Expert Systems with Applications

(1)

(2)

Taiwanese 3G mobile phone demand forecasting by SVR with hybrid

evolutionary algorithms

Wei-Chiang Hong

a,*

, Yucheng Dong

b

, Li-Yueh Chen

c

, Chien-Yuan Lai

a a

Department of Information Management, Oriental Institute of Technology, 58 Sec. 2, SiChuan Rd., Panchiao, Taipei 220, Taiwan b

Department of Management Science, School of Management, Xi’an Jiaotong University, Xi’an 710049, PR China c

Department of Hospitality Management, MingDao University, 369 Wen-Hua Rd., Peetow, Changhua, 52345, Taiwan

a r t i c l e

i n f o

Keywords:

Demand forecasting

Genetic algorithm–simulated annealing (GA–SA)

Support vector regression (SVR) Autoregressive integrated moving average (ARIMA)

General regression neural networks (GRNN) Third generation (3G) mobile phone

a b s t r a c t

Taiwan is one of the countries with higher mobile phone penetration rate in the world, along with the increasing maturity of 3G relevant products, the establishments of base stations, and updating regula-tions of 3G mobile phones, 3G mobile phones are gradually replacing 2G phones as the mainstream prod-uct. Therefore, accurate 3G mobile phones demand forecasting is desirable and necessary to communications policy makers and all enterprises. Due to the complex market competitions and various subscribers’ demands, 3G mobile phones demand forecasting reveals highly non-linear characteristics. Recently, support vector regression (SVR) has been successfully employed to solve non-linear regression and time-series problems. This investigation employs genetic algorithm–simulated annealing hybrid algorithm (GA–SA) to choose the suitable parameter combination for a SVR model. Subsequently, exam-ples of 3G mobile phones demand data from Taiwan were used to illustrate the proposed SVRGA–SA model. The empirical results reveal that the proposed model outperforms the other two models, namely the autoregressive integrated moving average (ARIMA) model and the general regression neural net-works (GRNN) model.

1. Introduction

1.1. Historical overview of mobile communications

Before explaining the importance and solid foundation of 3G mobile phones demand for economic growth and relevant business markets developments, it is worthwhile to take a brief historical overview of mobile communications. The matured characteristics of mobile communications (such as mobility, security, roaming on Internet, and improved voice/video service) are the most impor-tant attractiveness of current and potential subscribers.

First generation (1G) mobile phones were analog in nature, de-signed with primary focus on voice communications and provided localized wireless services. Only the basic but necessary communi-cational demands could be satisﬁed. By the late 1990s, the second generation (2G) mobile phones were deployed. 2G mobile phones were digital in nature, had improved voice capability (with short messaging services (SMS)), spectrum management, wider coverage area, and better mobility, in addition, added capability of text deliv-ery. During this time period, even the market also experienced the emergence of the Internet, however, it was far from reality to

re-ceive the same service by wireless approach. By the end of 2000, wireless voice services were already matured. By 2001, 2.5G tech-nologies were introduced, 2.5G techtech-nologies were also digital in nature, offering circuit and packed switched data services, such as voicemail, e-mail, location-based services (LBS), and other e-com-merce services. 2006 was year zero of the third generation (3G) mo-bile phone, which could provide content-rich applications independently of user location to reach the following main features for each subscriber: always-on connectivity, all IP network, global roaming, and value added services (Selian, 2002). More details and application designs for 3G mobile technologies could be re-ferred inTanguturi and Harmantzis (2006), Gerstheimer and Lupp (2004) and Yoo et al. (2005). 3G mobile phone thereby has been ex-pected to accelerate the development of mobile commerce and ser-vices (Yuan et al., 2006). For example, in Korea, only within a year after the launch of its 3G services in 2001, about 7% of the mobile phone subscribers had signed up for 3G services. At the same time, the average revenue per user (ARPU) for these users is nearly three times higher than that of 2G users (Yoo, Lyytinena, & Yang, 2005).

In Taiwan, the establishment of base stations, and updating of regulations governing 3G mobile phones, in addition, the relevant 3G products are also gradually mature, thus, 3G mobile phones are gradually replacing 2G phones as the market’s mainstream prod-uct. Therefore, mobile phone is currently going to the age of high

*Tel.: +886 2 7738 0145x5316; fax: +886 2 7738 6310. E-mail address:[email protected](W.-C. Hong)

Contents lists available atScienceDirect

Expert Systems with Applications

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e s w a

(3)

speed data communications, which combines personal mobile multimedia tools, computers, and television. They can provide well quality (always-on connectivity) in mobile television, videophone, and Internet functions. However, for telecommunications busi-nesses, it is important to understand the development trend of the 3G market and the growth of the 3G mobile phone penetration rate to allocate their investments in base stations and launched services; for policy makers, it is much more important to have the picture that which factors such as social economic degrees/lev-els, products/functions restrictions, economic disturbances, or costs block the rapid growth of 3G mobile phones. Therefore, an accurate forecast of 3G mobile phones demand, usually measured as a number of 3G mobile phones subscribers, is important to help researchers, telecommunications companies, and policy makers (or potential investors) with investigating future 3G development trends and making operational, tactical and marketing strategic decisions. Examples of operational decisions include business scheduling and staff training-on-job; tactical decisions relate to the preparation of 3G added-value services brochures, and strate-gic decisions are to do with base stations investments. Similarly, government bodies need accurate forecasts about 3G mobile phones demand in order to plan for telecommunication governing regulations. The beneﬁts of accurate forecasting are undisputed.

1.2. Reviews of forecasting approaches

Conventional quantitative forecasting models include two cate-gories, regression models and time-series models. Regression mod-els, also known as econometric models which are based on traditional statistical theory, are focused on constructing high rela-tionships among mobile phone subscribers and socio-economic fac-tors, such as income, living expenditure, and same generations’ effects. Those socio-economic factors are often found out that their coefficients are insignificant, thus, they play a minor role in coeffi-cients determination even they may have strengthened influences on mobile phone subscribers. On the other hand, a lot of scrapping redundant variables may sometimes increase the explanation abil-ity (the higher values of adjustedR2), however, co-linearity problem will also be suffered in the same time. This fact is one of the major limitations of econometric models. Therefore, more attention should be paid to collecting such variables. The second category ap-proaches are time-series forecasting models, developed byBox and Jenkins (1976), the ARIMA (autoregressive moving integrated mov-ing average) models have been one of the most popular approaches in time series forecasting, and are often employed when data are insufficient to build econometric models or when knowledge of the structure of regression models is limited. In some cases, as in short-term forecasting, time-series models are likely to outperform regression models (Witt & Witt, 1992). However, a fundamental limitation for time-series forecasting models is their inability to predict changes that are not evident in historical data, particularly for the non-linearity of mobile phone subscribers’ patterns.

Recently, due to the significant progress in the fields of pattern recognition methodology, artificial neural networks (ANNs) are pos-sible to be employed to forecast business demands. Many research-ers had applied ANNs concepts to construct appropriate forecasting models to implement forecasting works, such as original ANNs mod-els (_{Abdel-Aal, 2004; Law, 2000; Valverde Ramı´rez, de Campos} Vel-ho, & Ferreira, 2005; Vlahogianni, Karlaftis, & Golias, 2005; Yao & Tan, 2000; Zhang & Hu, 1998), and general regression neural net-works (GRNN) (Kamo & Dagli, 2009; Leung, Chen, & Daouk, 2000; Pai & Hong, 2005b; Wu & Lin, 2009). ANNs are primarily based on a model of emulating the processing of human neurological system to find out related spatial and temporal characteristics from the his-torical data patterns (especially for non-linear and dynamic evolu-tions), therefore, ANNs are able to approximate any degree of

complexity and without prior knowledge of problem solving, partic-ularly without understanding any assumptions of traditional statis-tical/econometric approaches required. As mentioned above that the process underlying mobile phone subscribers is complicated to be captured by a single linear statistical algorithm, ANNs have re-ceived much attention and been considered as alternatives for solv-ing mobile phone subscribers forecastsolv-ing. However, the trainsolv-ing procedure of ANNs models is not only time consuming but also pos-sible to get trapped in local minima and subjectively in selecting the model architecture (Suykens, 2001).

Proposed byVapnik (1995), support vector machines (SVMs) are one of the signiﬁcant developments in overcoming shortcom-ings of ANNs mentioned above. Rather than by implementing the empirical risk minimization (ERM) principle to minimize the train-ing error, SVMs apply the structural risk minimization (SRM) prin-ciple to minimize an upper bound on the generalization error. SVMs could theoretically guarantee to achieve the global optimum, instead of trapping local optimum like ANNs models. Thus, the solution of a non-linear problem in the original lower dimensional input space could ﬁnd its linear solution in the higher dimensional feature space. For more detailed mechanisms introduction of SVMs, it is referred toCortes and Vapnik (1995) and Vapnik et al. (1996), among others.

SVMs have found wide application in the ﬁeld of pattern recog-nition, bio-informatics, and other artiﬁcial intelligence relevant applications. Particularly, along with the introduction of Vapnik’s

e

-insensitive loss function, SVMs also have been extended to solve non-linear regression estimation problems (Drucker, Burges, Kauf-man, Smola, & Vapnik, 1997), which are so-called support vector regression (SVR). SVR have been successfully employed to solve forecasting problems in many fields, such as financial time-series (stocks index and exchange rate) forecasting (Cao, 2003; Cao & Gu, 2002; Huang, Nakamori, & Wang, 2005; Pai & Lin, 2005a, 2005b; Tay & Cao, 2001, 2002), engineering and software field (pro-duction values and reliability) forecasting (Pai & Hong, 2006), atmo-spheric science forecasting (Hong & Pai, 2007; Mohandes, Halawani, Rehman, & Hussain, 2004; Pai & Hong, 2007; Wang, Xu, & Lu, 2003), electric load forecasting (Pai & Hong, 2005a, 2005b), and so on. Meanwhile, SVR model had also been successfully ap-plied to forecast tourist arrivals (Pai & Hong, 2005c; Pai, Hong, Chang, & Chen, 2006). The empirical results indicated that the selec-tion of the three parameters (C,

e

, and

r

) in a SVR model inﬂuences the forecasting accuracy signiﬁcantly. Numerous publications in the literature had given some recommendations on appropriate setting of SVR parameters (Cherkassky & Ma, 2004), however, those approaches do not simultaneously consider the interaction effects among the three parameters. There is no general consensus and many contradictory opinions, thus, evolutionary algorithms are employed to determine appropriate parameter values.

1.3. GA–SA algorithm in parameter determination of a SVR model

Genetic algorithms (GAs) are auto-adaptive stochastic search techniques (Holland, 1975) that are based on the Darwinian sur-vival-of-the-ﬁttest philosophy and generate new individuals with selection, crossover and mutation operators. GAs start with a cod-ing of the parameter set of all types of objective functions, thus, GAs have the ability in solving problems those traditional algo-rithms are not easily to solve. InPai et al. (2006) and Pai and Hong (2005c), SVR with GAs is superior to other competitive forecasting models (ARIMA and ANNs). GAs are able to reserve a few best ﬁtted members of the whole population for the next generation in the operation process, however, after some generations GAs might lead to a premature convergence to a local optimum in the searching the suitable parameters of a SVR model.

(4)

Simulated annealing (SA) is a stochastic based general search tool that mimics the annealing process of material physics ( Kirkpa-trick, Gelatt, & Vecchi, 1983). When the system in the original state sold with energy is greater than that of the new generated state, this new state is automatically accepted. In contrast, the new state is accepted by Metropolis criterion with a probability function. The performance of SA is dependent on the cooling schedule. Thus, SA has some institution to be able to escape from local minima and reach global minimum (Lee & Johnson, 1983). In Pai and Hong (2005b) and Pai and Hong (2006), SVR with SA is also superior to other competitive forecasting models (Weibull Bayes, ARIMA, and GRNN). However, SA costs more computation time. To ensure the efﬁciency of SA, a proper temperature cooling rate (stop criterion) should be considered.

To overcome these drawbacks from GAs and SA, it is necessary to find some effective approach and improvement to avoid lead-ing to misleadlead-ing local optimum and to search optimum objective function efficiently. Genetic algorithm–simulated annealing (GA– SA) hybrid algorithm is a novel trial in dealing with the chal-lenges mentioned above. The GA–SA can firstly employ the supe-riority of SA algorithm to escape from local minima and approximate to the global minimum, and secondly apply the mutation process of GAs to improve searching ability in the range of values. In addition,Juang (2004)indicated that the hybrid of a GAs with existing algorithms can always produce a better algo-rithm than either the GAs or the existing algoalgo-rithms alone. There-fore, the hybrid algorithm has been applied to the fields of system design (Shieh & Peralta, 2005), system and network optimization (Ponnambalam & Reddy, 2003; Zhao & Zeng, 2006), query to information retrieval system (Cordón, Moya, & Zarco, 2002), con-tinuous-time production planning (Ganesh & Punniyamoorthy, 2005; Wang, Wong, & Rahman, 2004), and electrical power dis-tricting problem (Bergey, Ragsdale, & Hoskote, 2003). However, there is little application of the GA–SA to SVR’s parameter deter-mination. This investigation presented in this paper is motivated by a desire to solve the problem of maintaining the premature convergence to a local optimum of GAs and the efficiency of SA mentioned above in determining the three free parameters in the SVR 3G mobile phones demand forecasting model. Therefore, the GA–SA algorithm is employed in the SVR model, namely SVRGA–SA, to provide good forecasting performance in capturing non-linear 3G mobile phones demand changes tendency.

The remainder rest of the paper is organized as follows. The fun-damental principle and formulation of SVR and the GA–SA algo-rithm which is used to select the parameters of the SVR model are presented in the Section2, in addition, other alternative fore-casting models are also introduced. Numerical examples to dem-onstrate the forecasting performance of the proposed model and corresponding comparison results with the other forecasting mod-els are provided in Section3. Conclusions are ﬁnally made in Sec-tion4.

2. Forecasting models

2.1. Support vector regression

The brief ideas of SVMs for the case of regression are intro-duced. A non-linear mapping

u

ðÞ:Rn!Rnh _{is deﬁned to map} the input data (training data set)fðxi;yiÞg

N

i¼1into a so-called high dimensional feature space (which may have inﬁnite dimensions), Rnh₍_{Fig. 1}_{(a) and (b)). Then, in the high dimensional feature space,} there theoretically exists a linear function,f, to formulate the non-linear relationship between input data and output data. Such a lin-ear function, namely SVR function, is as,

fðxÞ ¼wT

u

_ð_x_{Þ þ}_b _ð₁_Þ

where fðxÞ denotes the forecasting values; the coefﬁcients wðw2Rnh_Þ _and _b_ð_b₂_R_Þ _{are adjustable. As mentioned above,} SVM method one aims at minimizing the empirical risk,

RempðfÞ ¼1 N XN i¼1

H

eðyi;w T

u

ðxiÞ þbÞ ð2Þ

whereHeðy;fðxÞÞis the

e

-insensitive loss function (as thick line in Fig. 1(c)) and deﬁned as Eq.(3),

He

ðy;fðxÞÞ ¼ jfðxÞ yj

e

; ifjfðxÞ yjP

e

0; otherwise

ð3Þ

In addition,Heðy; fðxÞÞis employed to ﬁnd out an optimum hy-per plane on the high dimensional feature space (Fig. 1(b)) to max-imize the distance separating the training data into two subsets. Thus, the SVR focuses on ﬁnding the optimum hyper plane and minimizing the training error between the training data and the

e

-insensitive loss function.

Then, the SVR minimizes the overall errors,

Minw;b;n_;_n R_eðw;n;nÞ ¼1 2w T_w þCX N i¼1 ðni þniÞ ð4Þ

with the constraints

yiwT

u

ðxiÞ b6

e

þn i; i¼1;2;. . .;N yiþw T

u

ðxiÞ þb6

e

_þ_n_i; i¼1;2;. . .;N ni P0; i¼1;2;. . .;N niP0; i¼1;2;. . .;N

The ﬁrst term of Eq.(4), employed the concept of maximizing the distance of two separated training data, is used to regularize weight sizes, to penalize large weights, and to maintain regression function ﬂatness. The second term penalizes training errors offðxÞ andyby using the

e

-insensitive loss function.Cis a parameter to trade off these two terms. Training errors above

e

are denoted as ni, whereas training errors below

e

are denoted asni(Fig. 1(b)).

After the quadratic optimization problem with inequality con-straints is solved, the parameter vectorwin Eq.(1)is obtained,

w¼X N

i¼1

ðbibiÞ

u

ðxiÞ ð5Þ

whereb_i;biare obtained by solving a quadratic program and are the

Lagrangian multipliers. Finally, the SVR regression function is ob-tained as Eq.(6)in the dual space,

fðxÞ ¼X N

i¼1

ðbi biÞKðxi;xÞ þb ð6Þ

whereKðxi;xjÞis called the kernel function, and the value of the

ker-nel equals the inner product of two vectors,xiandxj, in the feature

space

u

ðxiÞand

u

ðxjÞ, respectively; that is,Kðxi;xjÞ ¼

u

ðxiÞ

u

ðxjÞ.

Any function that meets Mercer’s condition (Vapnik, 1995) can be used as the kernel function.

There are several types of kernel function. The most used kernel functions are the Gaussian radial basis functions (RBF) with a width of

r

:Kðxi;xjÞ ¼expð0:5kxixjk2=r2Þand the polynomial

kernel with an order of d and constants a1 and

a2: Kðxi;xjÞ ¼ ða1xixjþa2Þd. If the value of

r

is very large, the RBF kernel approximates the use of a linear kernel (polynomial with an order of 1). Till now, it is hard to determine the type of ker-nel functions for speciﬁc data patterns (Amari & Wu, 1996). How-ever, the Gaussian RBF kernel is not only easier to implement, but also is capable to non-linearly map the training data into an inﬁ-nite dimensional space, thus, it is suitable to deal with non-linear relationship problems. Therefore, the Gaussian RBF kernel function

(5)

is speciﬁed in this study. The forecasting process of a SVR model is illustrated as inFig. 2.

The selection of the three positive parameters,C,

e;

r

of a SVR model is important to the accuracy of the forecasting. However, there is no structural method or any shortage opinions on efﬁcient setting of SVR parameters. The GA–SA algorithm is used in the pro-posed SVR model to optimize the parameter selection.

2.2. Genetic algorithms–simulated annealing hybrid algorithm (GA– SA)

2.2.1. Implementation structure of GA–SA

To overcome the drawbacks from GAs and SA, this study pro-pose a hybrid GA–SA algorithm by applying the superiority of SA to escape from local minima and approximate to the global min-imum, in addition, by using the mutation process of GAs to im-prove searching ability in the range of values. On the other hand, to avoid computation executing time consuming, only the optimal individual of GAs population will be delivered to the SA for further improving. The proposed GA–SA algorithm consists of the GAs part and the SA part. GAs evaluates the initial

popula-tion and operates on the populapopula-tion using three basic genetic operators to produce new population (best individual), then, for each generation of GAs, it will be delivered to SA for further pro-cessing. After finishing all the processes of SA, the modified indi-vidual will be sent back to GAs for the next generation. These computing iterations will be never stopped till the termination condition of the algorithm is reached. The proposed procedure of GA–SA is illustrated as follow and the flowchart is shown as Fig. 3.

The procedure of the GAs part is illustrated as follow:

Step 1: Initialization. Construct randomly the initial population of chromosomes. The three parameters,C,

r

, and

e

in a SVR model in theith generation are encoded into a bin-ary format; and represented by a chromosome that is composed of ‘‘genes” of binary numbers (Fig. 4). Each chromosome has three genes, which represent three parameters. Each gene has 40 bits. For instance, if each gene contains 40 bits, a chromosome contains 120 bits. More bits in a gene correspond to ﬁner partition of the search space.

Fig. 1.Transformation process illustration of a SVR model.

(6)

Step 2: Evaluating fitness. Evaluate the fitness of each chromo-some. Due to forecasting accuracy required, in this paper, a negative mean absolute percentage error (-MAPE) for forecasting errors calculation is used as the fitness func-tion. The MAPE is as Eq.(7),

MAPE¼1 N XN i¼1 aifi ai 100% ð7Þ

whereai andfi represent the actual and forecast values,

andNis the number of forecasting periods.

Step 3: Selection operation. Based on ﬁtness functions, chromo-somes with higher ﬁtness values are more likely to yield offspring in the next generation. The roulette wheel

selection principle (Holland, 1975) is applied to choose chromosomes for reproduction.

Step 4: Crossover operation and mutation operation. Mutations are performed randomly by converting a ‘‘1” bit into a ‘‘0” bit or a ‘‘0” bit in to a ‘‘1” bit. In crossover operation, chromosomes are paired randomly. The single-point-crossover principle is employed herein. Segments of paired chromosomes between two determined break-points are swapped. For simplicity, suppose a gene has four bits, thus, a chromosome contains 12 bits (Fig. 5). Before crossover is performed, the values of the three parameters in #1 parent are 1.5, 1.25 and 0.34375, respectively. For #2 parent, the three values are 0.625,

Fig. 3.The GA–SA algorithm ﬂowchart.

1 1 0 0 1 0 1 0 0

0 1 1

…

σ

C

ε

(7)

8.75 and 0.15625, accordingly. After crossover, for #1 off-spring, the three values are 1.625, 3.75 and 0.40625, accordingly. For #2 offspring, the three values are 0.5, 6.25 and 0.09375, respectively. Finally, decode the cross-over three parameters in a decimal format.

Step 5: Stop condition. If the number of generation is equal to a given scale, then the best chromosomes are presented as a solution, otherwise go to the Step 1 of the SA part. In the proposed GA–SA algorithm process, GAs will deliver its best individual to SA for further processing. After the optimal indi-vidual of GAs being improved, SA sends it back to GAs for the next generation. These computing iterations will be never stopped till the termination condition of the algorithm is reached. The proce-dure of the SA part is illustrated as follow:

Step 1: Generate initial current state. Receive values of the three parameters from GAs. The values of forecasting error, MAPE, shown as Eq.(7), is deﬁned as the system state (E). Here, the initial state (E0) is obtained.

Step 2: Provisional state. Make a random move to change the existing system state to a provisional state. Another set of three positive parameters are generated in this stage. Step 3: Metropolis criterion tests. The following Metropolis cri-terion equation is employed to determine the acceptance or rejection of provisional state (Metropolis, Rosenbluth, Rosenbluth, & Teller, 1953):

Accept the provisional state;ifEðsnewÞ>EðsoldÞ;andp<PðacceptsnewÞ;06p61: Accept the provisional state;ifEðsnewÞ6EðsoldÞ

Reject the provisional state; otherwise

8 > < > : ð8Þ

where thepis a random number to determine the acceptance of the provisional state,PðacceptsnewÞ, the probability of accepting the new

state, is given by the following probability function,

PðacceptsnewÞ ¼exp EðsoldÞkTEðsnewÞ

(T is the thermal equilibrium temperature,k represents the Boltzmann constant). If the provi-sional state is accepted, then set the proviprovi-sional state as the current state.

Step 4: Incumbent solutions. If the provisional state is not accepted, then return to step 2. Furthermore, if the cur-rent state is not superior to the system state, then repeat steps 2 and 3 until the current state is superior to the sys-tem state, and set the current state as new syssys-tem state.

Previous studies (Kirkpatrick et al., 1983) indicated that the maximum number of loops (Nsa) is 100dto avoid inﬁ-nitely repeated loops, where d denotes the problem dimension. In this investigation, three parameters (

r

,C, and

e

Þare used to determine the system states. There-fore,Nsais state to 300.

Step 5: Temperature reduction. After the new system state is obtained, reduce the temperature. The new temperature reduction is obtained by the Eq.(9):

New temperature¼ ðCurrent temperatureÞ

q

; where 0<

q

<1: ð9Þ

The

q

is set at 0.9 in this study (Dekkers & Aarts, 1991). If the pre-determined temperature is reached, then stop the algorithm and the latest state is an approximate optimal solution. Otherwise, go to step 2.

2.3. Other alternative forecasting models

In this study, for forecasting accuracy comparison with SVRGA– SA model, other alternative forecasting models, namely the autore-gressive integrated moving average (ARIMA) model and the gen-eral regression neural network (GRNN) model were employed to forecast Taiwanese 3G mobile phones demand. The introduction of these two modes is as follows.

2.3.1. ARIMA model

Introduced byBox and Jenkins (1976), the ARIMA model has been one of the most popular approaches in forecasting. In an AR-IMA model, the future value of a variable is supposed to be a linear combination of past values and past errors, expressed as follows:

yt¼h0þ/1yt1þ/2yt2þ þ/pytp

þ

e

th1

e

t1h2

e

t2 hq

e

tq ð10Þ

whereytis the actual value and

e

tis the random error at timet;/i

andhjare the coefﬁcients;pandqare integers and often referred to

as autoregressive and moving average polynomials, respectively. In addition, the difference ðrÞ is used to solve the non-stationary problem, and deﬁned as follows:

r

d yt¼

r

d1 yt

r

d1 yt1 ð11Þ

Basically, three phases are included in an ARIMA model: model identiﬁcation, parameter estimation and diagnostic checking. Fur-thermore, the backward shift operator,B, is deﬁned as follows:

B1yt¼yt1;B 2 yt¼yt2;. . . .;B p yt¼ytp ð12Þ B1

e

t¼

e

t1;B2

e

t¼

e

t2;. . . .;Bp

e

t¼

e

tp ð13Þ

then/pðBÞandhqðBÞcan be written as follows respectively:

/pðBÞ ¼1/1B 1 /2B 2 . . . ./pB p ð14Þ h1ðBÞ ¼1h1B1h2B2. . . .hqBq ð15Þ

Hence, Eq.(10)can be rewritten as Eq.(16), /pðBÞrd

yt¼C0þhqðBÞ

e

t ð16Þ

Eq.(16)is denoted as ARIMA (p,d,q) with non-zero constant,C0. For example, the ARIMA (2, 2, 1) model can be represented as Eq. (17).

/2ðBÞr2yt¼C0þh1ðBÞ

e

t ð17Þ

In general, the values ofp,d,qthen need to be estimated by autocorrelation function (ACF) and partial autocorrelation function (PACF) of the differenced series.

1 1 0 0 0 0 1 0 1 0 1 1 0 1 0 1 1 1 1 0 0 1 0 1 Parent 1 Parent 2 Crossover Point=1 Parameter 1 1 0 0 0 0 1 0 0 0 1 1 0 1 0 1 1 1 1 0 0 1 0 1 Offspring 1 Offspring 2 Parameter Parameter Parameter Parameter Parameter before crossover after crossover

(8)

2.3.2. GRNN model

The general regression neural network (GRNN) model, proposed bySpecht (1991), can approximate any arbitrary function from his-torical data. The foundation of GRNN operation is based on the the-ory of kernel regression. The procedure of the GRNN model can be equivalently represented as follows:

E½NjM ¼ R1 1NfðM;NÞdN R1 1fðM;NÞdN ð18Þ

where N is the predicted value of GRNN, M the input vector

ðM1;M2;. . .;MnÞwhich consists ofnvariables,E½NjMthe expected

value of the outputNgiven an input vectorM, andf(M,N) the joint probability density function ofMandN.

The GRNN primarily has four layers (Fig. 6). Each layer is as-signed with a speciﬁc computational function when non-linear regression function, Eq. (19), is performed. The ﬁrst layer of the network is to receive information. The input neurons then feed the data to the second layer. The primary task of the second layer is to memorize the relationship between the input neuron and its proper response. Therefore, the neurons in the second layer are also called pattern neurons. A multivariate Gaussian function of hi is given in Eq. (19), and the data from the input

neurons are used to compute an output hi by a typical pattern

neuroni, hi¼exp ð MUiÞ0ðMUiÞ 2

r

2 ð19Þ

whereUiis a speciﬁc training vector represented by pattern neuron

i, and

r

is the smoothing parameter. In the third layer, the neurons, namely the summation neurons, receive the outputs of the pattern neurons. The outputs from all pattern neurons are augmented. Basi-cally, two summations, the simple summation and the weighted summation, are conducted in the neurons of the third layer. The simple summation and the weighted summation operations can be represented as Eqs.(20) and (21)respectively.

Ss¼ X i hi ð20Þ Sw¼ X i wihi ð21Þ

wherewiis the pattern neuroniconnected to third layer of weights.

The summations of neurons in the third layer are then fed into the fourth layer. The GRNN regression outputQ is calculated as follows:

Q¼Ss Sw

ð22Þ

3. Numerical examples

3.1. The data set and index of performance evaluation

3G mobile phones demand data (2006–2008) is obtained from the revenue reports section of Chunghwa Telecom Co. Ltd. ﬁnancial information service which is published monthly (Chunghwa Tele-com Co. Ltd., 2008). Table 1 lists the total 27 data used in this example. The data exhibit a steady increasing tendency since Jan-uary 2006, and seems to follow 3-month cycles with increasing peaks. This study employs the Changhwa Telecom’s monthly 3G mobile phone demand data to compare the forecasting perfor-mances of the proposed SVRGA–SA model with those of ARIMA model and GRNN model. To be based on the same compared con-dition, in this paper, these 3G mobile phone demand data is di-vided into the three periods (training period, validation period, and testing period), particularly for the ratio of validation data to training data, it is recommended by Schalkoff’s (1997) to be approximately one to four. Therefore, the data set is divided as: training (January 2006–May 2007, 17 monthly 3G mobile phone subscribers), validation (June 2007–October 2007, 5 monthly 3G phone mobile subscribers), and testing (November 2007–March 2008, 5 monthly 3G phone mobile subscribers), accordingly.

(9)

The accuracy of the proposed 3G phone demand forecasting model is measured as mean absolute percentage error (MAPE), gi-ven by Eq.(7). The minimum values of MAPE indicate that the devi-ations between actual values and forecast values are very small.

3.2. Parameters determination of the three forecasting models

In this investigation, the free parameters of the three models are essential to obtain good forecasting results. For ARIMA models, the statistical package identiﬁed the most suitable model for the train-ing data as ARIMA (1, 1, 1) with constant term. The ARIMA (1, 1, 1) model can be expressed as follows:

ð10:0647B1Þryt¼98:91þ ð1þ0:3947B

1

Þ

e

t ð23Þ

After determining the suitable parameters of the ARIMA model, it is important to examine how closely the proposed model ﬁts a given time series. The autocorrelation function (ACF) was calcu-lated to verify the parameters.Fig. 7plots the estimated residual ACF and indicates that the residuals are not autocorrelated. PACF, the partial autocorrelation function, displayed inFig. 8, is also used to check the residuals and indicates that the residuals are not correlated.

For the GRNN model,Fig. 9shows the MAPE values of the GRNN with various

r

. Clearly, when

r

exceeds 0.42, the value of MAPE subsequently also increases. Therefore, the limit of

r

is 0.42. In this study, the value of

r

was set at 0.04.

For SVRGA–SA model, in the training stage, the rolling-based forecasting procedure is conducted, takingFig. 10as example, in which dividing training data into two subsets, namely fed-in (13 demand data) and fed-out (4 demand data) respectively. Firstly, the primary 13 demand data of fed-in subset are feeding into the proposed model, the structural risk minimization principle is em-ployed to minimize the training error, then, obtain one-step ahead forecasting demand, namely the 14th forecasting demand. Sec-ondly, the next 13 demand data, including 12 of the fed-in subset data (from 2nd to 13th) pulsing the 14th data in the fed-out subset, are similarly again fed into the proposed model, the structural risk minimization principle is also employed to minimize the training error, then, obtain one-step ahead forecasting demand, namely the 15th forecasting demand. Repeat the rolling-based forecasting procedure till the 17th forecasting demand is obtained.

Meanwhile, training error in this training stage is also obtained. Several types of data rolling were applied during the training stage to conduct the rolling-based forecasting procedure. Different num-bers of the 3G mobile phone demand in a time series were fed into the SVRGA–SA model to forecast the 3G demand in the next

Table 1

Total number of 3G mobile phone subscribers of Chunghwa Telecom Co. (2006–2008) (unit: 1000 subscribers).

Month (year) Subscribers Month (year) Subscribers Month (year) Subscribers

January 2006 304 January 2007 1035 January 2008 2365

February 2006 313 February 2007 1138 February 2008 2478

March 2006 328 March 2007 1285 March 2008 2588

April 2006 340 April 2007 1415 May 2006 360 May 2007 1520 June 2006 408 June 2007 1685 July 2006 458 July 2007 1805 August 2006 524 August 2007 1909 September 2006 597 September 2007 1993 October 2006 673 October 2007 2111 November 2006 786 November 2007 2195 December 2006 943 December 2007 2291

Resource:Chunghwa Telecom Co. Ltd. (2008).

1 2 3 4 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Lag Autocorrelation

ACF of Residuals for the ARIMA(1,1,1) model (with 95% confidence limits for the autocorrelations)

Fig. 7.Estimated residual ACF.

4 3 2 1 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Lag Partial Autocorrelation

PACF of Residuals for the ARIMA(1,1,1) model (with 95% confidence limits for the partial autocorrelations)

Fig. 8.Estimated residual PACF.

6 6.2 6.4 6.6 6.8 7 7.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 MAPE σ: [0.04,1]

(10)

validation period. While training errors improvement occurs, the three kernel parameters,

r

, C, and

e

of the SVRGA–SA model ad-justed by GA–SA algorithm are employed to calculate the valida-tion error. Then, the adjusted parameters with minimum validation error are selected as the most appropriate parameters. Finally, a ﬁve-step-ahead policy is employed to forecast 3G mobile phone demand. And, the kernel parameters,

r

, C, and

e

, in the pro-posed model with the smallest testing MAPE value is used as the most suitable model in this example.Table 2lists the free param-eters of the different models, in which the SVRGA model performs best when 10 fed-in data are used; the SVRGA–SA model performs best when 13 fed-in data are used.

Table 3lists the free parameters of the different models. These suitable parameters of the different models were used for forecast-ing the Taiwan 3G mobile phone demand in the testforecast-ing data set.

3.3. Forecasting results

The well-trained models, ARIMA (1, 1, 1), GRNN, SVRGA, and SVRGA–SA, are applied to forecast the 3G mobile phone demand from November 2007 to March 2008.Table 4shows the actual val-ues and the forecast valval-ues obtained using various forecasting

Fig. 10.The rolling-base forecasting procedure (training stage).

Table 2

Forecasting results of SVRGA and SVRGA–SA models.

SVRGA SVRGA–SA

Number of fed-in data Parameters Testing MAPE Number of fed-in data Parameters Testing MAPE

r C e r C e 10 0.7411 3057.6 5.143 2.142 10 0.8597 4007.1 5.044 1.254 11 0.7722 1005.90 47.980 3.506 11 0.6217 3330.9 0.569 1.053 12 3.9378 842.88 53.250 3.614 12 0.5380 5041.1 39.347 1.988 13 0.3959 1492.00 54.855 3.108 13 0.9010 7681 21.242 0.720 14 3.4412 1431.10 54.642 2.992 14 0.3295 9376.6 87.94 0.882 Table 3

Suitable values of parameters for different models.

Models Suitable parameter combinations ARIMA p¼1;d¼1;q¼1

GRNN r¼0:04

SVRGA r¼0:7411;C¼3;057:6;e¼5:143 SVRGA–SA r¼0:9010;C¼7681;e¼21:242

Table 4

Forecasting Taiwan 3G demand from November 2007 to March 2008 (unit: 1000 subscribers).

Months (years) Actual demand ARIMA GRNN SVRGA SVRGA–SA November 2007 2195 2226 2111 2143 2207 December 2007 2291 2333 2195 2259 2281 January 2008 2365 2439 2291 2353 2358 February 2008 2478 2544 2365 2426 2449 March 2008 2588 2650 2478 2476 2559 MAPE 2.287% 3.991% 2.142% 0.720%

(11)

models in this example. The MAPE values for each month are cal-culated to compare fairly the proposed models with other alterna-tive models. The proposed SVRGA–SA model has smaller MAPE values than the ARIMA (1, 1, 1), GRNNð

r

¼0:04Þ, and SVRGA mod-els to capture the 3G demand patterns on ﬁve-month average basis.

In addition, the GA–SA algorithm also helps to avoid trapping into local minimum than GAs did, thus, outperform the SVRGA model. For example, inTable 2, the GA–SA algorithm is then excel-lently to shift the local solution of SVRGA model by 13 fed-in data rolling type, ð

r;

C;eÞ ¼ ð0:3959;1492;54:855Þ with local optimal forecasting errors, in terms of MAPE (3.108%), to be improved by GA–SA algorithm to another better solution, ð

r;

C;eÞ ¼ ð0:9010;

7681;21:242Þto be the appropriate local optimal forecasting error in terms of MAPE (0.720%). Thus, it once again reveals that GA–SA algorithm is much appropriate than GAs in parameter adjustments to achieve forecasting accuracy improvement by integrated into the SVR model.Fig. 11illustrates the forecasting 3G mobile phone demand of ARIMA, GRNN, SVRGA and SVRGA–SA models.

To verify the signiﬁcance of accuracy improvement of SVRGA– SA model, Mann-WhitneyUtest (Mann & Whitney, 1947) is con-ducted. Mann-Whitney U test is an approach assessing the signif-icance of a difference in central tendency of two data series. These two data error series are ranked from the smallest value to the largest value. The test statisticUis given by Eq.(24):

U¼minfU1;U2g ð24Þ where U1¼n1n2þn1ðn1þ1Þ 2 R1; ð25Þ U2¼n1n2þ n2ðn2þ1Þ 2 R2 ð26Þ

Then1andn2are the sizes of data series I and data series II, respectively. TheR1andR2are the rank sums of data series I and data series II, correspondingly.

Mann-WhitneyUtest is performed at the 0.025 and 0.05 signif-icance levels in one-tail-test. The test results (Table 5) showed that the SVRGA–SA model almost yields improved forecast results and signiﬁcantly outperforms the other three forecasting models.

4. Conclusions

As mentioned above, in Taiwan, the establishment of 3G related infrastructures, governing regulations of 3G telecommunication, and the 3G relevant products are gradually mature, therefore, 3G telecommunication is currently going to the age of high speed data communications. However, there are several important issues such as whether the growth of the 3G mobile phone penetration rate will proceed quickly in keeping up with the 2G model, which socio-eco-nomic factors (such as social levels, inherent product restrictions, economic disturbances, or costs) will block the rapid growth of 3G phones. In other words, accurate 3G mobile phone demand forecast-ing will not only well investigate the future development trends of 3G, but also provide important guide for effective implementations of 3G related businesses nurturing. In addition, for telecommunica-tion businesses, it will help them plan their future marketing strate-gies. Particularly, in the highly technological changes and innovations of telecommunication market make 3G mobile phone demand forecasting more difﬁcult. Thus, it is worth analyzing where these forecasts fail and how forecasting accuracy is improved.

In this investigation, SVRGA–SA model is proposed to predict Taiwan 3G phone demand. The numerical example of Chunghwa Telecom Co. Ltd. is used to elucidate the forecasting accuracy of proposed model. This study evaluates the feasibility of GA–SA algorithm in parameter determination to achieve forecasting accuracy improvement by integrated into the SVR model.Tables 2 and 4illustrated that the SVRGA–SA model had given better re-sults than other forecasting models; in the meanwhile, particu-larly, the SVRGA–SA model had avoided being trapped in local optimum like SVRGA (hybrid SVR and GAs) did. The superior per-formance of the SVRGA–SA model is caused of: (1) non-linear mapping capabilities and so can more easily capture data patterns of tourist arrivals than can ARIMA models;(2)SVR model applies structural risk minimization rather than minimizing the training errors, this minimization of an upper bound on the generalization error provides better generalization performance than that of AR-IMA and GRNN models; finally,(3) the parameter selection in a SVR model heavily influences their forecasting performance, thus, improper selection of these three parameters will lead to either over-fitting or under-fitting of a SVR model. The GA–SA algorithm employed the superiority of SA algorithm to escape from local minima to the global minimum, and then, applied the mutation process of GAs’ searching capability to determine proper parame-ters combination.

The favorable results obtained in this work reveal that the pro-posed model is a valid alternative for use in high technological products demand forecasting science. In the future, other novel hy-brid evolutionary algorithms should be further applied to obtain more appropriate parameter combination, and then, to achieve more improvable, satisfactory accurate high technological prod-ucts demand forecasting if it exists. Other socio-economic factors, such as market prices and gross domestic expenditure per person, could be included in the SVRGA–SA model to further forecast 3G relevant products.

Acknowledgment

This research was conducted with the support of National Sci-ence Council, Taiwan (NSC 97-2410-H-161-001, NSC 98-2410-H-161-001, and NSC 98-2811-H-161-001). 2000 2100 2200 2300 2400 2500 2600 2700

Nov. 2007 Dec. 2007 Jan. 2008 Feb. 2008 Mar. 2008 3G demand Actual ARIMA GRNN SVRGA SVRGA-SA

Fig. 11.Forecasting demands by ARIMA, GRNN, SVRGA and SVRGA–SA models (November 2007 to March 2008). Table 5 Mann-WhitneyUtest. Mann-WhitneyUtest a¼0:025 a¼0:05 U¼2 U¼4 SVRGA–SA vs. ARIMA (1, 1, 1) 0 0 SVRGA–SA vs. GRNNðr¼0:04Þ 0 0 SVRGA–SA vs. SVRGA 2.5 2.5

(12)

References

Abdel-Aal, R. E. (2004). Short-term hourly load forecasting using abductive networks.IEEE Transactions on Power Systems, 19(1), 164–173.

Amari, S., & Wu, S. (1996). Improving support vector machine classiﬁers by modifying kernel functions.Neural Networks, 12(6), 783–789.

Bergey, P. K., Ragsdale, C. T., & Hoskote, M. (2003). A simulated annealing genetic algorithm for the electrical power districting problem.Annals of Operations Research, 121(1–4), 33–55.

Box, G. E. P., & Jenkins, G. M. (1976).Time series analysis: Forecasting and control. San Francisco: Holden-Day.

Cao, L. (2003). Support vector machines experts for time series forecasting. Neurocomputing, 51(1–4), 321–339.

Cao, L., & Gu, Q. (2002). Dynamic support vector machines for non-stationary time series forecasting.Intelligent Data Analysis, 6(1), 67–83.

Cherkassky, V., & Ma, Y. (2004). Practical selection of SVM parameters and noise estimation for SVM regression.Neural Networks, 17(1), 113–126.

Chunghwa Telecom Co. Ltd., (2008).Monthly revenue reports. Financial Information Service.http://www.cht.com.tw/CompanyCat.php?Page=FileDownload&CatID=798. Cordón, O., Moya, F., & Zarco, C. (2002). A new evolutionary algorithm combining simulated annealing and genetic programming for relevance feedback in fuzzy information retrieval systems.Soft Computing, 6(5), 308–319.

Cortes, C., & Vapnik, V. (1995). Support vector networks.Machine Learning, 20(3), 273–297.

Dekkers, A., & Aarts, E. H. L. (1991). Global optimization and simulated annealing. Mathematical Programming, 50(1–3), 367–393.

Drucker, H., Burges, C. J. C., Kaufman, L., Smola, A., & Vapnik, V. N. (1997). Support vector regression machines.Advances in Neural Information Processing Systems, 9(2), 155–161.

Ganesh, K., & Punniyamoorthy, M. (2005). Optimization of continuous-time production planning using hybrid genetic algorithms–simulated annealing. International Journal of Advanced Manufacturing Technology, 26(1–2), 148–154. Gerstheimer, O., & Lupp, C. (2004). Needs versus technology—the challenge to design third-generation mobile applications. Journal of Business Research, 57(12), 1409–1415.

Holland, J. (1975).Adaptation in natural and artiﬁcial system. Ann Arbor: University of Michigan Press.

Hong, W. C., & Pai, P. F. (2007). Potential assessment of the support vector regression technique in rainfall forecasting.Water Resources Management, 21(2), 495–513.

Huang, W., Nakamori, Y., & Wang, S. Y. (2005). Forecasting stock market movement direction with support vector machine. Computers & Operations Research, 32(10), 2513–2522.

Juang, C. F. (2004). A hybrid of genetic algorithm and particle swarm optimization for recurrent network design. IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, 34(2), 997–1006.

Kamo, T., & Dagli, C. (2009). Hybrid approach to the Japanese candlestick method for ﬁnancial forecasting.Expert Systems with Applications, 36(3 Part), 5023–5030.

Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing.Science, 220(4598), 671–680.

Law, R. (2000). Back-Propagation Learning in Improving the Accuracy of Neural Network-Based Tourism Demand Forecasting. Tourism Management, 21(4), 331–340.

Lee, J., & Johnson, G. E. (1983). Optimal tolerance allotment using a genetic algorithm and truncated Monte Carlo simulation.Computer Aided Design, 25(9), 601–611.

Leung, M. T., Chen, A. S., & Daouk, H. (2000). Forecasting exchange rates using general regression neural networks.Computers and Operations Research, 27(11– 12), 1093–1110.

Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other.Annals of Mathematical Statistics, 18(1), 50–60.

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., & Teller, A. H. (1953). Equations of state calculations by fast computing machines.Journal of Chemical Physics, 21(6), 1087–1092.

Mohandes, M. A., Halawani, T. O., Rehman, S., & Hussain, A. A. (2004). Support vector machines for wind speed prediction.Renewable Energy, 29(6), 939–947. Pai, P. F., & Hong, W. C. (2005a). Forecasting regional electric load based on

recurrent support vector machines with genetic algorithms.Electric Power Systems Research, 74(3), 417–425.

Pai, P. F., & Hong, W. C. (2005b). Support vector machines with simulated annealing algorithms in electricity load forecasting.Energy Conversion and Management, 46(17), 2626–2669.

Pai, P. F., & Hong, W. C. (2005c). An improved neural network model in forecasting arrivals.Annals of Tourism Research, 32(4), 1138–1141.

Pai, P. F., & Hong, W. C. (2006). Software reliability forecasting by support vector machines with simulated annealing algorithms.Journal of Systems and Software, 79(6), 747–755.

Pai, P. F., & Hong, W. C. (2007). A recurrent support vector regression model in rainfall forecasting.Hydrological Processes, 21(6), 819–827.

Pai, P. F., Hong, W. C., Chang, P. T., & Chen, C. T. (2006). The application of support vector machines to forecast tourist arrivals in Barbados: An empirical study. International Journal of Management, 23(2), 375–385.

Pai, P. F., & Lin, C. S. (2005a). Using support vector machines in forecasting production values of machinery industry in Taiwan.International Journal of Advanced Manufacturing Technology, 27(1–2), 205–210.

Pai, P. F., & Lin, C. S. (2005b). A hybrid ARIMA and support vector machines model in stock price forecasting.Omega, 33(6), 497–505.

Ponnambalam, S. G., & Reddy, M. M. (2003). A GA–SA multiobjective hybrid search algorithm for integrating lot sizing and sequencing in ﬂow-line scheduling. International Journal of Advanced Manufacturing Technology, 21(2), 126–137. Schalkoff, R. J. (1997).Artiﬁcial neural networks. New York: McGraw-Hill. Selian, A. (2002).From GSM to IMT-2000—A comparative analysis. International

Telecommunications Union (ITU).

Shieh, H. J., & Peralta, R. C. (2005). Optimal in situ bioremediation design by hybrid genetic algorithm–simulated annealing.Journal of Water Resources Planning and Management, 131(1), 67–78.

Specht, D. A. (1991). A general regression neural network.IEEE Transactions on Neural Networks, 2(6), 568–576.

Suykens, J. A. K. (2001). Nonlinear modelling and support vector machines. In Proceedings of IEEE instrumentation and measurement technology conference(pp. 287–294).

Tanguturi, V. P., & Harmantzis, F. C. (2006). Migration to 3G wireless broadband internet and real options: The case of an operator in India.Telecommunications Policy, 30(7), 400–419.

Tay, F. E. H., & Cao, L. (2001). Application of support vector machines in ﬁnancial time series forecasting.Omega, 29(4), 309–317.

Tay, F. E. H., & Cao, L. (2002). Modiﬁed support vector machines in ﬁnancial time series forecasting.Neurocomputing, 48(1–4), 847–861.

Valverde Ramı´ rez, M. C., de Campos Velho, H. F., & Ferreira, N. J. (2005). Artiﬁcial neural network technique for rainfall forecasting applied to The São Paulo Region.Journal of Hydrology, 301(1–4), 146–162.

Vapnik, V. (1995).The nature of statistical learning theory. New York: Springer-Verlag. Vapnik, V., Golowich, S., & Smola, A. (1996). Support vector machine for function approximation, regression estimation, and signal processing.Advances in Neural Information Processing Systems, 9(4), 281–287.

Vlahogianni, E. I., Karlaftis, M. G., & Golias, J. C. (2005). Optimized and meta-optimized neural networks for short-term trafﬁc ﬂow prediction: A genetic approach.Transportation Research Part C, 13(3), 211–234.

Wang, W., Xu, Z., & Lu, J. W. (2003). Three improved neural network models for air quality forecasting.Engineering Computations, 20(2), 192–210.

Wang, Z. G., Wong, Y. S., & Rahman, M. (2004). Optimisation of multi-pass milling using genetic algorithm and genetic simulated annealing.International Journal of Advanced Manufacturing Technology, 24(9-10), 727–732.

Witt, S. F., & Witt, C. A. (1992).Modeling and forecasting demand in tourism. London: Academic Press.

Wu, J. D., & Lin, B. F. (2009). Speaker identiﬁcation based on the frame linear predictive coding spectrum technique.Expert Systems with Applications, 36(4), 8056–8063.

Yao, J., & Tan, C. L. (2000). A case study on using neural networks to perform technical forecasting of forex.Neurocomputing, 34(1–4), 79–98.

Yoo, Y., Lyytinena, K., & Yang, H. (2005). The role of standards in innovation and diffusion of broadband mobile services: The case of South Korea.Journal of Strategic Information Systems, 14(3), 323–353.

Yuan, Y., Zheng, W., Wang, Y., Xue, Z., Yang, Q., & Gao, Y. (2006). Xiaolingtong versus 3G in China: Which will be the winner?Telecommunications Policy, 30(5-6), 297–313.

Zhang, G., & Hu, M. Y. (1998). Neural network forecasting of the British Pound/US Dollar Exchange Rate.Omega, 26(4), 495–506.

Zhao, F., & Zeng, X. (2006). Simulated annealing—genetic algorithm for transit network optimization.Journal of Computing in Civil Engineering, 20(1), 57–68.

10.1016/j.eswa.2009.12.066

ScienceDirect

w w w . e l s e v i e r . c o m / l o c a t e / e s w a

http://www.cht.com.tw/CompanyCat.php?Page=FileDownload&CatID=798.