Estimation of Smoothing Constant with Optimal Parameters of Weight in the Medical Case of Blood Extracorporeal Circulation Apparatus

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1 Inernaional Journal of Engineering and Technology Volume No. 0, Ocoer, 0 Esimaion of Smoohing Consan wih Opimal Parameers of Weigh in he Medical Case of Blood Exracorporeal Circulaion Apparaus 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 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. Furhermore, 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 Blood exracorporeal circulaion apparaus for hree cases ( apparaus, Dialyzer, and Blood circui). 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, lood exracorporeal circulaion apparaus. INTRODUCTION 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 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. In his paper, uilizing aove saed mehod, a revised forecasing mehod is proposed. 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 Blood exracorporeal circulaion apparaus for hree cases ( apparaus, Dialyzer, and Blood circui). 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 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. ISSN: IJET Pulicaions UK. All righs reserved. 949

2 Inernaional Journal of Engineering and Technology (IJET) Volume No. 0, Ocoer, 0. DESCRIPTION OF ESM USING ARMA MO DEL [] a (7) In ESM, forecasing a ime + is saed in he 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 mod el. x x e e () Generally, p, q order ARMA model is saed as Here, : x x p q ai x i e i j e j j Sample process of Saionary Ergodic Gaussian Process,,, N, e :Gaussian Whie Noise wih 0 mean (4) e variance x MA process in (4) is supposed o saisfy converiiliy condiio n. 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 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 [][]. Comparing wih () and (4), we oain From (), (6), Therefore, we ge (6) From aove, we can ge esimaion of smoohing consan afer we idenify he parameer of MA par of ARMA model. Bu, ge nerally MA par of ARMA model ecome non-linear equaions which are descried elow. Le (4) e p i ~ x x a x (8) q j i i ~ x e e (9) j j We express he auocorrelaion funcion of x~ as ~ r k and from (8), (9), 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 ( k q) ( k q ) (0) 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) (7) (0), we ge If we se q a ~ r0 e ~ r e 0 () ~ r ~ k k () r he following equaion is derived. () We can ge as follows. 4 (4) In order o have real roos, mus saisfy a ISSN: IJET Pulicaions UK. All righs reserved. 950

3 Inernaional Journal of Engineering and Technology (IJET) Volume No. 0, Ocoer, 0 (5) From inveriiliy condiion, mus saisfy From (), using he nex relaion, (5) always holds. As 0 0 is wihin he range of 0 Finally we ge 4 4 (6) which saisfy aove condiion. Thus we can oain a heoreical soluion y a simple way. Here mus saisfy 0 (7) 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 rend removal mehod, we descrie he cominaion of linea r and non-linear funcion. [] Linear funcion We se as a linear funcion. [] Non-linear funcion We se y a (9) x x c y a (0) x x cx d as a nd and a rd order non-linear funcion. 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 () 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 () are descried in secion 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 (4) xij L i j Monhly rend is removed y dividing he daa y (4). Numerical examples oh of monhly rend removal case and non-removal case are discussed in FORECASTING THE SHIPPING DATA OF MANUFACTURER 5. Analysis Procedure Sum oal daa of Blood exracorporeal circulaion apparaus for hree cases ( apparaus, Dialyzer, and Blood y a x (8) circui) 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.,,. [] The cominaion of linear and non-linear funcion We se ISSN: IJET Pulicaions UK. All righs reserved. 95

4 Inernaional Journal of Engineering and Technology (IJET) Volume No. 0, Ocoer, 0 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 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 (5) i i i Fig. : Sum oal daa of apparaus N (6) N i Variance of forecasing error is calculaed y: i N i (7) N i 5. Trend Removing Trend is removed y dividing original daa y,(),(),(). The paerns of rend removal are exhiied in Tale. Fig. : Sum oal daa of Dialyzer 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 () Fig. : Sum oal daa of Blood circui Analysis procedure is as follows. There are 6 monhly daa for each case. We use 4 daa( o 4) and remove rend y he 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 () which saisfy he range 0. 00, The es soluion is seleced which minimizes he variance of forecasing error. Esimaion resuls of coefficien of (8), (9) and (0) are exhiied in Tale. Esimaion resuls of weighs of (), () and () are exhiied in Tale. Tale : Coefficien of (8),(9) and (0) a s nd rd a c a c d Hemodialy sis ISSN: IJET Pulicaions UK. All righs reserved. 95

5 Inernaional Journal of Engineering and Technology (IJET) Volume No. 0, Ocoer, 0 apparaus Dialyzer Blood circui Tale : Weighs of (), () and () apparaus Dialyzer Blood circui 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. Fig. 4: Trend of apparaus ISSN: IJET Pulicaions UK. All righs reserved. 95

6 Inernaional Journal of Engineering and Technology (IJET) Volume No. 0, Ocoer, 0 Fig. 5: Trend of Dialyzer Fig. 6: Trend of Blood circui 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 apparaus Dialyzer Blood circui ISSN: IJET Pulicaions UK. All righs reserved. 954

7 Inernaional Journal of Engineering and Technology (IJET) Volume No. 0, Ocoer, 0 Tale 5: Monhly raio (Paern) Monh apparaus Dialyzer Blood circui Tale 6: Monhly raio (Paern) Monh apparaus Dialyzer Blood circui Tale 7: Monhly raio (Paern4) Monh apparaus Dialyzer Blood circui Tale 8: Monhly raio (Paern5) Monh apparaus Dialyzer Blood circui 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 ( 6). There are cases ha we canno oain a heoreical soluion ecause hey do no saisfy he condiion of (5). 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. Tale 9: Esimaed Smoohing Consan wih Minimum Variance Monhly Paern Paern raio apparaus Dialyzer Used No used Used No used Blood circui Used ISSN: IJET Pulicaions UK. All righs reserved. 955

8 Inernaional Journal of Engineering and Technology (IJET) Volume No. 0, Ocoer, 0 No used apparaus Dialyzer Blood circui 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 (7). Forecasing resuls are exhiied in Fig. 7, 8, 9 for he cases ha monhly raio is used. Fig. 7: Forecasing Resuls of apparaus Fig. 8: Forecasing Resuls of Dialyzer ISSN: IJET Pulicaions UK. All righs reserved. 956

9 Inernaional Journal of Engineering and Technology (IJET) Volume No. 0, Ocoer, 0 Variance of forecasing error is exhiied in Tale 0. Fig. 9: Forecasing Resuls of Blood circui Tale 0: Variance of Forecasing Error Monhly raio Paern Paern Paern Paern4 Paern5 apparaus Used 40,565,40, ,4,890,78.980,980,07,8.009,980,07,8.009,980,07,8.009 No used 46,46,46, ,,65, ,65,677,7.7 46,65,677,7.7 46,65,677,7.7 Dialyzer Used,6,45,909, No used,,5,5,89.000,44,78,6, ,00,809,55, ,74,565,86,7.000,985,9,086,0.7 0,049,97,858,670.80,9,9,856,54.090,049,97,858, ,9,9,856, Blood circui Used 57,66,, ,695,798, ,,44, ,456,400, ,40,486, No used 97,449,7, ,765,945, ,075,48, ,859,085, ,075,48, Remarks Comparing he resul of Tale0 wih hose of Tale, we can oserve he following. apparaus had a good resul in s order wih he case ha monhly raio is used. Dialyzer in s + rd order wih he case ha monhly raio is no used. Blood exracorporeal circulaion apparaus had good resul in s + nd + rd order wih he cases ha monhly raio is 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 arirarily. 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. 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 Blood exracorporeal circulaion apparaus for hree cases ( apparaus, Dialyzer, and Blood circui). 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. Comparing he resul of Tale0 wih hose of Tale, we can ISSN: IJET Pulicaions UK. All righs reserved. 957

10 Inernaional Journal of Engineering and Technology (IJET) Volume No. 0, Ocoer, 0 oserve he following. apparaus had a good resul in s order wih he case ha monhly raio is used. Dialyzer in s + rd order wih he case ha monhly raio is no used. Blood exracorporeal circulaion apparaus had good resul in s + nd + rd order wih he cases ha monhly raio is used. Various cases should e examined hereafer. REFERENCES [] Box Jenkins. (994) Time Series Analysis Third Ediion, Prenice Hall. [] R.G. Brown. (96) Smoohing, Forecasing and Predicion 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 ISSN: IJET Pulicaions UK. All righs reserved. 958