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Forecasting innovation diffusion of products using trend-weighted fuzzy time-series model
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Expert Systems with Applications Expert Systems with Applications 36 (2009) 1826–1832 www.elsevier.com/locate/eswa
Forecasting innovation diffusion of products using trend-weighted fuzzy time-series model Ching-Hsue Cheng a, You-Shyang Chen a,*, Ya-Ling Wu b a
Department of Information Management, National Yunlin University of Science and Technology, 123, Section 3, University Road, Touliu, Yunlin 640, Taiwan b Department of Applied English, National Chin-Yi University of Technology, 35 Lane 215 Chung-Shan Road Section 1, Taiping, Taichung 411, Taiwan
Abstract The time-series models have been used to make reasonably accurate predictions in weather forecasting, academic enrolment, stock price, etc. This study proposes a novel method that incorporates trend-weighting into the fuzzy time-series models advanced by Chen’s and Yu’s method to explore the extent to which the innovation diffusion of ICT products could be adequately described by the proposed procedure. To verify the proposed procedure, the actual DSL (digital subscriber line) data in Taiwan is illustrated, and this study evaluates the accuracy of the proposed procedure by comparing with different innovation diffusion models: Bass model, Logistic model and Dynamic model. The results show that the proposed procedure surpasses the methods listed in terms of accuracy and SSE (Sum of Squares Error). Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Innovation diffusion models; Trend-weighting; Fuzzy time series; ICT (Information and communication technologies) products
1. Introduction The product of information and communication technologies (ICT) on Internet is an outstanding issue to scholars and practitioners because the adoption of product of ICT has altered the operation of business model and has driven an innovative diffusion (Gharavi, Love, Sor, & Irani, 2004). Especially the diffusion of the Internet has resulted in heavily rapid changes to most enterprises. Therefore, diffusion of ICT products can be considered a phenomenon exceeding the assumption of markets with a rational manner (Gharavi, Love, & Cheng, 2004) and encompass relationships that the ICT products on Internet are interdependence. The phenomenon of innovation diffusion of a new product is a stylized form regarded as cumulative adoption or period-by-period adoption, which of these two representations is according to its application. For example, in the *
Corresponding author. E-mail address: [email protected] (Y.-S. Chen).
0957-4174/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2007.12.041
diffusion of mobile phones, a service provider is concerned with the demand on the infrastructure and is seen as cumulative adoptions (Meade & Islam, 2006). Rogers said ‘‘diffusion is the process by which an innovation is communicated through certain channels over time among the members of a social system (Rogers, 1995)”, and we know practically and theoretically that the successful pervasion of an innovation follows an S-shaped curve. The purpose of this study is to explore the extent to which the diffusion of the ICT products could be adequately represented by a trend-weight fuzzy time-series model. In order to test the prediction accuracy of the proposed procedure, the study offers that the adoption pattern of digital subscriber line (DSL) is on the adoption pattern of ICT products (on Internet), and take the actual number of cumulated DSL subscribers in Taiwan as empirical data. Sales managers of the enterprise attempt to predict sales activities in market based on either their professional knowledge or analyzing tools, or even both. The higher accuracy is most concerned since more profits will be made if high accurate predictions are given. Thus, they have
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perennially strived to discover ways to predict sales quantity accurately. DSL is a communication technology for bringing highspeed broadband information to homes or businesses. The xDSL refers to different types of DSL, such as ADSL, ADSL2, HDSL, IDSL, SDSL, SHDSL, RADSL, VDSL, VADSL and WDSL. In general, they are all called DSL. DSL installations began in 1998 in the US and continuously and enormously increase in a number of countries in next decade. DSL was introduced in Taiwan in 1999 and rapidly gets the most popular broadband product in 2001(Status of broadband development, 2006). It is necessary to install DSL for exploring information on Internet. For more than one decade, different fuzzy time-series models have also been applied to solve various domain problems, such as financial forecasting (Faff, Brooks, & Kee, 2002; Huarng & Yu, 2005; Yu, 2004), university enrolment forecasting (Chen, 1996; Song & Chissom, 1993b; Song & Chissom, 1994), and temperature forecasting. As Dourra and Siy notes, it is common practice to ‘‘deploy fuzzy logic engineering tools in the finance arena, specifically in the technical analysis field, since technical analysis theory consists of indicators used by experts to evaluate stock price (Dourra & Siy, 2002).” So far, few papers use fuzzy time-series models for forecasting ICT products; therefore, in addition to that model, this study proposes a trend-weighted fuzzy time-series model and compares it with Bass model, Logistic model and Dynamic model. In this study, a trend-based, fuzzy, time-series model is proposed to improve the forecast accuracy in ICT products. In this model, several factors such as the fuzzy relationships of trend-weighted, a reasonable universe of discourse, a reliable length of intervals and past patterns of actual number of cumulated DSL subscriber are all considered together for forecasting. Moreover, three refined processes are employed in the forecasting algorithm. Actual number of cumulated DSL subscriber in Taiwan is used as the datasets for training and testing. The remaining content of this paper is organized as follows: Section 2 introduces the related literature of innovation diffusion models and fuzzy time-series model; Section 3 demonstrates the proposed procedure and algorithm; Section 4 evaluates the model’s performance; and Section 5 offers conclusions. 2. Related works This section briefly reviews the related literature, including three sub-sections: literature reviews of innovation diffusion models, literature reviews of time-series model and fuzzy time-series definitions and algorithm. 2.1. Literature reviews of innovation diffusion model Innovation diffusion models can be used to forecast the growth of a new product (Mahajan, Muller, & Bass, 1990;
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Yu & Wang, 2006), and continually attracted the attention of researchers for modeling the process and validate the models in the context of a whole ranging applications. One example of the best-known diffusion models in marketing is the Bass model suggested by Bass (1969), which derives from a hazard function of probability in Eq. (1) (Meade & Islam, 2006). f ðtÞ ¼ ½p þ qF ðtÞ½1 F ðtÞ:
ð1Þ
where p is the coefficient of innovation (or the coefficient of external influence), q is the coefficient of imitation (or the coefficient of internal influence), f(t) is the probability density function of adoption at time t, and F(t) is the cumulated density function of adoption at time t. From the Bass model, it is easily understandable that the effect of external influence and internal influence is immediate and the maximal market potential will be eventually reached (Wang, Fergola, Lombardo, & Mulone, 2006). For simplicity, the Bass model describes the diffusion process by a first-order differential equation as follows (Osaki, Gemba, & Fumio, 2001; Wang et al., 2006): i dN ðtÞ h q ¼ p þ N ðtÞ ½m N ðtÞ: ð2Þ dt m Thus, Eq. (1) becomes Eq. (2). Where F(t) = N (t)/m is the fraction of the potential adopters by time t. N(t) is the cumulative number of adopters at time t, and m is the ceiling or the population of potential adopters. With a special case, when the coefficient p is zero, the Bass model in Eq. (2) becomes Eq. (3). The model is referred as the Logistic model, which is as follows (Lu, Ku, & Lin, 1993): dN ðtÞ q ¼ N ðtÞ½m N ðtÞ: dt m
ð3Þ
The Logistic model, another example of the diffusion models, is resembled to the internal influence model of Mahajan and Peterson (1985). In general, both the Bass model and the Logistic model have S-shaped patterns. The Dynamic model equation, dN ðtÞ q ¼ pþ N ðtÞ ½mðtÞ N ðtÞ: dt mðtÞ
ð4Þ
where m(t) = m0egt, which a general exponential form represents the time-variant effect motivated by the user growth on Internet, was supported by Rai, Ravichandran, and Samaddar (1998). The model is referred as the Dynamic model in (Lu et al., 1993; Sharif & Ramanathan, 1981). Recently, an interesting issue (Beifus, Proskurowski, & Udwadia, 1997) is to join fuzzy rule-based method into the modelling framework of innovation diffusion to broaden predict strategies. This enables the modeler to design fuzzy controller to reach forecasting goals when strategies depend on linguistic terms which are decided by managers. This framework sets up a base of fuzzy rule, which assists in obtaining desired market pervasion by controlling the parameter (Debasree & Karmeshu, 2004).
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2.2. Literature reviews of fuzzy time-series model Fuzzy theory was originally developed to deal with the problems involving human linguistic terms (Zadeh, 1975a; Zadeh, 1975b; Zadeh, 1976). Time-series methods had failed to consider the application of this theory until fuzzy time-series was defined by Song and Chissom (1993a). In 1993, Song and Chissom proposed the definitions of fuzzy time-series and methods to model fuzzy relationships among observations (Song & Chissom, 1993a). In the following research, they continued to discuss the difference between time-invariant and time-variant models (Song & Chissom, 1994). Besides these researchers, Chen proposed another method to apply simplified arithmetic operations in forecasting algorithm rather than the complicated max–min composition operations presented in Chen (1996). In time-series model, when unexpected conditions happen, the fluctuations cannot be recorded into the historical data immediately. This would probably results in terrible inaccurate forecast by using the out-of-date data. To deal with the problem, a group decision-making method was employed to integrate the subjective forecast values of all decision makers. Fuzzy weighted method was then combined with subjective forecast values to produce the aggregated forecast value. Huarng pointed out that the length of intervals affects forecast accuracy in fuzzy time-series and proposed a method with distribution-based length and average-based length to reconcile this issue (Huarng, 2001). The method applied two different lengths of intervals to Chen’s model and the conclusions showed that distribution-based and average-based lengths could improve the accuracy of forecast. Although this method has excellent performance, it creates too many linguistic values to be identified by analysts. According to Miller, establishing linguistic values and dividing intervals would be a trade-off between human recognition and forecasting accuracy (Miller, 1956). It becomes apparent that the major drawback of these methods is the lack of consideration in determining a reasonable universe of discourse and the length of intervals. Moreover, the researchers find that the neglected information, which indicates the patterns of trend changes in history, should be considered in the processes of forecasting. To reconcile these problems above, a novel method is hereby proposed.
2.3. Fuzzy time-series definitions and algorithm Over the past 14 years, many fuzzy time-series models have been proposed by following Song and Chissom’s definitions (Song & Chissom, 1993a). Among these models, Chen’s model is very conventional one because of easy calculations and good forecasting performance (Chen, 1996). Therefore, Song and Chissom’s definitions and the algorithm of Chen’s model are used for illustrations as follows:
Definition 1 (Fuzzy time-series). Let Y(t) (t = . . ., 0, 1, 2, . . .), a subset of real numbers is the universe of discourse by which fuzzy sets fj(t) are defined. If F(t) is a collection of f1(t), f2(t). . . then F(t) is called a fuzzy time-series defined on y(t). Following Definition 1, fuzzy relationships between two consecutive observations can be defined as follows: Definition 2 (Fuzzy time-series relationships). If there exists a fuzzy relationship R(t 1, t), such that F(t) = F(t 1)x R(t 1, t), where x represents an operator, then F(t) is said to be caused by F(t 1). The relationship between F(t) and F(t 1) ? F(t). Definition 3 (Fuzzy logical relationship (FLR)). Let F(t 1) = Ai and F(t) = Aj. The relationship between two consecutive observations, F(t) and F(t 1), referred to as a fuzzy logical relationship (FLR) (Chen, 1996; Song & Chissom, 1993b; Song & Chissom, 1994), can be denoted by Ai ? Aj, where Ai is called the left-hand side (LHS) and Aj the right-hand side (RHS) of the FLR. Definition 4 (Fuzzy logical relationship groups). Fuzzy logical relationships can be further grouped together into fuzzy logical relationship groups according to the same left-hand sides of the fuzzy logical relationships. For example, there are fuzzy logical relationships with the same left-hand sides (Ai): Ai ! Aj1 Ai ! Aj1; Aj2 . . . ... These fuzzy logical relationships can be grouped into a fuzzy logical relationship as follows: Ai ! Aj1; Aj2 . . . Definition 5 (A time-invariant or time-variant fuzzy timeseries). Suppose F(t) is caused by F(t 1) only, and F(t) = F(t 1, t) for any t. If R(t 1, t) is independent of t, then F(t) is named a time-invariant fuzzy time series, otherwise a time-variant fuzzy time-series. 3. Trend-weighted fuzzy time-series model The proposed procedure, trend-weighted fuzzy time-series model, can be fitted in innovation diffusion models as well. Therefore, this study presents research procedure of this model in Fig. 1 and its algorithm in this section. From review of the literature, there are two major drawbacks: (1) The lack of consideration in determining a reasonable universe of discourse and the length of intervals; and (2) Many researchers neglect the information which indicates patterns of trend changes in the past history. In order to reconcile these problems, three refined processes are factored into the model:
C.-H. Cheng et al. / Expert Systems with Applications 36 (2009) 1826–1832
Define the universe of discourse U and partition it into several intervals.
Establish fuzzy sets for observations and fuzzify historical data.
Build fuzzy logic relationships.
Establish fuzzy relationship groups into trends.
Assign trend-weights.
Aggregate weights to calculate forecast results.
Apply α to adapt the forecast result. Fig. 1. Research procedure of trend-weighted fuzzy time-series model.
(1) To define a reliable length of intervals for linguistic values. (2) To classify recurrent fuzzy relationships into three different types of trends and assign a proper weight to individual fuzzy relationships. (3) To modify the forecasting equation of Chen’s model and assign a adaptive value, alpha (a), to make the forecast results more reliable. Initially, in the first refined process, the universe of discourse should be partitioned into seven linguistic values (Miller, 1956), and if the data amount of a given linguistic value is larger than the average amount, then the original linguistic value should be further partitioned in half. Because the data occur more frequently in the linguistic value, using once-divided linguistic value to present the data is supposed to be less reliable than twice-divided. However, it would be undesirable to create too many linguistic values and, thereby, ignoring the meaning of fuzzy application. The second comes from the belief of the researchers that classifying these relations into three different types of trends should enhance the performance of prediction. Traditionally, fuzzy relationship weights are determined either based on knowledge which could be elicited from domain experts or their chronological order. Since each fuzzy relationship will reoccur, from the researcher’s perspective, classifying them into different trends and converting the counts of trends to incremental weights is reasonable for making more accurate predictions; hence, the trendweighted method is proposed. The details describing the assignment of weights are listed in Table 1. For example,
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Table 1 Assign weights to different trends (t = 1) (t = 2) (t = 3) (t = 4) (t = 5) (t = 6)
A 1 ? A1 A1 ? A 2 A2 ? A 1 A1 ? A 1 A1 ? A 1 A1 ? A 3
No change Up trend Down trend No change No change Up trend
Assign Assign Assign Assign Assign Assign
weight weight weight weight weight weight
1 1 1 2 3 1
*t denotes time point.
it is clear that among the Fuzzy Logical Relationships (FLRs), when t = 5 (t denotes time point), then it is assigned the highest value of 3, which means that the probability of its appearance in the near future is 3 times higher than in any of the other cases. The merits of the trendweighted model are that they can foresee the cycles and events, which will eventually occur and relate to the fuzzy relationship in a more reasonable manner. The third is to modify forecasting equation with a proper alpha (a) value. Here the value represents the confidence level of the investors in the whole data set, ranging from 0 1 but not equal 0. If the investor is very confident in the predicted variant, then a is assigned 1; conversely, if the investor is cautious, 0.1 may be assigned. By way of summarization, a detailed algorithm for the proposed procedure is illustrated below: Step 1: Define the universe of discourse and partition it into intervals. By the problem used in forecasting, the universe of discourse for observations is defined as: U = [starting, ending]. Then the average datum that should be in each linguistic value may be calculated. The linguistic value which the amount of the data falls in is larger than the average amount of all linguistic values and should further be split into smaller linguistic values by dividing them into two. Step 2: Establish fuzzy sets for observations. Each linguistic observation Ai can be defined by the intervals: u1, u2, . . ., un. Each Ai can be represented as following Eq. (5). And the value, kj, is determined by the situation as follows: if j = i 1, then kj = 0.5; if j = i, then kj = 1; if j = i + 1, then kj = 0.5; elsewhere kj = 0; n X k j =uj : ð5Þ Ai ¼ j¼1
Step 3: Build fuzzy logic relationships. Two consecutive fuzzy sets Ai(t 1) and Aj(t) can be established into a single FLR as Ai ? Aj. Step 4: Establish fuzzy relationship groups into corresponding trends. The FLRs with the same LHSs (left hand sides) can be grouped to form a FLRG. For example, Ai ? Aj, Ai ? Ak, Ai ? Am can be group as Ai ? Aj, Ak, Am. Step 5: Assign trend-weights. The FLRs are grouped into the trend to which they belong. For example, A1 ? A2, will be grouped into the ‘‘up trend,” A1 ? A1 into the ‘‘no change trend,” and A2 ? A1, into the ‘‘down trend”.
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These weights should be standardized to obtain the weight matrix, W ðtÞ ¼ ½W 01 ; W 02 ; . . . ; W 0j . The standardized weight matrix equation is defined in Eq. (6)
Table 2 Actual data of cumulate DSL subscriber in quarter Quarter
DSL subscriber (thousand)
Quarter
DSL subscriber (thousand)
W ðtÞ ¼ ½W 01 ; W 02 ; . . . ; W 0j " # W1 W2 Wk ; Pj ; ; Pj : ¼ Pj k¼1 W k k¼1 W k k¼1 W k
Sep-99 Mar-00 Sep-00 Mar-01 Sep-01 Mar-02 Sep-02
2 9 54 220 720 1180 1630
Dec-99 Jun-00 Dec-00 Jun-01 Dec-01 Jun-02
5 13 120 450 920 1400
ð6Þ
For example, if the forecast data is Table 1, the weights are specified as follows: W1 = 1, W2 = 1, W3 = 2, W4 = 3, W5 = 1. By Eq. (6) the weight matrix, W(t), is determined as following Eq. (7): 1 1 ; ; W ðtÞ ¼ 1þ1þ2þ3þ1 1þ1þ2þ3þ1 2 3 ; ; 1þ1þ2þ3þ1 1þ1þ2þ3þ1 1 : ð7Þ 1þ1þ2þ3þ1 Step 6: Calculate forecast value. From step 5, we can obtain the standardized weight matrix. Hereby, apply Eq. (8) to generate the forecast value (Ldf(t 1) is deffuzified matrix, Wn (t 1) is weight matrix)
order to have better identity due to human recognition. That is: U 1 ¼ ½0; 233Þ;
U 2 ¼ ½233; 468Þ;
U 4 ¼ ½701; 934Þ;
U 3 ¼ ½468; 701Þ;
U 5 ¼ ½934; 1167Þ;
U 6 ¼ ½1167; 1400Þ;
U 7 ¼ ½1400; 1633: Then, DSL subscriber linguistic observation of each quarter is as Table 3. Step 2: Establish fuzzy sets for observations.
ð8Þ
Each linguistic variable A1 to A7 can be defined as:
Step 7: Apply a to adapt the forecast value. The adapted-forecast Eq. (9) is generated from the modified forecast Eq. (8)
A1 ¼ 1=U 1 þ 0:5=U 2 þ 0=U 3 þ 0=U 4 þ 0=U 5 þ 0=U 6 þ 0=U 7
Adapted forecastðtÞ ¼ Actualðt 1Þ þ aðForecastðtÞ
þ 0=U 7 A3 ¼ 0=U 1 þ 0:5=U 2 þ 1=U 3 þ 0:5=U 4 þ 0=U 5 þ 0=U 6
ForecastðtÞ ¼ Ldf ðt 1Þ W n ðt 1Þ:
Actualðt 1ÞÞ:
ð9Þ
4. Verifications and comparisons To illustrate the forecasting performance of the proposed procedure, we use actual number of DSL subscriber in Taiwan (Status of broadband development, 2006). The data set covered from September 1999 to September 2002, which is used for training and December 2002 to June 2004 used for testing. Furthermore, the comparison of each model is listed.
A2 ¼ 0:5=U 1 þ 1=U 2 þ 0:5=U 3 þ 0=U 4 þ 0=U 5 þ 0=U 6
þ 0=U 7 A4 ¼ 0=U 1 þ 0=U 2 þ 0:5=U 3 þ 1=U 4 þ 0:5=U 5 þ 0=U 6 þ 0=U 7 A5 ¼ 0=U 1 þ 0=U 2 þ 0=U 3 þ 0:5=U 4 þ 1=U 5 þ 0:5=U 6 þ 0=U 7 A6 ¼ 0=U 1 þ 0=U 2 þ 0=U 3 þ 0=U 4 þ 0:5=U 5 þ 1=U 6 þ 0:5=U 7 A7 ¼ 0=U 1 þ 0=U 2 þ 0=U 3 þ 0=U 4 þ 0=U 5 þ 0:5=U 6 þ 1=U 7
4.1. Forecasting for cumulated DSL subscribers From the proposed algorithm in Section 3, the actual dataset is calculated as follows: Step 1: Define the universe of discourse and partition into intervals. According to actual data of cumulate DSL subscriber in quarter in Table 2, the universe of discourse for observations, U, is defined as [0, 1633]. According to Miller (1956), this study defines our intervals into seven levels in
Table 3 DSL subscriber linguistic observation of each quarter Quarter
DSL subscriber (thousand)
Linguistic value
Quarter
DSL subscriber (thousand)
Linguistic value
Sep-99 Mar-00 Sep-00 Mar-01 Sep-01 Mar-02 Sep-02
2 9 54 220 720 1180 1630
U1 U1 U1 U1 U4 U6 U7
Dec-99 Jun-00 Dec-00 Jun-01 Dec-01 Jun-02
5 13 120 450 920 1400
U1 U1 U1 U2 U4 U7
C.-H. Cheng et al. / Expert Systems with Applications 36 (2009) 1826–1832
From the comparisons in Table 4, the average forecasting error of Bass model is 22.35%, with SSE (sum of squares error) of 2,856,500; in Logistic model, the average error is 29.38%, and SSE is 4,500,800; in Dynamic model, the average error is 26.34%, and SSE is 5,211,500; in the proposed procedure, the average error is 10.67%, and SSE is 456,115. It is obvious that the proposed procedure has a smaller SSE and less average error than the models listed.
5000 Actual Bass Model Logistic Model Dynamic Model Proposed Model
Cumulated DSL Subscribers
4500 4000 3500
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3000 2500 2000 1500
5. Conclusions and future research
1000 500
Step 3: Build fuzzy logic relationships. For example, A6 ? A6; A6 ? A6; A6 ? A7; A6 ? A4. Step 4: Establish fuzzy relationships groups into corresponding trends. Up trend: A6 ? A7; No change: A6 ? A6, A6; Down trend: A6 ? A4. Step 5: Assign trend-weights. If F(t 1) = A6, the forecasting of f(t) is equal A7, A6, A6, A4. Thus, the weight matrix is [1/5, 1/5, 2/5, 1/5]. Step 6: Calculate forecast results. After iterative calculation by F(1) to F(7), the forecast values are 129, 816, 0, 1048, 0, 1513 and 1513, respectively. Step 7: Apply alpha to smoothen the forecast results.
As results in Table 4 and Fig. 2, this study confirms that the trend-weighted fuzzy time-series model can be good fit in innovation diffusion models. Therefore, this study has proposed a trend-weighted fuzzy time-series model to enhance the prediction accuracy in innovation diffusion of ICT products and to overcome the problems mentioned in the literature. From outcomes, this study can conclude that the proposed procedure would lead to better performance than the models listed in forecasting: (1) The proposed procedure is rule-based method, which have four advantages: highly expressive, easy to interpret, easy to generate, and predicting new instances rapidly; (2) assigning proper weights to classify recurrent fuzzy relationships, the proposed procedure can make more reasonable descriptions for the past patterns and more accurate predictions for the future; and (3) the reasonable alpha value adapted by user opinion can make more precise adjustments to match the past trends in the data set. However, there is still room for testing and improving the model as follows:
Finally, the alpha value is 0.1, which leads to the better outcome in prediction accuracy. The open data set from December 2002 to June 2004 is utilized to verify the researchers’ model. The comparisons with different models, Bass model, Logistic model, Dynamic model and the proposed procedure (Using a = 0.1 to generate the forecast results), are listed in Fig. 2 and the forecasting performances is listed in Table 4.
1. Using more empirical data of actual number of cumulated DSL subscriber in Taiwan to evaluate the accuracy and performance of the proposed procedure. 2. Applying different types of data sets to test the proposed procedure. 3. Reconsidering the factors affecting the diffusion of innovation and updating the proposed procedure to increase the prediction accuracy.
Table 4 Performances of the forecast results for cumulated DSL subscribers (thousand) with different models
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0 2-Sep 2-Dec 3-Mar 3-Jun 3-Sep 3-Dec 4-Mar 4-Jun Quarters Fig. 2. Comparisons of the forecast results for cumulated DSL subscribers (thousand) with different models.
Quarters
Actual data
Bass model
Logistic model
Dynamic model
Proposed procedure
Sep-02 Dec-02 Mar-03 Jun-03 Sep-03 Dec-03 Mar-04 Jun-04
1630 1820 1990 2200 2400 2550 2710 2840
1700 1760 1800 1820 1830 1830 1840
1600 1620 1630 1640 1640 1640 1640
1890 2190 2530 2920 3370 3890 4490
1411 1618 1789 1942 2131 2311 2446 2590
22.35 2,856,500
29.38 4,500,800
26.34 5,211,500
10.67 456,115
Average error (%) SSE
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Expert Systems with Applications Expert Systems with Applications 36 (2009) 1826–1832 www.elsevier.com/locate/eswa
Forecasting innovation diffusion of products using trend-weighted fuzzy time-series model Ching-Hsue Cheng a, You-Shyang Chen a,*, Ya-Ling Wu b a
Department of Information Management, National Yunlin University of Science and Technology, 123, Section 3, University Road, Touliu, Yunlin 640, Taiwan b Department of Applied English, National Chin-Yi University of Technology, 35 Lane 215 Chung-Shan Road Section 1, Taiping, Taichung 411, Taiwan
Abstract The time-series models have been used to make reasonably accurate predictions in weather forecasting, academic enrolment, stock price, etc. This study proposes a novel method that incorporates trend-weighting into the fuzzy time-series models advanced by Chen’s and Yu’s method to explore the extent to which the innovation diffusion of ICT products could be adequately described by the proposed procedure. To verify the proposed procedure, the actual DSL (digital subscriber line) data in Taiwan is illustrated, and this study evaluates the accuracy of the proposed procedure by comparing with different innovation diffusion models: Bass model, Logistic model and Dynamic model. The results show that the proposed procedure surpasses the methods listed in terms of accuracy and SSE (Sum of Squares Error). Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Innovation diffusion models; Trend-weighting; Fuzzy time series; ICT (Information and communication technologies) products
1. Introduction The product of information and communication technologies (ICT) on Internet is an outstanding issue to scholars and practitioners because the adoption of product of ICT has altered the operation of business model and has driven an innovative diffusion (Gharavi, Love, Sor, & Irani, 2004). Especially the diffusion of the Internet has resulted in heavily rapid changes to most enterprises. Therefore, diffusion of ICT products can be considered a phenomenon exceeding the assumption of markets with a rational manner (Gharavi, Love, & Cheng, 2004) and encompass relationships that the ICT products on Internet are interdependence. The phenomenon of innovation diffusion of a new product is a stylized form regarded as cumulative adoption or period-by-period adoption, which of these two representations is according to its application. For example, in the *
Corresponding author. E-mail address: [email protected] (Y.-S. Chen).
0957-4174/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2007.12.041
diffusion of mobile phones, a service provider is concerned with the demand on the infrastructure and is seen as cumulative adoptions (Meade & Islam, 2006). Rogers said ‘‘diffusion is the process by which an innovation is communicated through certain channels over time among the members of a social system (Rogers, 1995)”, and we know practically and theoretically that the successful pervasion of an innovation follows an S-shaped curve. The purpose of this study is to explore the extent to which the diffusion of the ICT products could be adequately represented by a trend-weight fuzzy time-series model. In order to test the prediction accuracy of the proposed procedure, the study offers that the adoption pattern of digital subscriber line (DSL) is on the adoption pattern of ICT products (on Internet), and take the actual number of cumulated DSL subscribers in Taiwan as empirical data. Sales managers of the enterprise attempt to predict sales activities in market based on either their professional knowledge or analyzing tools, or even both. The higher accuracy is most concerned since more profits will be made if high accurate predictions are given. Thus, they have
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perennially strived to discover ways to predict sales quantity accurately. DSL is a communication technology for bringing highspeed broadband information to homes or businesses. The xDSL refers to different types of DSL, such as ADSL, ADSL2, HDSL, IDSL, SDSL, SHDSL, RADSL, VDSL, VADSL and WDSL. In general, they are all called DSL. DSL installations began in 1998 in the US and continuously and enormously increase in a number of countries in next decade. DSL was introduced in Taiwan in 1999 and rapidly gets the most popular broadband product in 2001(Status of broadband development, 2006). It is necessary to install DSL for exploring information on Internet. For more than one decade, different fuzzy time-series models have also been applied to solve various domain problems, such as financial forecasting (Faff, Brooks, & Kee, 2002; Huarng & Yu, 2005; Yu, 2004), university enrolment forecasting (Chen, 1996; Song & Chissom, 1993b; Song & Chissom, 1994), and temperature forecasting. As Dourra and Siy notes, it is common practice to ‘‘deploy fuzzy logic engineering tools in the finance arena, specifically in the technical analysis field, since technical analysis theory consists of indicators used by experts to evaluate stock price (Dourra & Siy, 2002).” So far, few papers use fuzzy time-series models for forecasting ICT products; therefore, in addition to that model, this study proposes a trend-weighted fuzzy time-series model and compares it with Bass model, Logistic model and Dynamic model. In this study, a trend-based, fuzzy, time-series model is proposed to improve the forecast accuracy in ICT products. In this model, several factors such as the fuzzy relationships of trend-weighted, a reasonable universe of discourse, a reliable length of intervals and past patterns of actual number of cumulated DSL subscriber are all considered together for forecasting. Moreover, three refined processes are employed in the forecasting algorithm. Actual number of cumulated DSL subscriber in Taiwan is used as the datasets for training and testing. The remaining content of this paper is organized as follows: Section 2 introduces the related literature of innovation diffusion models and fuzzy time-series model; Section 3 demonstrates the proposed procedure and algorithm; Section 4 evaluates the model’s performance; and Section 5 offers conclusions. 2. Related works This section briefly reviews the related literature, including three sub-sections: literature reviews of innovation diffusion models, literature reviews of time-series model and fuzzy time-series definitions and algorithm. 2.1. Literature reviews of innovation diffusion model Innovation diffusion models can be used to forecast the growth of a new product (Mahajan, Muller, & Bass, 1990;
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Yu & Wang, 2006), and continually attracted the attention of researchers for modeling the process and validate the models in the context of a whole ranging applications. One example of the best-known diffusion models in marketing is the Bass model suggested by Bass (1969), which derives from a hazard function of probability in Eq. (1) (Meade & Islam, 2006). f ðtÞ ¼ ½p þ qF ðtÞ½1 F ðtÞ:
ð1Þ
where p is the coefficient of innovation (or the coefficient of external influence), q is the coefficient of imitation (or the coefficient of internal influence), f(t) is the probability density function of adoption at time t, and F(t) is the cumulated density function of adoption at time t. From the Bass model, it is easily understandable that the effect of external influence and internal influence is immediate and the maximal market potential will be eventually reached (Wang, Fergola, Lombardo, & Mulone, 2006). For simplicity, the Bass model describes the diffusion process by a first-order differential equation as follows (Osaki, Gemba, & Fumio, 2001; Wang et al., 2006): i dN ðtÞ h q ¼ p þ N ðtÞ ½m N ðtÞ: ð2Þ dt m Thus, Eq. (1) becomes Eq. (2). Where F(t) = N (t)/m is the fraction of the potential adopters by time t. N(t) is the cumulative number of adopters at time t, and m is the ceiling or the population of potential adopters. With a special case, when the coefficient p is zero, the Bass model in Eq. (2) becomes Eq. (3). The model is referred as the Logistic model, which is as follows (Lu, Ku, & Lin, 1993): dN ðtÞ q ¼ N ðtÞ½m N ðtÞ: dt m
ð3Þ
The Logistic model, another example of the diffusion models, is resembled to the internal influence model of Mahajan and Peterson (1985). In general, both the Bass model and the Logistic model have S-shaped patterns. The Dynamic model equation, dN ðtÞ q ¼ pþ N ðtÞ ½mðtÞ N ðtÞ: dt mðtÞ
ð4Þ
where m(t) = m0egt, which a general exponential form represents the time-variant effect motivated by the user growth on Internet, was supported by Rai, Ravichandran, and Samaddar (1998). The model is referred as the Dynamic model in (Lu et al., 1993; Sharif & Ramanathan, 1981). Recently, an interesting issue (Beifus, Proskurowski, & Udwadia, 1997) is to join fuzzy rule-based method into the modelling framework of innovation diffusion to broaden predict strategies. This enables the modeler to design fuzzy controller to reach forecasting goals when strategies depend on linguistic terms which are decided by managers. This framework sets up a base of fuzzy rule, which assists in obtaining desired market pervasion by controlling the parameter (Debasree & Karmeshu, 2004).
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2.2. Literature reviews of fuzzy time-series model Fuzzy theory was originally developed to deal with the problems involving human linguistic terms (Zadeh, 1975a; Zadeh, 1975b; Zadeh, 1976). Time-series methods had failed to consider the application of this theory until fuzzy time-series was defined by Song and Chissom (1993a). In 1993, Song and Chissom proposed the definitions of fuzzy time-series and methods to model fuzzy relationships among observations (Song & Chissom, 1993a). In the following research, they continued to discuss the difference between time-invariant and time-variant models (Song & Chissom, 1994). Besides these researchers, Chen proposed another method to apply simplified arithmetic operations in forecasting algorithm rather than the complicated max–min composition operations presented in Chen (1996). In time-series model, when unexpected conditions happen, the fluctuations cannot be recorded into the historical data immediately. This would probably results in terrible inaccurate forecast by using the out-of-date data. To deal with the problem, a group decision-making method was employed to integrate the subjective forecast values of all decision makers. Fuzzy weighted method was then combined with subjective forecast values to produce the aggregated forecast value. Huarng pointed out that the length of intervals affects forecast accuracy in fuzzy time-series and proposed a method with distribution-based length and average-based length to reconcile this issue (Huarng, 2001). The method applied two different lengths of intervals to Chen’s model and the conclusions showed that distribution-based and average-based lengths could improve the accuracy of forecast. Although this method has excellent performance, it creates too many linguistic values to be identified by analysts. According to Miller, establishing linguistic values and dividing intervals would be a trade-off between human recognition and forecasting accuracy (Miller, 1956). It becomes apparent that the major drawback of these methods is the lack of consideration in determining a reasonable universe of discourse and the length of intervals. Moreover, the researchers find that the neglected information, which indicates the patterns of trend changes in history, should be considered in the processes of forecasting. To reconcile these problems above, a novel method is hereby proposed.
2.3. Fuzzy time-series definitions and algorithm Over the past 14 years, many fuzzy time-series models have been proposed by following Song and Chissom’s definitions (Song & Chissom, 1993a). Among these models, Chen’s model is very conventional one because of easy calculations and good forecasting performance (Chen, 1996). Therefore, Song and Chissom’s definitions and the algorithm of Chen’s model are used for illustrations as follows:
Definition 1 (Fuzzy time-series). Let Y(t) (t = . . ., 0, 1, 2, . . .), a subset of real numbers is the universe of discourse by which fuzzy sets fj(t) are defined. If F(t) is a collection of f1(t), f2(t). . . then F(t) is called a fuzzy time-series defined on y(t). Following Definition 1, fuzzy relationships between two consecutive observations can be defined as follows: Definition 2 (Fuzzy time-series relationships). If there exists a fuzzy relationship R(t 1, t), such that F(t) = F(t 1)x R(t 1, t), where x represents an operator, then F(t) is said to be caused by F(t 1). The relationship between F(t) and F(t 1) ? F(t). Definition 3 (Fuzzy logical relationship (FLR)). Let F(t 1) = Ai and F(t) = Aj. The relationship between two consecutive observations, F(t) and F(t 1), referred to as a fuzzy logical relationship (FLR) (Chen, 1996; Song & Chissom, 1993b; Song & Chissom, 1994), can be denoted by Ai ? Aj, where Ai is called the left-hand side (LHS) and Aj the right-hand side (RHS) of the FLR. Definition 4 (Fuzzy logical relationship groups). Fuzzy logical relationships can be further grouped together into fuzzy logical relationship groups according to the same left-hand sides of the fuzzy logical relationships. For example, there are fuzzy logical relationships with the same left-hand sides (Ai): Ai ! Aj1 Ai ! Aj1; Aj2 . . . ... These fuzzy logical relationships can be grouped into a fuzzy logical relationship as follows: Ai ! Aj1; Aj2 . . . Definition 5 (A time-invariant or time-variant fuzzy timeseries). Suppose F(t) is caused by F(t 1) only, and F(t) = F(t 1, t) for any t. If R(t 1, t) is independent of t, then F(t) is named a time-invariant fuzzy time series, otherwise a time-variant fuzzy time-series. 3. Trend-weighted fuzzy time-series model The proposed procedure, trend-weighted fuzzy time-series model, can be fitted in innovation diffusion models as well. Therefore, this study presents research procedure of this model in Fig. 1 and its algorithm in this section. From review of the literature, there are two major drawbacks: (1) The lack of consideration in determining a reasonable universe of discourse and the length of intervals; and (2) Many researchers neglect the information which indicates patterns of trend changes in the past history. In order to reconcile these problems, three refined processes are factored into the model:
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Define the universe of discourse U and partition it into several intervals.
Establish fuzzy sets for observations and fuzzify historical data.
Build fuzzy logic relationships.
Establish fuzzy relationship groups into trends.
Assign trend-weights.
Aggregate weights to calculate forecast results.
Apply α to adapt the forecast result. Fig. 1. Research procedure of trend-weighted fuzzy time-series model.
(1) To define a reliable length of intervals for linguistic values. (2) To classify recurrent fuzzy relationships into three different types of trends and assign a proper weight to individual fuzzy relationships. (3) To modify the forecasting equation of Chen’s model and assign a adaptive value, alpha (a), to make the forecast results more reliable. Initially, in the first refined process, the universe of discourse should be partitioned into seven linguistic values (Miller, 1956), and if the data amount of a given linguistic value is larger than the average amount, then the original linguistic value should be further partitioned in half. Because the data occur more frequently in the linguistic value, using once-divided linguistic value to present the data is supposed to be less reliable than twice-divided. However, it would be undesirable to create too many linguistic values and, thereby, ignoring the meaning of fuzzy application. The second comes from the belief of the researchers that classifying these relations into three different types of trends should enhance the performance of prediction. Traditionally, fuzzy relationship weights are determined either based on knowledge which could be elicited from domain experts or their chronological order. Since each fuzzy relationship will reoccur, from the researcher’s perspective, classifying them into different trends and converting the counts of trends to incremental weights is reasonable for making more accurate predictions; hence, the trendweighted method is proposed. The details describing the assignment of weights are listed in Table 1. For example,
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Table 1 Assign weights to different trends (t = 1) (t = 2) (t = 3) (t = 4) (t = 5) (t = 6)
A 1 ? A1 A1 ? A 2 A2 ? A 1 A1 ? A 1 A1 ? A 1 A1 ? A 3
No change Up trend Down trend No change No change Up trend
Assign Assign Assign Assign Assign Assign
weight weight weight weight weight weight
1 1 1 2 3 1
*t denotes time point.
it is clear that among the Fuzzy Logical Relationships (FLRs), when t = 5 (t denotes time point), then it is assigned the highest value of 3, which means that the probability of its appearance in the near future is 3 times higher than in any of the other cases. The merits of the trendweighted model are that they can foresee the cycles and events, which will eventually occur and relate to the fuzzy relationship in a more reasonable manner. The third is to modify forecasting equation with a proper alpha (a) value. Here the value represents the confidence level of the investors in the whole data set, ranging from 0 1 but not equal 0. If the investor is very confident in the predicted variant, then a is assigned 1; conversely, if the investor is cautious, 0.1 may be assigned. By way of summarization, a detailed algorithm for the proposed procedure is illustrated below: Step 1: Define the universe of discourse and partition it into intervals. By the problem used in forecasting, the universe of discourse for observations is defined as: U = [starting, ending]. Then the average datum that should be in each linguistic value may be calculated. The linguistic value which the amount of the data falls in is larger than the average amount of all linguistic values and should further be split into smaller linguistic values by dividing them into two. Step 2: Establish fuzzy sets for observations. Each linguistic observation Ai can be defined by the intervals: u1, u2, . . ., un. Each Ai can be represented as following Eq. (5). And the value, kj, is determined by the situation as follows: if j = i 1, then kj = 0.5; if j = i, then kj = 1; if j = i + 1, then kj = 0.5; elsewhere kj = 0; n X k j =uj : ð5Þ Ai ¼ j¼1
Step 3: Build fuzzy logic relationships. Two consecutive fuzzy sets Ai(t 1) and Aj(t) can be established into a single FLR as Ai ? Aj. Step 4: Establish fuzzy relationship groups into corresponding trends. The FLRs with the same LHSs (left hand sides) can be grouped to form a FLRG. For example, Ai ? Aj, Ai ? Ak, Ai ? Am can be group as Ai ? Aj, Ak, Am. Step 5: Assign trend-weights. The FLRs are grouped into the trend to which they belong. For example, A1 ? A2, will be grouped into the ‘‘up trend,” A1 ? A1 into the ‘‘no change trend,” and A2 ? A1, into the ‘‘down trend”.
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These weights should be standardized to obtain the weight matrix, W ðtÞ ¼ ½W 01 ; W 02 ; . . . ; W 0j . The standardized weight matrix equation is defined in Eq. (6)
Table 2 Actual data of cumulate DSL subscriber in quarter Quarter
DSL subscriber (thousand)
Quarter
DSL subscriber (thousand)
W ðtÞ ¼ ½W 01 ; W 02 ; . . . ; W 0j " # W1 W2 Wk ; Pj ; ; Pj : ¼ Pj k¼1 W k k¼1 W k k¼1 W k
Sep-99 Mar-00 Sep-00 Mar-01 Sep-01 Mar-02 Sep-02
2 9 54 220 720 1180 1630
Dec-99 Jun-00 Dec-00 Jun-01 Dec-01 Jun-02
5 13 120 450 920 1400
ð6Þ
For example, if the forecast data is Table 1, the weights are specified as follows: W1 = 1, W2 = 1, W3 = 2, W4 = 3, W5 = 1. By Eq. (6) the weight matrix, W(t), is determined as following Eq. (7): 1 1 ; ; W ðtÞ ¼ 1þ1þ2þ3þ1 1þ1þ2þ3þ1 2 3 ; ; 1þ1þ2þ3þ1 1þ1þ2þ3þ1 1 : ð7Þ 1þ1þ2þ3þ1 Step 6: Calculate forecast value. From step 5, we can obtain the standardized weight matrix. Hereby, apply Eq. (8) to generate the forecast value (Ldf(t 1) is deffuzified matrix, Wn (t 1) is weight matrix)
order to have better identity due to human recognition. That is: U 1 ¼ ½0; 233Þ;
U 2 ¼ ½233; 468Þ;
U 4 ¼ ½701; 934Þ;
U 3 ¼ ½468; 701Þ;
U 5 ¼ ½934; 1167Þ;
U 6 ¼ ½1167; 1400Þ;
U 7 ¼ ½1400; 1633: Then, DSL subscriber linguistic observation of each quarter is as Table 3. Step 2: Establish fuzzy sets for observations.
ð8Þ
Each linguistic variable A1 to A7 can be defined as:
Step 7: Apply a to adapt the forecast value. The adapted-forecast Eq. (9) is generated from the modified forecast Eq. (8)
A1 ¼ 1=U 1 þ 0:5=U 2 þ 0=U 3 þ 0=U 4 þ 0=U 5 þ 0=U 6 þ 0=U 7
Adapted forecastðtÞ ¼ Actualðt 1Þ þ aðForecastðtÞ
þ 0=U 7 A3 ¼ 0=U 1 þ 0:5=U 2 þ 1=U 3 þ 0:5=U 4 þ 0=U 5 þ 0=U 6
ForecastðtÞ ¼ Ldf ðt 1Þ W n ðt 1Þ:
Actualðt 1ÞÞ:
ð9Þ
4. Verifications and comparisons To illustrate the forecasting performance of the proposed procedure, we use actual number of DSL subscriber in Taiwan (Status of broadband development, 2006). The data set covered from September 1999 to September 2002, which is used for training and December 2002 to June 2004 used for testing. Furthermore, the comparison of each model is listed.
A2 ¼ 0:5=U 1 þ 1=U 2 þ 0:5=U 3 þ 0=U 4 þ 0=U 5 þ 0=U 6
þ 0=U 7 A4 ¼ 0=U 1 þ 0=U 2 þ 0:5=U 3 þ 1=U 4 þ 0:5=U 5 þ 0=U 6 þ 0=U 7 A5 ¼ 0=U 1 þ 0=U 2 þ 0=U 3 þ 0:5=U 4 þ 1=U 5 þ 0:5=U 6 þ 0=U 7 A6 ¼ 0=U 1 þ 0=U 2 þ 0=U 3 þ 0=U 4 þ 0:5=U 5 þ 1=U 6 þ 0:5=U 7 A7 ¼ 0=U 1 þ 0=U 2 þ 0=U 3 þ 0=U 4 þ 0=U 5 þ 0:5=U 6 þ 1=U 7
4.1. Forecasting for cumulated DSL subscribers From the proposed algorithm in Section 3, the actual dataset is calculated as follows: Step 1: Define the universe of discourse and partition into intervals. According to actual data of cumulate DSL subscriber in quarter in Table 2, the universe of discourse for observations, U, is defined as [0, 1633]. According to Miller (1956), this study defines our intervals into seven levels in
Table 3 DSL subscriber linguistic observation of each quarter Quarter
DSL subscriber (thousand)
Linguistic value
Quarter
DSL subscriber (thousand)
Linguistic value
Sep-99 Mar-00 Sep-00 Mar-01 Sep-01 Mar-02 Sep-02
2 9 54 220 720 1180 1630
U1 U1 U1 U1 U4 U6 U7
Dec-99 Jun-00 Dec-00 Jun-01 Dec-01 Jun-02
5 13 120 450 920 1400
U1 U1 U1 U2 U4 U7
C.-H. Cheng et al. / Expert Systems with Applications 36 (2009) 1826–1832
From the comparisons in Table 4, the average forecasting error of Bass model is 22.35%, with SSE (sum of squares error) of 2,856,500; in Logistic model, the average error is 29.38%, and SSE is 4,500,800; in Dynamic model, the average error is 26.34%, and SSE is 5,211,500; in the proposed procedure, the average error is 10.67%, and SSE is 456,115. It is obvious that the proposed procedure has a smaller SSE and less average error than the models listed.
5000 Actual Bass Model Logistic Model Dynamic Model Proposed Model
Cumulated DSL Subscribers
4500 4000 3500
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3000 2500 2000 1500
5. Conclusions and future research
1000 500
Step 3: Build fuzzy logic relationships. For example, A6 ? A6; A6 ? A6; A6 ? A7; A6 ? A4. Step 4: Establish fuzzy relationships groups into corresponding trends. Up trend: A6 ? A7; No change: A6 ? A6, A6; Down trend: A6 ? A4. Step 5: Assign trend-weights. If F(t 1) = A6, the forecasting of f(t) is equal A7, A6, A6, A4. Thus, the weight matrix is [1/5, 1/5, 2/5, 1/5]. Step 6: Calculate forecast results. After iterative calculation by F(1) to F(7), the forecast values are 129, 816, 0, 1048, 0, 1513 and 1513, respectively. Step 7: Apply alpha to smoothen the forecast results.
As results in Table 4 and Fig. 2, this study confirms that the trend-weighted fuzzy time-series model can be good fit in innovation diffusion models. Therefore, this study has proposed a trend-weighted fuzzy time-series model to enhance the prediction accuracy in innovation diffusion of ICT products and to overcome the problems mentioned in the literature. From outcomes, this study can conclude that the proposed procedure would lead to better performance than the models listed in forecasting: (1) The proposed procedure is rule-based method, which have four advantages: highly expressive, easy to interpret, easy to generate, and predicting new instances rapidly; (2) assigning proper weights to classify recurrent fuzzy relationships, the proposed procedure can make more reasonable descriptions for the past patterns and more accurate predictions for the future; and (3) the reasonable alpha value adapted by user opinion can make more precise adjustments to match the past trends in the data set. However, there is still room for testing and improving the model as follows:
Finally, the alpha value is 0.1, which leads to the better outcome in prediction accuracy. The open data set from December 2002 to June 2004 is utilized to verify the researchers’ model. The comparisons with different models, Bass model, Logistic model, Dynamic model and the proposed procedure (Using a = 0.1 to generate the forecast results), are listed in Fig. 2 and the forecasting performances is listed in Table 4.
1. Using more empirical data of actual number of cumulated DSL subscriber in Taiwan to evaluate the accuracy and performance of the proposed procedure. 2. Applying different types of data sets to test the proposed procedure. 3. Reconsidering the factors affecting the diffusion of innovation and updating the proposed procedure to increase the prediction accuracy.
Table 4 Performances of the forecast results for cumulated DSL subscribers (thousand) with different models
References
0 2-Sep 2-Dec 3-Mar 3-Jun 3-Sep 3-Dec 4-Mar 4-Jun Quarters Fig. 2. Comparisons of the forecast results for cumulated DSL subscribers (thousand) with different models.
Quarters
Actual data
Bass model
Logistic model
Dynamic model
Proposed procedure
Sep-02 Dec-02 Mar-03 Jun-03 Sep-03 Dec-03 Mar-04 Jun-04
1630 1820 1990 2200 2400 2550 2710 2840
1700 1760 1800 1820 1830 1830 1840
1600 1620 1630 1640 1640 1640 1640
1890 2190 2530 2920 3370 3890 4490
1411 1618 1789 1942 2131 2311 2446 2590
22.35 2,856,500
29.38 4,500,800
26.34 5,211,500
10.67 456,115
Average error (%) SSE
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