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Afuzzy seasonal ARIMAmodel for forecasting

Fang-Mei Tseng

a; ∗

, Gwo-Hshiung Tzeng

b

aDepartment of Finance, Hsuan Chuang University, No. 48 Hsuan Chuang Rd, Hsin-chu, Taiwan bEnergy and Environmental Research Group, Institute of Management of Technology, College of Management,

National Chiao Tung University, No. 1001 Ta-hsieh Rd, Hsin-chu, Taiwan

Received 20 August 1999; received in revised form 20 November 2000; accepted 3 January 2001

Abstract

This paper proposes a fuzzy seasonal ARIMA (FSARIMA) forecasting model, which combines the advantages of the seasonal time series ARIMA (SARIMA) model and the fuzzy regression model. It is used to forecast two seasonal time series data of the total production value of the Taiwan machinery industry and the soft drink time series. The intention of this paper is to provide business which are a2ected by diversi3ed management with a new method to conduct short-term forecasting. This model includes both interval models with interval parameters and the possible distribution of future value. Based on the results of practical application, it can be shown that this model makes good forecasts and is realistic. Furthermore, this model makes it possible for decision makers to forecast the best and worst estimates based on fewer observations than the SARIMA model. c 2002 Elsevier Science B.V. All rights reserved.

Keywords: SARIMA; Fuzzy regression; Fuzzy SARIMA; Fuzzy time series; Time series

1. Introduction

Modern enterprises are confronted with new tech-nologies and 3erce competition worldwide. The envi-ronment is becoming progressively more dynamic and is expanding globally. These factors make decision-making di8cult and critical. E2ective forecasting is fundamental to future technology development and customer demand, and time series is one of the meth-ods we can use for prediction. The seasonal time series ARIMA (SARIMA) model was initially presented by Box–Jenkins [1] and was successfully used in forecast-ing economic, marketforecast-ing, social problems, etc. While

Corresponding author.

this model has the advantage of accurate forecasting over short periods, it also has the limitation that at least 50 and preferably 100 observations or more should be used [1]. In addition, this model uses the concept of measurement error to deal with the deviations be-tween estimators and observations, but the data it uses are precise values that do not include measurement errors. Tanaka [14]. Tanaka and Ishibuchi [15] and Tanaka et al. [16] suggested the use of fuzzy regres-sion to solve the fuzzy environment problem and avoid modeling error. This model is basically an interval prediction model with the disadvantage that the pre-diction interval can be very wide if extreme values are present. An application that uses fuzzy regression to fuzzy time series analysis was presented by Watada

0165-0114/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved.

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[11], but this model did not include the concept of the Box–Jenkins’s model [1]. Tseng et al. [17] proposed the fuzzy ARIMA (FARIMA) method which uses the fuzzy regression method to fuzzify the parameters of the ARIMA model, although this model did not deal with the problem of seasonality. This paper is an ex-tension of Tseng’s work [17], in which we combine the advantages of the SARIMA(p; d; q)(P; D; Q)s model

and the fuzzy regression model to develop the fuzzy SARIMA (FSARIMA) method.

From the results of practical application, the pro-posed method makes good forecasts and appears to be the most appropriate tool. Its advantages are as follows:

(a) To provide the decision makers with insight re-garding the possible best and worst estimates. (b) The required number of observations is less than

required by the SARIMA model (at least 50 and preferably more than 100 observations [1]). This paper is organized as follows: Concepts of the SARIMA model and the fuzzy regression model are reviewed in Section 2. In Section 3, the FSARIMA model is formulated and proposed. The FSARIMA model is applied to forecast the production value of Taiwan machinery industry and the soft drink time series [9] in Section 4, and 3nally conclusions are discussed in Section 5.

2. Review of the seasonal ARIMA model and the fuzzy regression model

Atime series {Zt| t = 1; 2; : : : ; k} is generated by a

SARIMA(p; d; q)(P; D; Q)s process with mean of

the Box–Jenkins’s model [1] if ’(B)(Bs)(1 − B)d(1 − Bs)D(Z

t− )

= (B)(Bs)a

t; (1)

where p; d; q; P; D and Q are integers, and s is periodicity, ’(B) = 1 − ’1B − ’2B2− · · · − ’pBp; (Bs) = 1 −  1Bs− 2B2s− · · · − PBPs; (B) = 1 − 1B − 2B2− · · · − qBq and (Bs) = 1 −1Bs− 2B2s− · · · − QBQs are polynomials in

B of degrees p; q; P, and Q; B is the backward shift

operator, d is the number of regular di2erences; D is the number of seasonal di2erences, and Ztdenotes

observed value of time series data, t = 1; 2; : : : ; k. The SARIMA model formulation includes four steps:

(a) Identi3cation of the SARIMA(p; d; q) (P; D; Q)s

structure: use autocorrelation function (ACF) and partial autocorrelation function (PACF) to develop the rough function.

(b) Estimation of the unknown parameters.

(c) Goodness-of-3t tests on the estimated residuals. (d) Forecast future outcomes based on the known

data.

The at, which are the estimated residuals at each

time period, should be independent and identically distributed as normal random variables with mean 0 and variance 2. The roots of ’(Z) = 0 and (Z) = 0

should all lie outside the unit circle. If possible, at least 50 and preferably 100 observations or more should exist in the SARIMA model. In the real world, how-ever, the environment is uncertain and there are rapid changes, so we usually must forecast future situations using little data in a short time-span, and it is hard to verify that the data has a normal distribution. Thus, the assumption of the SARIMA model has limitations. The current model uses the concept of measurement error to deal with the deviations between estimators and observations, these data are precise values and do not include measurement errors. This is the same as the basic concept of fuzzy regression model as sug-gested by Tanaka et al. [16].

The basic concept of fuzzy regression is that the residuals between estimators and observations are not produced by measurement errors, but rather by the parameter uncertainty in the model, and the possibility distribution is used to deal with real observations.

The following is a generalized model of fuzzy linear regression: Y = 0+ 1x1+ · · · + nxn= n  i=0 ixi= x; (2)

where x0= 1; x is the vector of independent variables;

superscript denotes the transformation operation; n

is the number of variables and i represents the ith

parameter of the model.

Instead of using crisp, fuzzy parameters i in the

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[3], (i; ci)L, possibility distribution is

Bi(i) = L{(i− i)=ci); (3)

where L is a function type. Fuzzy parameters in the form of triangular fuzzy numbers are used

Bi(i) =      1−|i−c i| i ; i−ci6i6i+ ci; 0 otherwise; (4) where Bi(i) is the membership function of the fuzzy

set which is represented by parameter i; i is the

center of the fuzzy number, and ci is the width or

spread around the center of the fuzzy number. Through the extension principle, the membership function of the fuzzy number Yt= xt can be de3ned

by the membership function using pyramidal fuzzy parameter  as follows: Y(yt) =      1−|yt−xt|=c|xt| for xt= 0; 1 for xt= 0; yt= 0; 0 for xt= 0; yt= 0; (5) where  and c denote vectors of model parameter val-ues and spreads, respectively, for all model parame-ters, t is the number of observations, t = 1; 2; : : : ; k.

Finally, the method uses the criterion of minimizing the total vagueness, S, de3ned as the sum of individual spreads of the fuzzy parameters of the model. minimize S = k  t=1 c|x t|: (6)

At the same time, this approach takes into account the condition that the membership value of each observation yt is greater than an imposed threshold

h ; h ∈ [0; 1]. This criterion simply expresses the fact that the fuzzy output of the model should ‘cover’ all the data points y1; y2; : : : ; ykto a certain h level. The

choice of the h level value inIuences the widths c of the fuzzy parameters

Y (yt) ¿ h for t = 1; 2; : : : ; k: (7)

The index t refers to the number of nonfuzzy data used in constructing the model. Then the problem of 3nding the fuzzy regression parameters was formu-lated by Tanaka and Ishibuchi [15] as a linear pro-gramming problem: minimize S = k  t=1 c|x t| subject to x t + (1 − h)c|xt| ¿ yt; t = 1; 2; : : : ; k; x t − (1 − h)c|xt| 6 yt; t = 1; 2; : : : ; k; c ¿ 0; (8) where = ( 1; 2; : : : ; n) and c= (c0; c1; : : : ; cn) are

vectors of unknown variables and S is the total vague-ness as previously de3ned.

Watada [18] suggested the use of fuzzy time se-ries analysis, which is formulated by possibility re-gression model but does not include the concept of Box–Jenkins [1] model. Also in the Watada model, the weight of the objective function does not contain criteria which may be somewhat subjective. We pro-pose the FSARIMA model, which uses the criterion of fuzzy regression model for its formulation and im-proves the limitations in Watada’s model.

3. Model formulation

In the previous section, the parameter of SARIMA (p; d; q) (P; D; Q)s, ’1; ’2; : : : ; ’p, 1; 2; : : : ; P;

1; 2; : : : ; q and 1; 2; : : : ; Q are all crisp values.

The SARIMA model is a precise forecasting model for short time periods, although it is limited by the large amount of historical data required. However, we usually have to forecast future situations using limited amounts of data in a short span of time. So this model addresses the limitations of real world applications. This model uses the concept of mea-surement error to deal with the di2erence between estimators and observations, but these data are correct values that do not include measurement errors. In-stead of using crisp, fuzzy parameters, ˜’1; ˜’2; : : : ; ˜’p, ˜1; ˜2; : : : ; ˜P; ˜1; ˜2; : : : ; ˜q and ˜1; ˜2; : : : ; ˜Q,

in the form of triangular fuzzy numbers are used. Finally, at has a fuzzy parameter ˜!.

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A FSARIMA(p; d; q)(P; D; Q)smodel is described

by a fuzzy function with fuzzy parameter ˜’(B) ˜(Bs)W t= ˜0+ ˜(B) ˜(Bs)at; (9) Wt= (1 − B)d(1 − Bs)DZt; (10) ˜ Wt= ˜0+ p  i=1 ˜’iWt−i+ P  i=1 ˜iWt−is p  i=1 ’i1Wt−s−i− p  i=1 ’i2Wt−2s−i − · · · − p  i=1 ’iPWt−Ps−i+ ˜!at q  i=1 ˜iat−i− Q  i=1 iat−is + q  i=1 ˜i1at−s−i+ q  i=1 ˜i2at−2s−i + · · · + q  i=1 ˜iQat−Qs−i; (11) where Zt is observations, ˜’1; ˜’2; : : : ; ˜’p, ˜1; ˜2; : : : ;

˜P, ˜!; ˜1; ˜2; : : : ; ˜qand ˜1; ˜2; : : : ; ˜Qare fuzzy

num-bers. The center value of ˜! is 1. Eq. (11) is modi3ed as ˜ Wt= ˜0+ p  i=1 ˜iWt−i+ P  i=1 ˜p+iWt−is P  j=1 p  i=1 ˜i˜p+jWt−js−i+ ˜p+P+1at q  i=1 ˜p+P+1+iat−i Q  i=1 ˜p+P+q+1+iat−is+ Q  j=1 q  i=1 ˜p+P+1+i˜p+P+q+1+jat−js−i: (12)

Fuzzy parameters in the form of triangular fuzzy numbers are used

˜Bi(i)

= 

1−|i−i|=ci if i−ci6i6i+ci;

0 otherwise; (13)

where ˜Bi(i) is the membership function of the fuzzy

set that represents parameter i; iis the center of the

fuzzy number, and ci is the width or spread around

the center of the fuzzy number.

From the extension principle [19], Laarhoven and Pedrycz [8] use Dubois and Prade’s [3] approximation formula to get fuzzy multiplication

Ai⊗ Aj= (cicj; aiaj; bibj);

where Ai= (ci; ai; bi) and Aj= (cj; aj; bj) are triangular

fuzzy numbers.

In Eq. (12), the fuzzy multiplication of ˜i˜p+j is given as

˜i˜p+j = (cicp+j; ip+j; cicp+j): (14)

Using fuzzy parameters iin the form of triangular

fuzzy numbers the membership of W in Eq. (12) is given as ˜w(Wt) =  1 − |Wt− Et|=Ft for Wt= 0; at = 0; 0 otherwise; (15) where Et= 0+ p  i=1 iWt−i+ P  i=1 p+iWt−is P  j=1 p  i=1 ip+jWt−js−i+ at q  i=1 p+P+iat−i− Q  i=1 p+P+q+iat−is + Q  j=1 q  i=1 p+P+i˜p+P+q+jat−js−i; (16) Ft= c0+ p  i=1 ci|Wt−i| + P  i=1 cp+i|Wt−is| +P j=1 p  i=1 cicp+j|Wt−js−i| + cp+P+1|at| + q  i=1 cp+P+1+i|at−i| + Q  i=1 cp+P+q+1+i|at−is| + Q  j=1 q  i=1 cp+P+1+icp+P+q+1+j|at−js−i|: (17)

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Simultaneously, Zt represents the tth observation,

and h-level is the threshold value representing the de-gree to which the model should satisfy all the data points Z1; Z2; : : : ; Zt. Achoice of the h value inIuences

the widths c of the fuzzy parameters

Z(Zt) ¿ h for t = 1; 2; : : : ; k: (18)

The index t refers to the number of nonfuzzy data used for constructing the model. On the other hand, the fuzziness S included in the model is de3ned by Eq. (17).

The problem of 3nding the fuzzy seasonal ARIMA parameters was formulated as the following linear pro-gramming problem: minimize S = k  t=1 Ft subject to Et+ (1 − h)Ft¿ Wt; t = 1; 2; : : : ; k; Et− (1 − h)Ft6 Wt; t = 1; 2; : : : ; k; ci¿ 0 for all i = 1; 2; : : : ; p + P + q + Q + 1: (19) In the SARIMA model, if possible, at least 50 and preferably 100 observations or more should be used [1]. This makes the number of the LP constraint func-tions to be twice the number of the observafunc-tions, lead-ing the LP model to be more complex. The method we develop here to combine the advantages of two methods is comprised of two basic phases that let the number of constraint functions remain the same as the number of observations (the concept derived by Savic and Pedrycz [10]):

Phase I: Determining the type of FSARIMA model by using observations. Fitting the SARIMA model by using the available information about the center points of the observations, i.e., input data is considered non-fuzzy. The results of Phase I, the optimum solution of the parameter, = (

1; 2; : : : ; p+P+q+Q); is used as

one of the input data sets in Phase II.

Phase II: Determining the minimal fuzziness by us-ing the same criterion as in Eq. (18), but without  being a vector of decision variable. The FSARIMA model is an interval prediction model. The model is

˜

Wt = Eq: (16) ± Eq: (17): (20)

In order that the model include all possible condi-tions, when data include a signi3cant di2erence, the FSARIMA processes it as possibly happening, so as to produce cj with a wide spread. Ishibuchi and Tanaka

[7] suggest deleting the data around the model up-per bound and lower bound, and then formulating the fuzzy regression model. Here the FSARIMA model also uses the same method.

There are some constraints when we use the FSARIMA model, as follows:

(a) The FSARIMA model cannot be used to forecast the future value when the p and P equal zero. This is because the white noise is the residual of the actual value and it does not exist in the future. (b) The FSARIMA model cannot be used when the

linear programming of Phase II has the situation that all of the spreads are equal to zero.

4. Experimental results

In the following, the performance of the FSARIMA model is compared with other models using two seasonal time series: the total production value of Taiwan machinery industry, and the sales volume of soft drinks quoted from Montgomery [9]. In Sec-tion 4.1, the results of the total producSec-tion value of Taiwan machinery industry are described. Section 4.2 describes the soft drink time series. Section 4.3 de-scribes the di2erences between SARIMA, FSARIMA and Watada’s fuzzy time series.

4.1. Production value of machinery industry in Taiwan

The machinery industry in Taiwan has made steady progress over the past decade, playing a critical sup-porting role as the foundation of Taiwan’s overall manufacturing industries. In addition, it is itself a ma-jor exporting industry. Projecting into the future, the potential of the machinery industry is almost limitless, considering the postulations of automation, system-ization, and precision for the integral manufacturing industry, and the backgrounds of current machinery industry is as follows [5,6]:

(a) The development of the machinery industry began from the assembly of parts, and the repair and maintenance of machinery then maturing into one

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of the main supplier countries of world machinery industry. For example, the scale of exports of fans and blowers has already topped all others, while carpentry machines, sewing machines, and o8ce machines are being ranked third.

(b) Taiwan leads the world in exports of ventilation fans, which are increasingly important devices in today’s information industry. In addition, the development of precision machinery, semicon-ductor manufacturing equipment, high-tech anti-pollution equipment, and crucial machinery parts are being actively sponsored under government promotion. By taking advantage of the domestic market to push for the localization of equipment, as well as enhancing the proportion of domesti-cally made crucial parts, thorough research and development for the precision machinery industry can be established. In addition, a strategic alliance system can be set up in coordination with other industries in order to expand overseas markets, and to facilitate the cooperation with partners for international technology and infrastructure investment.

(c) With the development of the semiconductor industry, the industry for manufacturing semi-conductor equipment provides vast business opportunities for Taiwan. However, in view of domestic input, the level of our skill is restricted to a certain peripheral ability on the later part of the manufacturing process, thus the stage is considered embryonic as we are far short of the manufacturing equipment to design and the abil-ity to manufacture. However, Taiwan’s govern-ment is now vigorously pushing to turn Taiwan into a machinery manufacturing center for the Asia–Paci3c region, actively promoting the de-velopment of IC manufacturing equipment, wafer slicing and packaging equipment, as well as de-tection equipment for semiconductors, it is hoped that a self-production rate of 25% for this kind of equipment can be achieved in 3ve years. With the employment of the domestic market to push for the localization policy of such machinery, there are abundant possibilities for development. (d) Of those manufacturers dedicated to the

machin-ery industry, 95% of them are of medium and small enterprises; in 95% of these enterprises the number of their sta2 employed is less than 100.

Fig. 1. Total production value of Taiwan Machinery Industry (Jan. 1994 to Dec. 1997).

With regard to business turnover, an approximate 90% of the entrepreneurs have an annual turnover of less than NT$50 million.

(e) The modes of production within the industry vary, but most of them involve assembly of parts pro-duced elsewhere.

(f) The entrepreneurs in the machinery industry are mainly focused on overseas markets, with 60– 70% of their market share in Mainland China, the United States, and South-east Asia. And the rate of reliance on imports remains high since the rate of self-containment is less than 30%. Thus, the ex-pansion of domestic market waits to be explored under active government initiative.

(g) In terms of production value, the machinery in-dustry comprises as much as 5% of the manufac-turing industry, but its expansion growth in export value is far less than that of the integral manufac-turing industry.

From the preceding data, it can be seen that, al-though there are very strong future business opportu-nities in the machinery industry, there might not be any immediate breakthrough in this area. However, Taiwan’s government is now vigorously pushing to turn Taiwan into a machinery manufacturing center in the Asia Paci3c region with active drives being ren-dered to promote IC manufacturing equipment, wafer slicing, packaging equipment, etc. According to the above description, the forecasting of the total produc-tion value of Taiwan’s machinery industry is suitable for time series forecasting. As Fig. 1 shows, the time series data of the total production value of Taiwan’s machinery industry in the period from January 1994 to December 1997 showed strong seasonality and growth trends. This experiment used a data set from January

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Fig. 2. The result of the estimated value of the FSARIMA model.

1994 to December 1996 to build the models, and fore-cast for the period from January 1997 to December 1997.

The FSARIMA model formulation is as follows: Phases I: Fitting the SARIMA (p; d; q)(P; D; Q)s

model. Using SAS package software, we acquire the best model of the production value of machinery in-dustry which is SARIMA(1; 1; 0)(0; 1; 1)12;

Wt= −0:2588Wt−1+ at− 0:73997at−12;

Wt= Zt− Zt−1− Zt−12+ Zt−13: (21)

Phase II: Determining the minimal fuzziness. From Phase I, we can get the FSARIMA model as

Wt= 1; c1 Wt−1+ 1; c2 at+ 2; c2 at−12;

Wt= Zt− Zt−1− Zt−12+ Zt−13: (22)

Table 1

Forecasted production value of the machinery industry by using the SARIMA model, the FSARIMA model and Watada’s fuzzy time series model

Date Actual SARIMA (1; 1; 0)(0; 1; 1)12 FSARIMA Watada’s fuzzy time series

value 95% CI 95% CI Lower Upper Lower Upper

Lower bound Upper bound bound bound bound bound

Jan-97 26910 15133 38288 24690 28731 18542 9010 Feb-97 20489 9558 33316 17930 24943 13763 24231 Mar-97 27489 15284 39969 26453 28800 20273 30741 Apr-97 27669 15072 405586 25672 29959 19283 29751 May-97 29737 14789 41076 27160 28705 20284 30752 Jun-97 29053 15143 42200 27110 30233 20389 30857 Jul-97 29279 15795 43603 26166 33232 20684 31152 Aug-97 29020 15177 43715 27622 31270 21184 31652 Sep-97 28251 14120 43371 26336 31155 20544 31012 Oct-97 30288 14671 44617 27338 31949 20852 31320 Nov-97 30188 15742 46368 30128 31981 22773 33241 Dec-97 35099 15736 470275 25194 37570 23866 34334

Set (0; 1) = (−0:259; −0:740) and h = 0. The

fol-lowing linear interval model is obtained and its results are shown in Fig. 2.

Wt= −0:259; 0:548 Wt−1+ 1; 0 at

+ −0:740; 1:640 at−12;

Wt= Zt− Zt−1− Zt−12+ Zt−13: (23)

Using Eq. (23), we forecast the future production value of the machinery industry over the next six months, with the results as shown in Fig. 2. We 3nd that the predictions are clari3ed.

At the same time, we apply the SARIMA model and Watada’s fuzzy time series model [18] to forecast the production value of the machinery industry, and show this in Table 1.

Based on the empirical results of this application, we 3nd that the prediction interval of the FSARIMA model is narrower than the 95% con3dence interval of the SARIMA model and Watada’s fuzzy time se-ries model. From this application we can see that FSARIMA is the most e2ective method to forecast the production value of the machinery industry.

4.2. The soft drinks time series

In order to demonstrate the performance of the FSARIMA model, the authors applied the models to another time series, the monthly sales volume of soft drinks from Montgomery’s book Forecasting

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Fig. 3. Monthly sales of a 32-oz drink in hundreds of cases.

and Time Series Analysis, [9, p. 364]. The time se-ries demonstrates growth trend and seasonality, as is shown in Fig. 3.

The FSARIMA model formulation is as follows: Phase I: Building the SARIMA model. The time series data was pre-processed using logarith-mic transformation, 3rst-order regular di2erencing, and 3rst-order seasonal di2erencing in order to sta-bilize the variance and remove the growth trend and seasonality. The authors used the SAS statis-tical package to formulate the SARIMA model. Akaike Information Criterion (AIC) [1,20] was used to determine the best model. The derived model is ARIMA(1; 1; 0)(0; 1; 0)12, and the equation is

(1 + 0:73B)(1 − B)(1 − B12)Z

t= at: (24)

Phase II: Determining the minimal fuzziness. Let wt= Zt− Zt−1− Zt−12+ Zt−13, we get the model that

is in Eq. (24). The results are shown in Table 2. ˜wt= 1; 1:63 at+ −0:73; 0:80 wt−1: (25)

Based on the empirical results of this application, we 3nd that the prediction interval of the FSARIMA model is narrower than the 95% con3dence interval of the SARIMA model.

The basic concept of SARIMA is used to formu-late the model and 3nd  by FSARIMA. The

out-put of FSARIMA is fuzziness leading to the assump-tion of white noise (at). This make FSARIMA require

fewer observations than SARIMA. In the real world, the environment is uncertain, and we must use limited amounts of data to forecast future situations in a short time. Under these circumstances, FSARIMA is more satisfactory than SARIMA. There are several aspects

which indicate that FSARIMA is the most appropriate tool, as follows:

(a) It provides the decision makers the best and worst possible situations.

(b) The required observations are less than those re-quired by the SARIMA model (prefer more than 100).

4.3. Comparison between SARIMA, FSARIMA and fuzzy time series

A comparison between SARIMA, FSARIMA and fuzzy time series [18] models are described as follow-ing and are shown in Table 3.

(a) The theoretical foundation of the SARIMA method is founded on the probability distribution of statistics, and the relationship between input and output is of precise function. In addition, the input and output information can only be man-aged through the relationship of their informa-tion, thus massive amounts of information would be required. Amethod of this kind is bene3cial for observing information that has trendal growth and seasonal cycles, with merely acceptable cost and expense.

(b) The FSARIMA method is revised from the SARIMA method, and is based on the concept of fuzzy regression, as the residual di2erence be-tween the prediction value and observation value is brought about because of the uncertainty of pa-rameter, in other words, the parameter is a fuzzy number. Also, the probability distribution of the coe8cient can be obtained from the distribution of information, thus the input of the model is of a certain domain embracing observation value, that is FSARIMA is a kind of prediction for the domain. Such a method is able to deal with the information observed, if that information is en-dowed with trendal growth and periodic cycle, and its cost is not excessive.

(c) Watada’s fuzzy time series uses the theory of fuzzy method, employs probability or fuzzy information management with regard to the information observed, and de3nes them as fuzzy numbers. Furthermore, the time series model is construed as the fuzzy function of time, while the parameter is construed as the fuzzy number. This method is capable of dealing with

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Table 2

Forecasted sales value of soft drinks using the SARIMA and the FSARIMA model

Date Actual value SARIMA (1; 1; 0)(0; 1; 1)12 FSARIMA

95% CI 95% CI Lower bound Upper bound

Lower bound Upper bound

Jan-75 52 39.12 55.68 35.09 91.62 Feb-75 60 49.32 60.54 39.54 106.38 Mar-75 66 55.502 71.99 42.60 126.94 Apr-75 80 69.002 83.26 50.74 157.46 May-75 85 69.59 100.55 51.62 172.64 Jun-75 95 76.19 95.73 57.60 201.46 Jul-75 100 74.39 107.78 58.69 218.15 Aug-75 104 81.74 113.94 61.86 240.64 Sep-75 101 61.84 117.78 58.99 242.01 Oct-75 94 58.56 116.666 54.86 235.31 Nov-75 81 32.38 108.636 45.68 205.55 Dec-75 70 42.68 99.36 38.10 179.00 Table 3

Comparison between SARIMA, FSARIMA and Watada’s fuzzy time series models

SARIMA FSARIMA Fuzzy time series

Theory Probability of statistic Fuzzy regression Fuzzy regression

The relationship of Previous function Fuzzy function Fuzzy function input and output

Forecasted interval Provide con3dence interval Provide possibility Provide possibility distribution distribution

Treatment of trend Yes Yes Yes

Treatment of Yes Yes Yes

seasonal cycle

Number of observations At least 50 and Less than SARIMA Less than SARIMA preferably 100 or more

Unit cost of forecast Low Low Low

information observations that have trendal growth as well as periodic cycles, and only an insigni3-cant amount of observation is required for it. 5. Conclusions

Based on the basic concepts of the SARIMA model and Tanaka’s fuzzy regression model, we combine the advantages of these two methods to present a new method (FSARIMA) and apply it to forecast the pro-duction value of Taiwan’s machinery industry and the sales volume of the soft drink. From the empirical re-sults of the production value of Taiwan’s machinery

industry, we 3nd that the prediction capability of the SARIMA model and the FSARIMA model are both rather encouraging but the preference of the fuzzy time series is not. In addition, the interval of the proposed method is narrower than SARIMA, and all of the three models have the capacity to handle growth trends and seasonal cycles, with the unit cost of forecasting being relatively low. From the empirical results of the sales volume of soft drinks, we 3nd that the prediction ca-pability of the FSARIMA model is rather encouraging and the interval of the proposed method is narrower than SARIMA.

In the real world, the environment is uncertain and we must use limited amount of data to provide

(10)

future forecasts in a short period. For this kind of situation, the FSARIMA is more satisfactory than the SARIMA. The FSARIMA appears to be the most ap-propriate tool according to the following two primary advantages:

(a) It provides the decision makers with best and worst possible situations.

(b) The number of required observations is less than required by the SARIMA model.

References

[1] G.P. Box, G.M. Jenkins, Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, CA, 1976. [2] S.M. Chen, Forecasting enrollments based on fuzzy time

series, Fuzzy Sets and Systems 81 (3) (1996) 311–319. [3] D. Dubois, H. Prade, Operations on fuzzy numbers, Internat.

J. Systems Sci. 9 (1978) 613–629.

[4] D. Dubois, H. Prade, in: Theory and Applications, Fuzzy Sets and Systems, Academic Press, New York, 1980.

[5] Institute of Mechanical Industry in Industrial Technology Research Institute, 1997, Present Position and Trend Analysis of Mechanical Industry, Research Report in Industrial Technology Research Institute, 1998.

[6] Institution for Economic Research, Taiwan Year Book of Mechanical Industry, Research Report in Institution for Economic Research, 1996.

[7] H. Ishibuchi, H. Tanaka, Interval regression analysis based on mixed 0–1 integer programming problem, J. Japan Soc. Industrial Engng. 40 (5) (1988) 312–319.

[8] P.J.M. Laarhoven, W. Pedrycz, Afuzzy extension of Saaty’s priority theory, Fuzzy Sets and Systems 11 (1983) 229–241.

[9] Montgomery, D.C.L.A. Johnson, J.S. Gardiner, Forecasting and Time Series Analysis, McGraw-Hill, New York, 1990, p. 364.

[10] D.A. Savic, W. Pedrycz, Evaluation of fuzzy linear regression models, Fuzzy Sets and Systems 39 (1) (1991) 51–63. [11] Q. Song, B.S. Chissom, Fuzzy time series and its models,

Fuzzy Sets and Systems 54 (3) (1993) 269–277.

[12] Q. Song, B.S. Chissom, Forecasting enrollments with fuzzy time series—Part I, Fuzzy Sets and Systems 54 (1) (1993) 1–9.

[13] Q. Song, B.S. Chissom, Forecasting enrollments with fuzzy time series—Part II, Fuzzy Sets and Systems 62 (1) (1994) 1–8.

[14] H. Tanaka, Fuzzy data analysis by possibility linear models, Fuzzy Sets and Systems 24 (3) (1987) 363–375.

[15] H. Tanaka, H. Ishibuchi, Possibility regression analysis based on linear programming, fuzzy regression analysis, in: J. Kacprzyk, M. Fedrizzi (Eds.), Omnitech Press, Warsaw and Physica-Verlag, Heidelberg, 1992, pp. 47–60.

[16] H. Tanaka, S. Uejima, K. Asai, Linear regression analysis with fuzzy model, IEEE Trans. Systems, Man Cybernet. 12 (6) (1982) 903–907.

[17] F.M. Tseng, G.H. Tzeng, H.C. Yu, B.J.C. Yuan, Fuzzy ARIMA model for forecasting the foreign exchange market, Fuzzy Sets and Systems 118 (2000) 9–19.

[18] J. Watada, Fuzzy time series analysis and forecasting of sales volume, fuzzy regression analysis, in: J. Kacprzyk, M. Fedrizzi (Eds.), Omnitech Press, Warsaw and Physica-Verlag, Heidelberg, 1992, pp. 211–227.

[19] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Parts 1–3, Inform. Sci. 8 (1975) 199–249; 8 (1975) 301–357; 9 (1976) 43–80. [20] H. Akaike, Maximum likelihood identi3cation of Gaussian

auto-regressive moving average models, Biometrika 60 (1973) 255–266.

數據

Fig. 1. Total production value of Taiwan Machinery Industry (Jan. 1994 to Dec. 1997).
Fig. 2. The result of the estimated value of the FSARIMA model.
Fig. 3. Monthly sales of a 32-oz drink in hundreds of cases.

參考文獻

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