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1 Title Estimation of Smoothing Constant of Mi tion to Industrial Data Author(s) Takeyasu, Kazuhiro; Imura, Keiko Editor(s) Citation Journal of Economics, Business and Law Issue Date URL Rights

2 67 Estimation of Smoothing Constant of Minimum Variance and Its Application to Industrial Data Kazuhiro Takeyasu*, Keiko Imura** ABSTRACT Focusing on the idea that the equation of exponential smoothing method (ESM) is equivalent to (1,1) order ARMA model equation, new method of estimation of smoothing constant in exponential smoothing method is proposed before by us which satisfies minimum variance of forecasting error. Generally, smoothing constant is selected arbitrary. But in this paper, we utilize above stated theoretical solution. Firstly, we make estimation of ARMA model parameter and then estimate smoothing constants. Thus theoretical solution is derived in a simple way and it may be utilized in various fields. Furthermore, combining the trend removing method with this method, we aim to upraise forecasting accuracy. An approach to this method is executed in the following method. Trend removing by a linear function is applied to the original shipping data of consumer goods. The combination of linear and non-linear function is also introduced in trend removal. For the comparison, monthly trend is removed after that. Theoretical solution of smoothing constant of ESM is calculated for both of the monthly trend removing data and the non monthly trend removing data. Then forecasting is executed on these data. The variance of forecasting error of linear trend removing and monthly trend removing data was minimum. The new method shows that it is useful especially for the time series that has stable characteristics and has rather strong seasonal trend. The effectiveness of this method should be examined in various cases. Key Words: minimum variance, exponential smoothing method, forecasting, trend * Professor of College of Economics, Osaka Prefecture University ** Takumatec Corporation

3 68 Journal of Economics, Business and Law Vol Introduction Many methods for time senes analysis have been presented such as Autoregressive model (AR Model), Autoregressive Moving Average Model (ARMA Model) and Exponential Smoothing Method (ESM)[I]-[4]. Among these, ESM is said to be a practical simple method. For this method, various improving method such as adding compensating item for time lag, coping with the time series with trend [5], utilizing Kalman Filter [6], Bayes Forecasting [7], adaptive ESM[8], exponentially weighted Moving Averages with irregular updating periods [9], making averages of forecasts using plural method [10] are presented. For example, Maeda [6] calculated smoothing constant in relationship with SIN ratio under the assumption that the observation noise was added to the system. But he had to calculate under supposed noise because he couldn't grasp observation noise. It can be said that it doesn't pursue optimum solution from the very data themselves which should be derived by those estimation. Ishii [11] pointed out that the optimal smoothing constant was the solution of infinite order equation, but he didn't show analytical solution. Based on these facts, we proposed a new method of estimation of smoothing constant in ESM before [13]. Focusing that the equation of ESM is equivalent to (1,1) order ARMA model equation, a new method of estimation of smoothing constant in ESM was derived. In this paper, utilizing above stated method, revised forecasting method is proposed. In making forecast such as shipping data, seasonal trend is removed by dividing the original time series by Monthly Ratio which is calculated using moving average data. Utilizing above stated method, forecasting is executed with this data in which seasonal trend is removed. Monthly Ratio is multiplied after that, and forecasting data is obtained. This is a revised forecasting method. Variance of forecasting error of this newly proposed method is assumed to be less than those of previously proposed method. The combination of linear and non-linear function is also introduced in trend removal. The rest of the paper is organized as follows. In section 2, ESM is stated

4 Estimation of Smoothing Constant of Minimum Variance and Its Application to Industrial Data 69 by ARMA model and estimation method of smoothing constant is derived using ARMA model identification. The combination of linear and nonlinear function is introduced for trend removing in section 3. The Monthly Ratio is referred in section 4. Forecasting is executed in section 5, and estimation accuracy is examined. 2. Description of ESM using ARMA model In ESM, forecasting at time t+ 1 is stated in the following equation. Here, x t + 1 = xt+a(xt-x t) ( 1 ) = ax t + (l-a)x t ( 2 ) Xt+l : forecasting at t+ 1 x t : realized value at t a : smoothing constant (0 < a < 1) (2) is re-stated as Xt+l = I:: a(l-ayx H 1=0 By the way, we consider the following (1,1) order ARMA model. Xt = X t- 1 = e t -!3e t - 1 Generally, (p, q) order ARMA model is stated as Here, p x t+ I; atxt- i = et- t = 1 j= 1 q I; bjet_j ( 3 ) ( 4 ) ( 5 ) {Xt} : Sample process of Stationary Ergodic Gaussian Process x(t)t = 1,2,,N, {ej : Gaussian White Noise with 0 mean a; variance MA process in (5) is supposed to satisfy convertibility condition. Utilizing the relation that E[e t le t - 1, e t - 2, J = 0 we get the following equation from (4). x = x t - 1 -!3e t - 1 Operating this scheme on t+ 1, we finally get ( 6 ) ( 6 )

5 70 Journal of Economics, Business and Law Vol.ll 2008 Xt+! = x t + (1 -j3)e t = x t +(1-j3)(x t -x t ) ( 7 ) If we set 1-13 = a, the above equation is the same with (1), i.e., equation of ESM is equivalent to (1,1) order ARMA model, or is said to be (0,1,1) order ARIMA model because 1st order AR parameter is _1[1J[3J. Comparing with (4) and (5), we obtain la! = -1 b! = -13 = a-i From (1), (7), a = Therefore, we get { a! = -1 b! = -13 = a-i ( 8 ) From above, we can get estimation of smoothing constant after we identify the parameter of MA part of ARMA model. But, generally MA part of ARMA model become non-linear equations which are described below. Let (5) be p xt+xt+ L: aixt-i t! q x t = et+.l: bjet_j J! ( 9 ) (10) We express the autocorrelation function of x t as fk and from (9), (10), we get the following non-linear equations which are well known[3j. fk = 1 0 ; ::Objbk+j (k :::;; q) (k:2 q+l) (11) For these equations, recursive algorithm has been developed. In this paper, parameter to be estimated is only bj, so it can be solved in the following way. From (4) (5) (8) (11), we get

6 Estimation of Smoothing Constant of Minimum Variance and Its Application to Industrial Data 71 q = 1 a l =-1 b l = -/3 = a-i fo = (1 +bdo; - b 2 r l = IOe If we set fk Pk =----=ro the following equation is derived. PI = b l l+b2 I We can get b l as follows. b l = l±jl-4p; 2 PI (12) (14) (15) In order to have real roots, PI must satisfy I I <- PI -- 2 From invertibility condition, b l must satisfy I bll < 1 From (14), using the next relation, (1-b l )2 ;;:: 0 (1 +b l )2 ;;:: 0 (16) always holds. As a = b l +l b l is within the range of -1<b l <O Finally we get b l = a= I-jl-4p 2 PI 1 +2PI-j 1-4p 2PI (16) (17) which satisfy above condition. Thus we can obtain a theoretical solution by

7 72 Journal of Economics, Business and Law Vol.ll 2008 a simple way. Focusing on the idea that the equation of ESM is equivalent to (1,1) order ARMA model equation, we can estimate smoothing constant after estimating ARMA model parameter. It can be estimated only by calculating Oth and 1st order autocorrelation function. 3. Trend Removal Method As trend removal method, we describe linear and non-linear function, and the combination of these. [IJ Linear function We set y = ajx+bj (18) as linear function. [2J Non-linear function We set y = azx z + bzx+cz y = a3x3+b3xz+c3x+d3 as 2nd and 3rd order non-linear function. (19) (20) [3J The combination of linear and non-linear function We set y = a j (ajx+b j) +azcazx z + b Z x+c3) (21) as the combination of linear and 2nd order non-linear function, and y = J3j(ajx+bj)+J3zCa3x3+b3Xz+C3X+d3) (22) as the combination linear and 3rd order non-linear function. Here, az = I-a j and J3z = I-J3j. Comparative discussion concerning (18), (21) and (22) is described in section Monthly Ratio For example, if there is the monthly data of L years as stated bellow

8 Estimation of Smoothing Constant of Minimum Variance and Its Application to Industrial Data 73 {Xi) (i = 1, "', L) (j = 1, "', 12) Monthly Ratio Xj is calculated as follows 1 L -L.2: xij,= 1 Xj = L Ī I; I; x L 12 i=lj=l ZJ (j = 1, "', 12) (23) 5. Forecasting the Shipping Data of Manufacturer 5.1 Analysis Procedure The shipping data of food manufacture for 3 cases from January 1999 to December 2001 are analysed. First of all, graphical charts of these time series data are exhibited in Figure 5-1,5-2,5-3. Analysis procedure is as follows. There are 36 monthly data for each case. We use 24 data(1 to 24) and remove trend by the method stated in 3. Then we calculate monthly ratio by the method stated in 4. After removing monthly trend, the method stated in 2 is applied and Exponential Smoothing Constant with minimum variance of forecasting error is estimated. 2,500 FOOD A 2,000 lj E c. 1,500 1,000 :> z 111 Original Data 500 o Figure 5-1. Shipping data of Food A

9 74 Journal of Economics, Business and Law VoLl ,500 FOOD B 3,000 2,500 [2,000 " 1l1,500 E :> z 1,000 II Original Data 500 o Figure 5-2. Shipping data of Food B 600 FOOD C 500 '" 400 1" " E 0- :<:.: 300 o.0 " E :> z o Figure 5-3. Shipping data of Food C

10 Estimation of Smoothing Constant of Minimum Variance and Its Application to Industrial Data 75 Then 1 step forecast is executed. Thus, data is shifted to 2nd to 25th and the forecast for 26th data is executed consecutively, which finally reaches forecast of 36th data. To examine the accuracy of forecasting, variance of forecasting error is calculated for the data of 25th to 36th data. Final forecasting data is obtained by multiplying monthly ratio and trend. Forecasting error is expressed as: ci = Xi-Xi _ 1 C = JjL:ci (24) (25) Variance of forecasting error is calculated by: 2 1 N ( -)2 ao = N-l i1 ci- c (26) 5.2 Trend Removing Trend is removed by dividing original data by (18), (21), (22). Parameter estimation result is exhibited in Table 5-1 for the case of calculation result of 1st to 24th data. Where weight of ai' a 2, /31' /32 are set 0.5 in (21), (22). Table 5-1. Parameter Estimation Result 1st 2nd 3rd parameter weight parameter weight parameter Q I b l a l a 2 Q 2 b 2 c2 /31 /32 Q) b J C J d J Food A FoodB Food C Graphical chart of trend is exhibited in Figure 5-4, 5-5, Removing trend of monthly ratio After removing trend, monthly ratio IS calculated by the method stated in 4. Calculation result for 1st to 24th data is exhibited in Table 5-2, 5-3,5-4.

11 76 Journal of Economics, Business and Law VoLl ,500 FOOD A 2,000 E 1, :;: b '" L " 1,000 ::l z _Original Data --1st order 1 st+2nd order -4-1 st+3rd order 500 3,500 Figure 5-4. Trend of Food A FOOD B 2,500 " hooo.h500 E ::l Z 1,000 _Original Data.. 1st order -&--1 st+2nd order xl st+3rd order 500 o Figure 5-5, Trend of Food B

12 Estimation of Smoothing Constant of Minimum Variance and Its Application to Industrial Data FOOD C '" 400 " E Co :.c: 'b'" 300 L.D" E :l Z 200 A 1 st+2nd order 1 st+3rd order 100 o Figure 5-6. Trend of Food C Table 5-2. Monthly ratio (1st order) Month Food A FoodB Food C Table 5-3. Monthly ratio (lst+2nd order) Month Food A FoodB Food C Table 5-4. Monthly ratio (lst+3rd order) Month Food A FoodB Food C

13 78 Journal of Economics, Business and Law Vol.ll Estimation of Smoothing Constant with Minimum Variance of Forecasting Error After removing monthly trend, Smoothing Constant with minimum variance of forecasting error is estimated utilizing (17). There are cases that we cannot obtain a theoretical solution because they do not satisfy the condition of (16). In those cases, Smoothing Constant with minimum variance of forecasting error is derived by shifting variablefrom 0.01 to 0.99 with 0.01 interval. Calculation result for 1st to 24th data is exhibited in Table 5-5, 5-6, 5-7. Table 5-5. Estimated Smoothing Constant with Minimum Variance (1st order) PI Food A FoodB (Does not satisfy (16) ) a Food C Table 5-6. Estimated Smoothing Constant with Minimum Variance (lst+2nd order) PI Food A ' FoodB Food C a Table 5-7. Estimated Smoothing Constant with Minimum Variance (lst+3rd order) PI Food A FoodB Food C a

14 Estimation of Smoothing Constant of Minimum Variance and Its Application to Industrial Data Forecasting and Variance of Forecasting Error Utilizing smoothing constant estimated in the previous section, forecasting is executed for the data of 25th to 36th data. Final forecasting data is obtained by multiplying monthly ratio and trend. Variance of forecasting error is calculated by (26). Forecasting results are exhibited in Figure 5-7, 5-8,5-9. Variance of forecasting error is exhibited in Table 5-8. Variance of forecasting error in the case that monthly ratio is not used is exhibited in Table Remarks Forecasting accuracy IS apparently improved by introducing monthly ratio. The combination of linear and non-linear function in trend removing is also examined. In Food A and C, 1st order trend removing has better fore FOOD A Original Data '" 1500 HillHI"--lI!---JlI- Co :;: '" 'b " E 1000 HllHlHtill-JIlHIr :J z. -a-- Forecasting (1st order) :.3.- Forecasting (lst+2nd order) ---e-forecasting (1 st+3rd order) Figure 5-7. Forecasting Result of Food A

15 80 Journal of Economics, Business and Law Vol.ll FOOD B [2000 :<: '" 1500 E :J Z Original Data Forecasting (1 st order)... _. Forecasting (lst+2nd order) ---m- Forecasting (lst+3rd order) 500 o Figure 5-8. Forecasting Result of Food B 600 FOOD C Original Data '" 400 E c. :<:..'" 300 o.ll E 200 Forecasting (1 st order) IA' Forecasting (1 st+2nd order) ----a-- Forecasting (1 st+3rd order) 100 o Figure 5-9. Forecasting Result of Food C

16 Estimation of Smoothing Constant of Minimum Variance and Its Application to Industrial Data 81 Table 5-8. Variance of Forecasting Error 1st order 1st + 2nd 1st + 3rd order order Food A 76, , , FoodB 245, , , Food C , Table 5-9. Variance of Forecasting Error (Monthly Ratio is not used) 1st order 1st + 2nd 1st + 3rd order order Food A 153, , , FoodB 254, , , Food C 2, , , casting accuracy than other methods. On the other hand, 1st+3nd order trend removing has better forecasting accuracy in Food B. This may be because Food B has the value of 0 at the last 3 months. Excluding this term, we obtain following results which correspond to Table 5-8. Food B 1st order: 82, st+2nd order: 91, st+3rd order: 92, st order trend removing has better forecasting accuracy as is in Food A and C. Thus, 1st order trend removal method would be effective for the time series data which have stable characteristics. 6. Conclusion Focusing on the idea that the equation of exponential smoothing method (ESM) was equivalent to (1,1) order ARMA model equation, new method of estimation of smoothing constant in exponential smoothing method was proposed before by us which satisfied minimum variance of forecasting error.

17 82 Journal of Economics, Business and Law Vol.ll 2008 Generally, smoothing constant was selected arbitrary. But in this paper, we utilized above stated theoretical solution. Firstly, we made estimation of ARMA model parameter and then estimated smoothing constants. Thus theoretical solution was derived in a simple way and it might be utilized in various fields. Furthermore, combining the trend removal method with this method, we aimed to increase forecasting accuracy. An approach to this method was executed in the following method. Trend removal by a linear function was applied to the original shipping data of consumer goods. The combination of linear and non-linear function was also introduced in trend removing. For the comparison, monthly trend was removed after that. Theoretical solution of smoothing constant of ESM was calculated for both of the monthly trend removing data and the non monthly trend removing data. Then forecasting was executed on these data. The variance of forecasting error of linear trend removing and monthly trend removing data was minimum. The new method showed that it was useful especially for the time series that had stable characteristics and has rather strong seasonal trend. The effectiveness of this method should be examined in various cases. REFERENCES [ 1 ] Box Jenkins; "Time Series Analysis Third Edition", Prentice Hall, 1994 [2] R. G. Brown; "smoothing, Forecasting and Prediction of Discrete -Time Series", Prentice Hall, 1963 [3] Hidekatsu TOKUMARU et al.; "Analysis and Measurement -Theory and Application of Random data Handling", Baifukan Publishing, 1982 [ 4 ] Kengo KOBAYASHI; "Sales Forecasting for Budgeting", Chuou-keizaisha Publishing [5] Peter R. Winters; "Forecasting Sales by Exponentially Weighted Moving Average", Mgt.Sci., Vol. 6, No.3, 1984 [6] Katsuro MAEDA; Smoothing Constant of Exponential Smoothing Method", Seikei University Report Faculty of Engineering No. 38, 1984 [7] M. West and P.J.Harrison; "Baysian Forecsting and Dynamic Models", Springer. Verlag, New York, 1989 [8] StenarEKern; "Adaptive exponential smoothing revisited", J. Opl. Res.

18 Estimation of Smoothing Constant of Minimum Variance and Its Application to Industrial Data 83 Soc. Vol. 32 pp , 1982 [9 J F.R.Johnston; "Exponentially Weighted Moving Average (EWMA) with Irregular Updating Period"; J. Opl. Res. Soc., Vol. 44, No.7 pp , 1993 [10J Spyros Makridakis and Robeat L. Winkler; "Averages of Forecasts; Some Empirical Results", Mgt. Sci., No.9, 1983 [l1j Naohiro Ishii, Akira Iwata and Nobuo Suzumura; "Bilateral exponential smoothing of time series", Int. J. System Sci., Vol. 12, No.8 pp , 1991 [12J Kazuhiro TAKEYASU; "System of Production, Sales and Distribution", Chouou-keizaisha Publishing, 1996 [13J Kazuhiro Takeyasu; "Estimation of Smoothing Constant in Exponential Smoothing Method." The 4th Asia- Pacific Conference on Industrial Engineering and Management Systems