing error is exhiied in Tale 0. 0

11 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// Tale 0: Variance of Forecasing Error A Tue and A Caheer The ue for serilized respiraory organs, and a caheer The ue for serilized lood vessels, and a caheer Monhly raio Used No used Used No used Used No used Paern Paern Paern Paern4 Paern5 8,5,09,85,,008,4,005, 6,9,856,4,,96,4,90,90,00, , , ,446,544,9,,4,894,908,,05,0,57,,6,46,8,,487, , ,750.0,90,87,60 6,77,,7 0,886,79,8 0,966,944, 0,886,79, ,967,597,50 4,8,9,4 8,76,05,96 8,8,9,5 8,76,05, ,948,,468,,46,008,9,,76,907,50,,6,456,56,,60, ,8.70 4, ,40,6,0,4,660,0, 48,894,57, 506,00,80, 48,894,57, Remarks These ime series have non-linear rend and rend y monh. Applying only an ESM does no make good f orecasing accuracy. The ue for serilized respiraory organs, and a caheer had a good resul in s + nd order wih he case ha monhly raio is used. A Tue and A Caheer in s + nd + rd order wih he case ha monhly raio is no used. The ue for serilized lood vessels, and a caheer had good resul in s + nd order wih he cases ha monhly raio is no used. 6.CONCLUSION Focusing on he idea ha he equaion of exponenial smoohing mehod(esm) was equivalen o (,) order ARMA model equaion, a new mehod of esimaion of smoohing consan in exponenial smoohing mehod was proposed efore y us which saisfied minimum variance of forecasing error. Generally, smoohing consan was seleced arirar ily. Bu in his paper, we uilized aove saed heoreical soluion. Firsly, we made esimaion of ARMA model parameer and hen esimaed smoohing consans. Thus heoreical soluion was derived in a simple way and i migh e uilized in various fields. Furhermore, comining he rend removal mehod wih his mehod, we aimed o improve forecasing accuracy. A mere applicaion of ESM does no make good forecasing accuracy for he ime series which has non-linear rend and/or rend y monh. A new mehod o cope wih his issue is required. Therefore, uilizing aove saed mehod, a revised forecasing mehod is proposed in his paper o improve forecasing accuracy. An approach o his mehod was execued in he following mehod. Trend removal y a linear funcion was applied o he original Sum oal daa of A Tue and A Caheer for hree Sum oal daa of A Tue and A Caheer for hree cases (A Tue and A Caheer, The ue for serilized respiraory organs and a caheer, and The ue for serilized lood vessels and a caheer). The cominaion of linear and non-linear funcion was also inroduced in rend removing. For he comparison, monhly rend was removed afer ha. Theoreical soluion of smoohing consan of ESM was calculaed for oh of he monhly rend removing daa and he non monhly rend removing daa. Then forecasing was execued on hese daa. The ue for serilized respiraory organs, and a caheer had a good resul in s + nd order wih

12 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// he case ha monhly raio is used. A Tue and A Caheer in s + nd + rd order wih he case ha monhly raio is no used. The ue for serilized lood vessels, and a caheer had good resul in s + nd order wih he cases ha monhly raio is no used. Various cases should e examined hereafer. APPENDIX: Esimaion of smoohing consan in Expone nial Smoohing Mehod [] Comparing wih () and (4), we oain From (), (6), Therefore, we ge From aove, we can ge esimaion of smoohing consan af er we idenify he parameer of MA par of ARMA model. Bu, generally MA par of ARMA model ecome non-linear equ aions which are descried elow. Le (4) e p i ~ x x a x (A-) q j i i ~ x e e (A-) j j We express he auocorrelaion funcion of x~ as ~ r k and from (A-), (A-), we ge he following non-linear equaions which are well known []. ~ r k ~ r 0 0 qk e j0 q e j0 j k j j a a ( k q) ( k q ) (A-) (A-4) For hese equaions, recursive algorihm has een developed. In his paper, parameer o e esimaed is only, so i can e solved in he following way. From () (4) (A-) (A-4), we ge If we se q a ~ r0 e ~ r e 0 (A-5) ~ r ~ k k (A-6) r he following equaion is derived. (A-7) We can ge as follows. 4 (A-8) In order o have real roos, mus saisfy (A-9) From inveriiliy condiion, mus saisfy From (A-7), using he nex relaion, (A-9) always holds. As 0 0 is wihin he range of 0 Finally we ge 4 4 which saisfy aove condiion. (A-0)

13 Aug 0.Vol., No.4 ISSN0708 Inernaional Journal of Research In Medical and Healh Sciences 0 IJRMHS & K.A.J. All righs reserved hp:// REFERENCES [] Box Jenkins. (994) Time Series Analysis Third Edi ion,prenice Hall. [] R.G. Brown. (96) Smoohing, Forecasing and Pre dic on of Discree Time Series, Prenice Hall. [] Hidekasu Tokumaru e al. (98) Analysis and Measuremen Theory and Applicaion of Random daa Handling, Baifukan Pulishing. [4] Kengo Koayashi. (99) Sales Forecasing for Budgeing, Chuokeizai-Sha Pulishing. [5] Peer R.Winers. (984) Forecasing Sales y Exponenially Weighed Moving Averages, Managemen Science,Vol6,No., pp [6] Kasuro Maeda. (984) Smoohing Consan of Exponenial Smoohing Mehod, Seikei Universiy Repor Faculy of Engineering, No.8, pp [7] M.Wes and P.J.Harrison. (989) Baysian Forecasingand Dynamic Models,Springer-Verlag,New York. [8] Seinar Ekern. (98) Adapive Exponenial SmoohingRevisied, Journal of he Operaional Research Sociey, Vol. pp [9] F.R.Johnson. (99) Exponenially Weighed Moving Average (EWMA) wih Irregular Updaing Periods, Journal of he Operaional Research Sociey,Vol.44, No.7 pp [0] Spyros Makridakis and Roea L.Winkler. (98) Averages of Forecass ; Some Empirical Resuls, Managemen Science,Vol.9, No.9, pp [] Naohiro Ishii e al. (99) Bilaeral Exponenial Smoohing of Time Series, In.J.Sysem Sci., Vol., No.8, pp [] Kazuhiro Takeyasu and Keiko Nagaa.(00) Esimaion of Smoohing Consan of Minimum Variance wih Opimal Parameers of Weigh, Inernaional Journal of Compuaional Science Vol.4,No.5, pp Geneic Algorihm And Is Applicaion o Indusrial Daa, NCSP'0, Honolulu,Hawaii,USA [5] Kazuhiro Takeyasu, Keiko Nagaa, Kana Takagi. (00) Esimaion of Smoohing Consan of Minimum Variance wih Opimal Parameers of Weigh, NCSP'0, Honolulu,Hawaii,USA [6] Kazuhiro Takeyasu, Keiko Nagaa, Tomoka Kuwahara. (00) Esimaion of Smoohing Consan of Minimum Variance Searching Opimal Parameers of Weigh, NCSP'0, Honolulu,Hawaii,USA [7] Kazuhiro Takeyasu, Keiko Nagaa, Mai Io, Yuki Higuchi (00). A Hyrid Mehod o Improve Forecasing Accuracy Uilizing Geneic Algorihm, The h APEIMS, Melaka, Malaysia [8] Kazuhiro Takeyasu, Keiko Nagaa, Kaori Masumura. (0) Esimaion of Smoohing Consan of Minimum Variance and Is Applicaion o Sales Daa, JAIMS, Honolulu, Hawaii, USA [9] Hiromasa Takeyasu, Yuki Higuchi, Kazuhiro Takeyasu. (0) A Hyrid Mehod o Improve Forecasing Accuracy in he Case of Bread, Inernaional Journal of Informaion and Communicaion Technology Research, Vol., No., pp.804~8. [] Kazuhiro Takeyasu, Keiko Nagaa, Yuki Higuchi. (009) Esimaion of Smoohing Consan of Minimum Variance And Is Applicaion o Shipping Daa Wih Trend Removal Mehod, Indusrial Engineering & Managemen Sysems (IEMS),Vol.8,No.4, pp.57-6, [4] Kazuhiro Takeyasu, Keiko Nagaa, Yui Nishisako. (00) A Hyrid Mehod o Improve Forecasing Accuracy Uilizing