Forecasting holiday daily tourist flow based on seasonal support vector regression with adaptive genetic algorithm

Applied Soft Computing 26 (2015) 435–443

Contents lists available at ScienceDirect

Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Review Article

Forecasting holiday daily tourist flow based on seasonal support vector regression with adaptive genetic algorithm Rong Chen a,b,c , Chang-Yong Liang a,c,∗ , Wei-Chiang Hong d , Dong-Xiao Gu a,c a

School of Management, Hefei University of Technology, Hefei, Anhui 230009, PR China Department of Economic and Management, Bengbu University, Bengbu, Anhui 233000, PR China Ministry of Education Engineering Research Center for Intelligent Decision-making & Information Systems Technologies, Hefei, Anhui 230009, PR China d Department of Information Management, Oriental Institute of Technology, Panchiao, New Taipei 220, Taiwan b c

a r t i c l e

i n f o

Article history: Received 15 March 2014 Received in revised form 12 July 2014 Accepted 16 October 2014 Available online 23 October 2014 Keywords: Holiday daily tourist flow forecasting Support vector regression Adaptive genetic algorithm Seasonal index adjustment Back-propagation neural network

a b s t r a c t Accurate holiday daily tourist flow forecasting is always the most important issue in tourism industry. However, it is found that holiday daily tourist flow demonstrates a complex nonlinear characteristic and obvious seasonal tendency from different periods of holidays as well as the seasonal nature of climates. Support vector regression (SVR) has been widely applied to deal with nonlinear time series forecasting problems, but it suffers from the critical parameters selection and the influence of seasonal tendency. This article proposes an approach which hybridizes SVR model with adaptive genetic algorithm (AGA) and the seasonal index adjustment, namely AGA-SSVR, to forecast holiday daily tourist flow. In addition, holiday daily tourist flow data from 2008 to 2012 for Mountain Huangshan in China are employed as numerical examples to validate the performance of the proposed model. The experimental results indicate that the AGA-SSVR model is an effective approach with more accuracy than the other alternative models including AGA-SVR and back-propagation neural network (BPNN). © 2014 Elsevier B.V. All rights reserved.

Contents 1.

2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Traditional tourist flow forecasting approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Support vector regression with evolutionary algorithms in tourist flow forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Seasonal adjustment in tourist flow forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The AGA-SSVR forecasting model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Support vector regression principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The selection of optimization parameters using AGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The AGA-SVR procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Seasonal index adjustment approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental model settings and comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. AGA-SSVR holiday daily tourist flow forecasting model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ Corresponding author at: School of Management, Hefei University of Technology, Hefei, Anhui 230009, PR China. Tel.: +86 551 62919150; fax: +86 551 62904962. E-mail address: [email protected] (C.-Y. Liang). http://dx.doi.org/10.1016/j.asoc.2014.10.022 1568-4946/© 2014 Elsevier B.V. All rights reserved.

436 436 436 436 437 437 437 438 438 438 438 442 442 443 443

436

R. Chen et al. / Applied Soft Computing 26 (2015) 435–443

1. Introduction With the implementation of the new official holiday regulation from 2007 in China, the number of tourists in holidays has increased rapidly. The huge market has been boosting by the business of holiday tourism. According to reports of the National Holiday Office (NHO), in 2011 the total number of tourists reached about 153 million during Spring Festival and 302 million during National Day, with year-on-year growth of more than 22.7% and 18.8% respectively, and tourism income added up to 82.05 billion yuan and 145.8 billion yuan, increasing 27% and 25.1% respectively [1]. The rapid growth of tourists flow in the short term will be a heavy burden for sightseeing spots, airports, and hotels during holidays. Since various important decisions, such as tourism planning, transportation and accommodation have been made based on the results of holiday daily tourist flow forecasting, an optimal solution of the allocation of social resources in such areas is crucial [2]. Accurate forecasting in holiday daily tourist flow can provide direct basis for tourism decision-makers, which can help them make scientific decisions. Then human, financial and material resources of scenic spots can be well planned and allocated ahead of time, and the over-consumption of tourism resources can be avoided as well. However, in China, there are seven legal holidays, including New Year, Spring Festival, Ching-Ming Festival, May Day, Dragon Boat Festival, Mid-Autumn Festival and National Day, as well as summer holidays. Each of them has its own fluctuation pattern from weather conditions and different holiday periods, which make holiday daily tourist flow present a complicated nonlinear characteristic and seasonal tendency. Because of their complicated nonlinearity and seasonality, currently existing methods cannot exactly deal with both issues. Accurate forecasting of holiday daily tourist flow remains a difficult task attracted attentions in the literatures, and it is greatly necessary to develop new forecasting techniques to obtain satisfied accurate level.

1.1. Traditional tourist flow forecasting approaches In recent years, various researches in tourist flow forecasting have resulted in the development of numerous forecasting approaches, which in general can be classified into two types, classical linear methods and nonlinear methods [3]. The most popular classical linear methods include autoregressive moving average (ARMA) [4,5], autoregressive integrated moving average (ARIMA) [6,7], and exponential smoothing (ES) [8,9], etc. These methods usually employ historical data to forecast the future tourist flow by a univariate or multivariate mathematical function, which mostly depend on linear assumptions. However there are some drawbacks in linear methods, for example, inability to capture the seasonal and nonlinear characteristics [10]. Thus, the nonlinear methods have been paid much more attention in recent years, for example artificial neural network (ANN) method. These methods emulate the processes of the human neurological system to process self-learning from the historical tourist flow patterns, especially for nonlinear and dynamic variations. And now the ANN methods have been widely used in tourist flow forecasting [11,12]. Empirical evidence shows that the ANNs generally outperform the classical linear methods (ES, ARIMA) in tourism forecasting [13]. But the methods lack a systematic procedure for model building, therefore, obtaining a reliable neural model involves selecting a large number of parameters experimentally through trial and error [14]. Additionally, ANN cannot successfully capture the changes of seasonality or trend [15].

1.2. Support vector regression with evolutionary algorithms in tourist flow forecasting In mid 1990s, Vapnik [16] developed a statistical learning algorithm – support vector machines (SVM), which adheres to the principle of structural risk minimization seeking to minimize the upper bound of the generalization error, rather than minimize the training error. SVM has been extended to solve nonlinear regression estimation problems, i.e., the so-called SVR. As a statistical theory-based method, SVR overcomes shortcomings of traditional tourist flow forecasting approaches. At present, SVR has been applied to financial time series forecasting [17–19], electric power load forecasting [20,21] and traffic flow forecasting [22,23]. It has also successfully been applied to tourist flow forecasting [24–27] which have been proved superior to ARIMA, ES and BPNN etc. However, the previous research results show parameter optimization in SVR plays an important role in building a prediction model with high prediction accuracy and stability [28]. And poor forecasting accuracy is mainly suffered from lacking knowledge of the selection of three parameters (C,  2 and ε) in a SVR model. But till now, no general rules are feasible to determine suitable parameters [29]. Therefore, the determination of optimal parameters is a critical procedure in the SVR research fields. In recent years, GAs have already been used to select optimization parameters of SVR in nonlinear forecasting successfully [30,31]. As an auto-adaptive stochastic search techniques, the core of this class of algorithms lies in the production of new genetic structures along selection, crossover and mutation operators, thereby providing innovations to solutions for the problem at hand [32]. However, fixed parameters of the crossover probability pc and the mutation probability pm in the GAs often affect performance directly, its parameter settings without tuning often lead to some questions, such as premature convergence and local optima [33]. In order to overcome these drawbacks, various adaptive techniques have been suggested to adjust parameters such as mutation probability and crossover probability in the process of running Gas [34]. Adaptive genetic algorithm (AGA) is one of them. In the AGA, the probabilities of crossover pc and mutation pm are adaptively varied depending on the fitness values of the solution. High fitness solutions are ‘protected’, while solutions with subaverage fitnesses are totally disrupted [35]. Recently, the AGA combined with SVR approach has been concerned in literatures [36]. But there are few articles in tourism forecasting which refer to this combined method. Therefore, SVR with AGA will be employed in this paper, by which, forecasting performance in capturing nonlinear and searching for parameters will to be expected to greatly improved. 1.3. Seasonal adjustment in tourist flow forecasting Seasonality is a notable characteristic of tourist flow forecasting, it affects the accuracy of tourist flow forecasting and cannot be ignored in the modeling process. Therefore, how to handle seasonal variations of tourism data has always been an important and complex issue in tourist flow forecasting analysis [37]. The common models used for dealing with seasonal variations are to eliminate seasonal variations by some seasonal adjustment approaches, and then the models are scaled back through the estimated seasonal effects for prediction. Generally speaking, the common seasonal adjustment method is to eliminate the seasonal factors, filtering the original data by differencing before forecasting. Seasonal ARIMA (SARIMA) method is the most widely used seasonal adjustment forecasting methods. As another most popular and important seasonal adjustment methods, X-11-ARIMA and X-12-ARIMA have also been applied to adjust seasonality in time series [38–40]. But all models mentioned above have often been criticized. Firstly,

R. Chen et al. / Applied Soft Computing 26 (2015) 435–443

differencing is not always an appropriate way to handle both seasonality and nonlinearity [41]; secondly, they are applied alone because of the lack of an explicit model concerning the decomposition of the original series, an essentially linear relationship assumed in these models limits their capability to model complex nonlinear problems commonly encountered in reality [42]. For example, Chen and Pai [26,43] considered the forecasting accuracy of SARIMA is often inferior in non-linear data when compared with SVR; thirdly, the methods require a large amount of previous data, which is not always possible. Therefore the application of the methods will sometimes limited [44]. Due to aforementioned reasons, seasonal index adjustment approach [45,46] is presented to deal with seasonal variations in this article. Seasonal index adjustment focuses on predicted values, not only adjusting seasonality but also avoiding the limitations of above methods. In summary, the proposed AGA-SSVR model is to apply SVR to deal with nonlinearity and AGA to optimize its parameters, as well as seasonal index adjustment to handle seasonality. This hybrid method is a new application in tourism forecasting. The holiday daily data set of the world geological park Mount Huangshan in China is used as examples. The experimental results demonstrate that the proposed model outperform AGA-SVR and back-propagation neural network (BPNN). The rest of paper is organized as follows. Section 2 describes the principles of the AGA-SSVR forecasting model. Section 3 presents two different numerical examples from Mountain Huangshan in China, validating the forecasting performance of the proposed model. Section 4 gives the conclusions.

2. The AGA-SSVR forecasting model 2.1. Support vector regression principle As a learning algorithm based on statistical learning theories, the essence of SVR is designed to solve a constrained quadric programming and provide a global optimization solution: given a data set (xi , yi ), i = 1, 2, . . ., N, yi ∈ R, xi = Rn , where xi is the ith input in ndimensional space, and yi is the actual output value corresponding to xi . Through a nonlinear transformation ϕ(x), SVR maps input data xi to high dimensional feature space (possibly infinite dimensional). Theoretically, there is a linear function f in the high dimensional space, which can describe the relationships between input data and output ones. This linear function is formulated as SVR function: f (x) = wT ϕ(x) + b

(1)

where f(x) denotes the forecasting values; w and b are coefficients determined by minimizing the regularized risk function as follows: 1 1 1 ||w||2 + Remp = ||w||2 + C × |yi − f (xi )|ε 2 2 n

From Eqs. (1)–(3), the function regression problem is equivalent to functional minimization:

 1 (i + i∗ ) ||w||2 + C 2 N

min

i=1

yi − wT ˚(x) − b ≤ ε + i∗ , i = 1, 2, . . ., N st

− yi

i=1

w=

N 

 |y − f (x)|ε =

0, |y − f (x)| ≤ ε

  y − f (x) − ε, else

(3)

≤ ε + i , i = 1, 2, . . ., N

(ˇi − ˇi∗ )ϕ(xi )

(5)

i=1

where ˇi and ˇi∗ are the Lagrangian multipliers and are gained by solving a quadratic program. Eventually, the SVR regression function is obtained in the dual space shown below: f (x) =

N 

(ˇi − ˇi∗ )k(xi , x) + b

(6)

i=1

where k(xi , x) is a kernel function which satisfies Mercer theorem [46]. An example of kernels which satisfies Mercer’s theorem is Gaussian radial basis function (RBF) kernel [47]. In machine learning theories, the approximating feature map for Gaussian RBF kernel not only performs non-linear mapping between the input space and a high-dimensional space, but also it is easier to implement [48], thus it is suitable to deal with nonlinear relationship problems [49]. Therefore, in the kernel functions of SVR, Gaussian RBF kernel is the most widely used [31,50]. In our article, the data set present obvious nonlinearity, clearly Gaussian RBF kernel is chosen in this article: f (x) =

N  i=1



(ai − a∗i ) exp

−||xi − x||2 2 2



+b

(7)

In Eq. (7),  is the width of RBF kernel function, xi , x are the input vector of the training data and the input vector of the testing data respectively. 2.2. The selection of optimization parameters using AGA Different from GAs, in AGA, the crossover probability pc and the mutation probability pm are varied adaptively based on the fitness values of the solution, which low pc and pm are distributed to high fitness solutions and excellent solutions are preserved. Meanwhile high pc and pm are assigned to low fitness solutions which makes the speed of new individual increased. In our approach, the adaptive pc and pm are employed as follows: pc =

pc1 − (pc1 − pc2 )(f  − favg )/(fmax − favg ) f  > favg

 where Remp represents the empirical risks; (1/2)||w||2 and C denote the Euclidean norm and a cost parameter measuring the empirical risk respectively; |yi − f(xi )|ε is an ε-insensitive loss function, which controls deviation and makes the estimation robust. The details are formulated as follows:

(4)

where i∗ , i are slack variables used to represent the sizes of the stated excess positive and negative deviation. After the quadratic optimization problem with inequality constraints is solved, the parameter vector w in Eq. (1) is obtained:



(2)

+ wT ˚(x) + b

i∗ , i ≥ 0, i = 1, 2, . . ., N

N

Rreg =

437

pm =

pc1

f  ≤ favg

pm1 − (pm1 − pm2 )(f − favg )/(fmax − favg ) f > favg pm1

f ≤ favg

(8)

(9)

where pc1 , pc2 – the maximum and minimum of crossover probability respectively; pm1 ,pm2 – the maximum and minimum of the mutation probability respectively. fmax – the maximum fitness value of the member; favg – the average fitness of the set of solutions; f – the larger fitness values of the solutions to be crossed; f – the larger fitness value of the solutions to be mutated; Eqs. (8)

438

R. Chen et al. / Applied Soft Computing 26 (2015) 435–443

and (9) show when f = fmax , f = fmax , pc and pm will be increased to pc2 and pm2 respectively, avoiding the possibility of local optimal. While f , f are both not greater than favg , pc , pm will be equal to pc1 , pm1 respectively, with higher values. Compared with GAs, in the AGA, the optimal individuals have higher pc and pm , and the best solutions are explicitly protected from disruption [35].

Initial Value of (C, σ2, ε)

Historical data set

Train SVR Model

Training set

Testing set

Calculate fitness value

2.3. The AGA-SVR procedure The optimization processes of the AGA-SVR model can be described as the following steps: Step 1 Population initialization. The relative real-valued parameters or variables can be directly employed to form a chromosome when the AGA deals with optimal problems. Therefore, three parameters of the SVR model (C,  2 and ε) can be produced through coding chromosome directly. The chromosome X here is defined as X = {P1 , P2 , P3 }, where P1 , P2 and P3 represent parameters C,  2 and ε respectively. Step 2 Evaluation fitness. Put the initialized chromosome into SVR, and evaluate the fitness value (forecasting errors) of each chromosome. In this article, the mean absolute percentage error (MAPE) defined in the training data set is adopted as the fitness function:





1   yi − fi  MAPE =  y  × 100% N i N

Test SVR model No

AGA-SVR forecast

Step 1 Calculate the seasonal index for each day of different holidays yt = ft

i=1

As mentioned above, In China, there are official holidays and summer holidays, and people prefer to travel during these periods. Affected by different length of holidays and the seasonal nature of climate, holiday daily tourist flow presents seasonal fluctuations trend which has great impact on the forecasting accuracy. Therefore, this seasonal component must be estimated carefully. According to the characteristics of the holiday daily tourist flow, we try to use seasonal index adjustment to adjust its seasonality. The details are described as follows:

Yes

Satisfy Stopping criteria

Fig. 1. The flowchart of AGA-SVR model.

(10)

2.4. Seasonal index adjustment approach

Train SVR model

Crossover, mutation ( adaptive pc and pm)

pt =

where yi , fi are the actual value and forecast value respectively; N is the number of training data samples. Step 3 Selection. According to the roulette wheel selection principle, the number of chromosomes is chosen to N (N is the even number) for reproduction. At the same time, calculate the favg and fmax of the populations respectively. Step 4 Crossover. Each chromosome of the populations is paired randomly, adding up to N/2 pairs. Then adaptive crossover probability pc is calculated according to Eq. (8). Meanwhile crossover operation will be performed with pc , That is, R (0, 1) is generated randomly. If R < pc , crossover operation is used on the current individual. Step 5 Mutation. To all individuals (N) in the populations, mutation probability pm is calculated according to Eq. (9). Meanwhile mutation operation will be performed with pm , That is, R (0, 1) is generated randomly. If R < pm , mutation operation is used on the current chromosome. Step 6 Elitist strategy. The fitness value of new individual is calculated by the each pair of the new population generated from crossover and mutation. New individual, together with parents’ generation, forms a new generation of group. Step 7 Stop criteria. If the evolutionary generation is equal to the specified one, the optimal parameters (C,  2 and ε) are obtained (here, the number of generations is equal to 100). They will be brought into the testing data set of SVR model to forecast, and then the algorithm goes to stop. If not, turn to Step 2. The flow chart of the AGA-SVR is shown as in Fig. 1.

optimal parameters (C, σ2, ε))

Generate new population

yt N 

(ai − a∗i ) exp(−||xi

(11) − x||2 /2 2 ) + b

i=1

where t = j, l + j, 2l + j, . . ., (m − 1) + j is the same day in each holiday period, yt and ft are the actual and forecasting value respectively. Then the seasonal index (SI) for each day of holidays is shown as follow: SIj =

1 (p + pl+j + · · · + p(m−1)l+j ) j = 1, 2, . . ., l m j

(12)

Step 2 Calculate the forecasting values of the AGA-SSVR model. Finally, the forecasting value of the AGA-SSVR is derived by the following formula: fn+k =

 N 

 (ai − a∗i ) exp

i=1

−||xi − xn+k ||2 2 2





+b

· SIk

(13)

where k = 1, 2,. . ., l means the same day in another holiday period (forecasting period in testing data). 3. Experimental model settings and comparison 3.1. Data set As both world natural and cultural heritage sites, Mountain Huangshan, located in Anhui Province of Southeast China, is famous for its “odd-shape pines, spectacular rocks, seas of clouds and hot springs”. Every year thousands of tourists come here particularly in Chinese official holidays. In our study, the data set of daily tourist flow volume in seven different holidays from 2008 to 2012 in Mountain Huangshan is collected and used. This data set includes three parts: holiday daily tourist flow; holiday daily tourist flow before 8 am; holiday daily human comfort index based on daily temperature, humidity, wind direction and wind speed [51]. Each of part, denoted as vectors {X1 , X2 , X3 }, contains 145 data points respectively. The data set applied in this example is listed in Figs. 2 and 3. The two figures show characteristics of this data set as follows: (1) Nonlinear features. Due to different fluctuations of daily tourist flow in each holiday, this data set presents a complex nonlinearity.

R. Chen et al. / Applied Soft Computing 26 (2015) 435–443

439

Tourist flow(×104)

6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

Daily tourist flow Daily tourist flow before 8 am

2008

2009

2010

2011

2012

Fig. 2. Holiday daily tourist flow in 2008–2012.

70

Human comfort index

60 50 40 30 20 10 0 2008

-10

2009

2010

2011

2012

-20 Fig. 3. Holidays daily human comfort index in 2008–2012.

Table 1 AGA parameters settings. Number of generations

Population sizes

100

20

Crossover probability

Mutation probability

pc1

pc2

pm1

pm2

0.9

0.6

0.1

0.001

Table 2 Each time point (day) of the seasonal indexes. Time point (day)

SI

New Year

1 2 3

0.9806 0.9503 0.8358

Spring Festival

1 2 3 4 5 6 7

0.8319 1.0300 1.1555 0.9845 0.9983 0.9805 1.0248

Time point (day) Ching-Ming Festival

May Day

Dragon Boat Festival

(2) Yearly periodicity and seasonal trend. Holiday daily data set appears periodically and seasonally pattern, which is primarily due to climate conditions, different holiday periods and the similar traveling behavior of people year after year. Meanwhile, both holiday daily tourist flow before 8 am and holiday daily human comfort index follow the similar periodicity and seasonal trend as the holiday daily tourist flow. It is clear that both

SI 1 2 3

1.1046 1.1012 0.8877

1 2 3 1 2 3

1.1317 0.9082 1.0273 1.1148 1.0633 1.0067

SI

Time point (day)

Mid-Autumn Festival

1 2 3

1.1334 0.9988 0.9462

National Day

1 2 3 4 5 6 7

1.0939 1.0394 1.0408 1.0221 0.9419 0.9088 0.9520

of them have significant correlations with holiday daily tourist flow. Besides these, due to the length of different holidays, Lee [52] found that nearly 43.62% of domestic tourists would travel on May Day and National Day, and approximately 25% of them would travel on Ching-Ming Festival, Dragon Boat Festival and Mid-Autumn

440

R. Chen et al. / Applied Soft Computing 26 (2015) 435–443

Fig. 4. Forecasting values from 2008 to 2011 in training set.

Table 3 Parameter settings of BPNN. Hidden layer network

Input layers nodes

Output layers nodes

Function of hidden nodes and output nodes

Training function

Learning function

Max. training iterations

Learning rate

Training target error

Perform function

1

4

1

Tansig

Traincgf

Learngdm

1000

0.01

0.001

MSE

MSE, mean square error.

Table 4 Normalized legal holidays forecasting values comparison of the AGA-SSVR, AGA-SVR and BPNN. Holiday

Day

Actual

SSVR

SVR

BPNN

New Year

1 2 3

0.1516 0.1498 0.0335

0.1654 0.1710 0.0297

0.1687 0.1800 0.0355

0.0968 0.1235 0.0222

Spring Festival

1 2 3 4 5 6 7

0.0088 0.0374 0.2104 0.3682 0.3841 0.3042 0.2014

0.0061 0.0379 0.1575 0.3724 0.3601 0.3839 0.1621

0.0073 0.0368 0.1363 0.3783 0.3607 0.3915 0.1582

0.0000 0.0162 0.1534 0.2493 0.2913 0.2289 0.1781

Ching-Ming Festival

1 2 3

0.2757 0.6708 0.1910

0.2631 0.6740 0.2305

0.2382 0.6668 0.2597

0.2616 0.6282 0.1689

May Day

1 2 3

0.4181 0.9364 0.3867

0.3799 0.8715 0.3880

0.3357 0.9596 0.3772

0.4945 0.6566 0.5287

Dragon Boat Festival

1 2 3

0.2775 0.3554 0.1046

0.2367 0.3671 0.0634

0.2123 0.3453 0.0630

0.3374 0.4222 0.0831

Mid-Autumn Festival

1 2 3

0.2387 0.3890 0.1363

0.1990 0.4180 0.1144

0.1756 0.4199 0.1209

0.3100 0.4183 0.1568

National Day

1 2 3 4 5 6 7

0.3583 0.9823 1.0000 0.9215 0.7143 0.4290 0.1670

0.3270 0.9221 0.9561 0.8461 0.7264 0.4792 0.1379

0.2989 0.8871 0.8999 0.8278 0.7818 0.5273 0.1448

0.3116 0.8278 0.9383 0.8343 0.7315 0.3857 0.2178

R. Chen et al. / Applied Soft Computing 26 (2015) 435–443

441

Fig. 5. Holiday daily tourist flow forecasting values in 2012 from different models.

Festival. So the length of different holidays is also one of important factors which can influence daily tourist flow. In order to turn the length of different holidays into decision variables, a binary virtual variable approach [53] is presented to identify different holidays.



Di =

1 the ith day of the holidays

i = 1, 2, . . ., 7

AGA-SSVR AGA-SVR BPNN

(14)

the other day of the holidays

0

Table 5 Performance criteria values of the three models in legal holidays.

Therefore, the vectors {X1 , X2 , X3 , D1 , . . ., D7 } are used as input variables in AGA-SVR model, and the forecasting value of holiday daily tourist flow in the same day as the output variable, denoted as y. In the experiment, the data set is divided into two parts: the training data set (116 days, 2008–2011) and the testing data set

MAE

MAPE

RMSE

R

1180 1697 2387

0.1182 0.1479 0.2319

1456 2119 3202

0.9985 0.9973 0.9819

(29 days, 2012). In order to improve the forecasting accuracy, all the data are normalized by (xt − xmin )/(xmax − xmin ) within the range of [0,1], where xt , xmin and xmax respectively represent the tourist flow at time t, minimum and maximum of the tourist flow during the period of data source.

Table 6 Normalized summer holidays forecasting values comparison of the AGA-SSVR, AGA-SVR and BPNN. Jun.

Actual

AGA-SSVR

AGA-SVR

BPNN

SI

Aug.

Actual

AGA-SSVR

AGA-SVR

BPNN

SI

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

0.2002 0.6760 0.4868 0.2714 0.3338 0.3268 0.3502 0.3958 0.7038 0.5325 0.3489 0.3336 0.3804 0.4213 0.4053 0.7524 0.5768 0.3612 0.4478 0.4168 0.5262 0.4225 0.8863 0.6514 0.3047 0.3981 0.4682 0.3908 0.3676 0.6706 0.5680

0.1688 0.5962 0.3713 0.2969 0.2587 0.3244 0.3353 0.3659 0.5553 0.5303 0.3497 0.3340 0.3024 0.5141 0.4485 0.7658 0.5506 0.2747 0.2782 0.4322 0.4106 0.4206 0.7756 0.6696 0.2899 0.2839 0.3819 0.4912 0.3916 0.6894 0.5917

0.1804 0.5918 0.3912 0.3104 0.2452 0.3213 0.3519 0.3633 0.5617 0.5397 0.3552 0.3500 0.3190 0.5425 0.4488 0.7876 0.5531 0.2818 0.2750 0.4400 0.4178 0.4255 0.7889 0.6701 0.2888 0.2809 0.3623 0.4477 0.3514 0.6033 0.5186

0.0000 0.4071 0.3831 0.1910 0.2160 0.2295 0.2813 0.3029 0.5798 0.4469 0.2324 0.2684 0.3510 0.2989 0.3477 0.6499 0.4612 0.2938 0.2496 0.2924 0.3492 0.3557 0.6937 0.5291 0.2211 0.2780 0.3165 0.2829 0.3313 0.5593 0.4578

0.9359 1.0075 0.949 0.9566 1.0552 1.0097 0.953 1.0071 0.9886 0.9826 0.9845 0.9543 0.9479 0.9477 0.9993 0.9723 0.9954 0.9748 1.0118 0.9823 0.9827 0.9884 0.9832 0.9993 1.0041 1.011 1.0541 1.0971 1.1145 1.1426 1.1409

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

0.4330 0.5032 0.4621 0.5602 0.4847 0.8007 0.5309 0.3990 0.4592 0.5057 0.4041 0.4472 1.0000 0.5736 0.4009 0.3678 0.4384 0.4136 0.4269 0.6471 0.5613 0.2818 0.2443 0.2534 0.2510 0.2405 0.5264 0.2756 0.1741 0.0472 0.0169

0.3036 0.5036 0.3903 0.4600 0.4434 0.8099 0.4649 0.4617 0.4430 0.4365 0.5690 0.4199 0.8297 0.6401 0.3130 0.3981 0.3837 0.4103 0.3130 0.6830 0.5528 0.2940 0.2312 0.2819 0.2424 0.3008 0.3682 0.2530 0.1765 0.0656 0.0162

0.3010 0.5021 0.4000 0.4440 0.4506 0.8141 0.4452 0.4766 0.4555 0.4508 0.5729 0.4117 0.8058 0.6269 0.3163 0.4120 0.3812 0.4226 0.3256 0.7155 0.5768 0.2979 0.2278 0.2898 0.2389 0.3078 0.3802 0.2489 0.1785 0.0718 0.0177

0.3015 0.3968 0.3457 0.3815 0.3717 0.8120 0.4478 0.2913 0.3087 0.3558 0.2483 0.3678 0.7732 0.6208 0.3069 0.3432 0.3460 0.3019 0.3243 0.7503 0.6270 0.2667 0.2305 0.2812 0.2280 0.2337 0.5678 0.3725 0.1672 0.0974 0.0311

1.0088 1.003 0.9758 1.0360 0.9839 0.9948 1.0442 0.9687 0.9726 0.9681 0.9932 1.0198 1.0297 1.0211 0.9895 0.9663 1.0067 0.9709 0.9614 0.9546 0.9584 0.9868 1.0148 0.9727 1.0148 0.9772 0.9685 1.0165 0.9888 0.9138 0.9138

442

R. Chen et al. / Applied Soft Computing 26 (2015) 435–443

3.2. AGA-SSVR holiday daily tourist flow forecasting model Before applying the seasonal index adjustment for the AGASSVR model, it is essential to apply the AGA algorithm to confirm appropriate values of the SVR’s three parameters. The selection of pc , pm and population sizes is based on the previous historical literatures and lots of experiments in the training data of SVR. Table 1 provides an overview of the AGA parameter settings, as those values provide the optimal parameters in the training data set: C = 9.9427,  2 = 9.1912 and ε = 0.0167. According to the AGA-SVR modeling procedure, optimal parameters are fed into SVR model, training in the training data. The forecasting values in the training data are shown in Fig. 4. Then seasonal index adjustment is applied into the SVR model. Since the total time points of seven holidays are 29 in each year, the seasonal length is chosen as 29. According to Eqs. (11) and (12), the seasonal indexes for each day point are computed based on the 116 forecasting values of the AGA-SVR model in training data (red line points in Fig. 4 from 2008 to 2011), as shown in Table 2. From Table 2, it can be seen some of seasonal indexes are smaller than 1, which means the average forecasting values (based on 116 training forecasts) are overestimated in the AGA-SVR model. On the contrary, seasonal indexes larger than 1 indicates the underestimation of daily tourist flow. In this situation, according to Eq. (13), the forecasting values in testing data are adjusted by seasonal indexes; overestimated and underestimated values are much closer to the actual ones. In order to validate the performance of AGA-SSVR approach, BPNN model is adopted as a comparative method on the same conditions. Additionally, the AGA-SVR model which has no seasonal mechanism is also applied to verify the effectiveness of seasonal index adjustment. Here the details of BPNN parameter settings are shown in Table 3. The predicted results of different models are shown in Table 4 and Fig. 5. It can be seen that of all 29 days in 2012 holidays, there are 19 days (marked in bold) in the AGA-SSVR model, whose predicted values are much closer to the actual ones, while only 6 days in the AGA-SVR and 4 days in the BPNN. It seems that our proposed model is more appropriate than the other two ones. In order to demonstrate the viewpoint, the mean absolute error (MAE), mean absolute percentage error (MAPE), root mean squared error (RMSE) and correlative coefficient (R) are introduced into the models. As shown in Table 5, the values of MAE, MAPE and RMSE given by the AGASSVR model are smaller than those by the other two models, while R value from AGA-SSVR is the highest amongst all models, which means low deviations between the predicted values and actual ones in AGA-SSVR. Therefore, once again, it proves the AGA-SSVR model follows the overall trend in holiday daily tourist flow forecasting, and it does capture the cyclic fluctuation for holiday data patterns, outperforming the AGA-SVR and BPNN with greater accuracy and few errors. In order to further prove the validity of proposed AGA-SSVR model, summer holiday daily data of China1 in 2008–2012 from Mountain Huangshan was also used as the other example. The model detail process is the same as that of the first one. The results of forecasting and their performance are shown in Tables 6 and 7, respectively. Clearly, in Table 6, the data marked in bold mean predicted values are much closer to the actual ones, which AGASSVR accounts for 41 days, AGASVR 15 days and BPNN 6 days. Besides, in Table 7,

1 Generally speaking, China’s traditional summer holiday is in July and August, added up to 62 days.

Table 7 Performance criteria values of the three models in summer holidays.

AGA-SSVR AGA-SVR BPNN

MAE

MAPE

RMSE

R

373 1101 1760

0.0337 0.0948 0.1534

1532 1643 2080

0.9990 0.9942 0.9938

the similar results are also proved that AGA-SSVR obtains better forecasting accuracy than AGA-SVR and BPNN. 4. Conclusion Since holiday tourism plays an important role in tourism industry, the accuracy of forecasting holiday daily tourist flow is expected to be increasingly significant for planning tourism-related resources, overestimated or underestimated predicted values will influence the reasonable distribution of tourism resources. As a representative scenic spot in China, the historical tourists daily flow data set from Mountain Huangshan present not only nonlinearity but also seasonality characteristics. Thus, this article presented a hybrid AGA-SSVR model for holiday daily tourist flow forecasting. The experiments of results demonstrate the AGA-SSVR has greater accuracy and few errors than AGA-SVR and BPNN models. The superior performance of the AGA-SSVR mainly comes from the following reasons. (1) Compared with BPNN, the SVR can capture data patterns of holiday daily tourist flow more easily due to its nonlinear mapping capabilities. Meanwhile, the SVR minimizes the structural risk instead of the empirical risk, which provides better generalization performance than BPNN. Furthermore, holiday daily tourist flow forecasting is a small sample problem, and the SVR is an effective forecasting method for the small samples, while BPNN is effective for the large ones. Therefore, the SVR-based methods in the article outperform BPNN, which have been proved in Tables 4–7 and Fig. 4. (2) The AGA played an important role in searching for proper parameters because of the adaptive probabilities of crossover and mutation. Therefore, the phenomena of local optimal and premature convergence from GA can be avoided in the AGA. (3) Seasonal index adjustment modifies the predicted values from AGA-SVR caused by the influence of seasonal trend, which makes overestimated or underestimated predicted values tend to be much closer to the actual ones. (4) For the reasons above, our proposed model AGA-SSVR can provide a more effective way for holiday daily tourist flow forecasting indeed. The predicted outcome regarding in holiday daily tourist flow volume is very important to China’s tourism industry. China is a country with the largest population in the world, and the number of tourists is increasing rapidly during recent years especially in the holidays. Accurate forecasting in holiday daily tourist flow will provide direct assistance to tourism management departments, and help them make scientific decisions ahead of time, thus avoiding the tourism resources being excessive consumption. Besides, tourist flow is also affected by crises and disasters. Once happen, the pattern of tourist flow will lost its inherent characteristic and post-event tourist flow forecasting will become more difficult. In recent years, many literatures mainly focused on the effects of crises and disasters, such as SARS and September 11, etc [54–56], but there is little attention about the ex post forecasts of crises and disasters. Therefore, how to deal with this kind of forecasting will be a great challenge, forecasting tools that qualitative

R. Chen et al. / Applied Soft Computing 26 (2015) 435–443

or quantitative, combined or not, SVR or others will be a guiding direction for further research. Acknowledgements This study is supported by the National Natural Science Foundation (71331002, 71271072, 71301037, 71301040, 71201045) of China, China Post-Doc Science Foundation (2013M541651), Anhui Provincial Research Institute of Humanities& Social Science Key Project (SK2013A148, Jiangsu Province Post-Doc Science and Technology Program (1302129C), and National Science Council, Taiwan (NSC 100-2628-H-161-001-MY4). We thank Mount Huangshan Scenic Spot for giving us valuable materials about holiday daily tourist flow. Specially, we are grateful to the senior editor, associate editor, and the anonymous reviewers whose comments have improved this paper considerably. References [1] G.R. Zhang, D.Q. Liu, R. Song, China’s Tourism Development Analysis and Prediction, Social Sciences Academic Press, China, 2012, pp. 123. [2] K.L. Wu, B. Dai, Golden week tourist flow forecasting based on neural network, IEEE Int. Conf. Ind. Technol. (2006) 2866–2871. [3] N. Sapankevych, R. Sankar, Time series prediction using support vector machines: a survey, IEEE Comput. Intell. Manag. 4 (2) (2009) 24–38. [4] F.L. Chu, Forecasting tourism demand with ARMA-based methods, Tour. Manag. 30 (5) (2009) 740–751. [5] P. Gustavsson, J. Nordstrom, The impact of seasonal unit roots and vector ARMA modelling on forecasting monthly tourism flows, Tour. Econ. 7 (2) (2001) 117–133. [6] C. Lim, M. McAleer, Time-series forecasts of international travel demand for Australia, Tour. Manag. 23 (4) (2002) 389–396. [7] C. Goh, R. Law, Modeling and forecasting tourism demand for arrivals with stochastic nonstationary seasonality and intervention, Tour. Manag. 23 (5) (2002) 499–510. [8] C.J.S.C. Burger, M. Dohnal, M. Kathrada, R. Law, A practitioners guide to timeseries methods for tourism demand forecasting – a case study of Durban, South Africa, Tour. Manag. 22 (4) (2001) 403–409. [9] J.H. Kim, M.T. Ngo, Modeling and forecasting monthly airline passenger flows among three major Australian cities, Tour. Econ. 7 (4) (2001) 397–412. [10] J.Z. Wang, S.L. Zhu, W.Y. Zhang, H.Y. Lu, Combined modeling for electric load forecasting with adaptive particle swarm optimization, Energy 35 (4) (2010) 1671–1678. [11] R. Law, Back-propagation learning in improving the accuracy of neural network-based tourism demand forecasting, Tour. Manag. 21 (4) (2000) 331–340. [12] P.F. Pai, W.C. Hong, An improved neural network model in forecasting arrivals, Ann. Tour. Res. 2 (4) (2005) 1138–1141. [13] V. Cho, A comparison of three different approaches to tourist arrival forecasting, Tour. Manag. 4 (3) (2003) 323–330. ˜ A. Sesé, Designing an artificial neural network for [14] A. Palmer, J. José Montano, forecasting tourism time-series, Tour. Manag. 27 (5) (2006) 781–790. [15] F.M. Tseng, H.C. Yu, G.H. Tzeng, Combining neural network model with seasonal time series ARIMA model, Technol. Forecast. Soc. Change 69 (1) (2002) 71–87. [16] V.N. Vapnik, The Nature of Statistical Learning Theory, Springer, New York, 1995. [17] K.J. Kim, Financial time series forecasting using support vector machines, Neurocomputing 55 (1–2) (2003) 307–319. [18] C.J. Lu, T.S. Lee, C.C. Chiu, Financial time series forecasting using independent component analysis and support vector regression, Decis. Supp. Syst. 47 (2) (2009) 115–125. [19] W.M. Hung, W.C. Hong, Application of SVR with improved ant colony optimization algorithms in exchange rate forecasting, Control Cybern. 38 (3) (2009) 863–891. [20] M. Mohandes, Support vector machines for short-term electrical load forecasting, Int. J. Energy Res. 26 (4) (2002) 335–345. [21] W.C. Hong, Application of chaotic ant swarm optimization in electric load forecasting, Energy Policy 38 (10) (2010) 5830–5839. [22] C.H. Wu, J.M. Ho, D.T. Lee, Travel-time prediction with support vector regression, IEEE Trans. Intell. Transp. Syst. 5 (4) (2004) 276–281. [23] W.C. Hong, Traffic flow forecasting by seasonal SVR with chaotic simulated annealing algorithm, Neurocomputing 74 (12–13) (2011) 2096–2107. [24] K.Y. Chen, C.H. Wang, Using genetic algorithms to optimize support vector regression in tourism demand forecasting, J. Outdoor Recreat. Study 18 (1) (2005) 47–72.

443

[25] P.F. Pai, W.C. Hong, P.T. Chang, The application of support vector machines to forecast tourist arrivals in Barbados: an empirical study, Int. J. Manag. 123 (2) (2006) 375–385. [26] K.Y. Chen, C.H. Wang, Support vector regression with genetic algorithms in forecasting tourism demand, Tour. Manag. 28 (1) (2007) 215–226. [27] K.Y. Chen, Combining linear and nonlinear model in forecasting tourism demand, Expert Syst. Appl. 38 (8) (2011) 10368–10376. [28] J.H. Min, Y.C. Lee, Bankruptcy prediction using support vector machine with optimal choice of kernel function parameters, Expert Syst. Appl. 28 (4) (2005) 603–614. [29] N. Cristianini, T.J. Shawe, An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods, Cambridge University Press, Cambridge, UK, 2000. [30] J. Huang, Y.C. Bo, H.Y. Wang, Electromechanical equipment state forecasting based on genetic algorithm-support vector regression, Expert Syst. Appl. 38 (7) (2011) 8399–8402. [31] J.R. Gu, M.C. Zhu, L.G.Y. Jiang, Housing price forecasting based on genetic algorithm and support vector machine, Expert Syst. Appl. 38 (4) (2011) 3383–3386. [32] C.F. Huang, A hybrid stock selection model using genetic algorithms and support vector regression, Appl. Soft Comput. 12 (2012) 807–818. [33] J. Andre, P. Siarry, T. Dognon, An improvement of the standard genetic algorithm fighting premature convergence in continuous optimization, Adv. Eng. Softw. 32 (1) (2001) 49–60. [34] Z. Bingul, Adaptive genetic algorithms applied to dynamic multiobjective problems, Appl. Soft Comput. 7 (3) (2007) 791–799. [35] M. Srinivas, L.M. Patnaik, Adaptive probabilities of crossover and mutation in genetic algorithms, IEEE Trans. Syst. Man Cybern. 24 (4) (1994) 656–667. [36] X. Luo, H. Ding, L. Zhang, L. Xia, H.F. Ding, L.W. Zhang, Forecasting of national expressway network scale based on support vector regression with adaptive genetic algorithm, in: Proceedings of Inernational Conference of Logistics Engineering and Management, 2012, pp. 434–441. [37] H.Y. Song, G. Li, Tourism demand modelling and forecasting — a review of recent research, Tour. Manag. 29 (2) (2008) 203–220. [38] J. Shiskin, A.H. Young, J.C. Musgrave, The X-11 Variant of the Census Method II Seasonal Adjustment Program, US Department of Commerce, Bureau of the Census, 1965. [39] Z.B. Zhou, X.C. Dong, Analysis about the seasonality of China’s crude oil import based on X-12-ARIMA, Energy 2 (1) (2012) 281–288. [40] A. Hayat, M.I. Bhatti, Masking of volatility by seasonal adjustment methods, Econ. Model. 33 (2013) 676–688. [41] C.R. Nelson, C.R. Plosser, Trends and random walks in macroeconmic time series: some evidence and implications, J. Monet. Econ. 10 (2) (1982) 139–162. [42] G.P. Zhang, M. Qi, Neural network forecasting for seasonal and trend time series, Eur. J. Oper. Res. 60 (2) (2005) 501–514. [43] P.F. Pai, K.P. Lin, C.S. Lin, P.T. Chang, Time series forecasting by a seasonal support vector regression model, Expert Syst. Appl. 7 (6) (2010) 4261–4265. [44] K. Kurtulus, U. Sevki, Measuring the seasonality in tourism with the comparison of different methods, Euro. Med. J. Bus. 5 (2) (2010) 191–214. [45] R. Deo, C. Hurvich, Y. Lu, Forecasting realized volatility using a long-memory stochastic volatility model: estimation, prediction and seasonal adjustment, J. Econ. 131 (1–2) (2006) 29–58. [46] A.J. Smola, B. Schölkopf, A tutorial on support vector regression, Stat. Comput. 4 (3) (2004) 199–222. [47] K. Hotta, Robust face recognition under partial occlusion based on support vector machine with local Gaussian summation kernel, Image Vis. Comput. 26 (11) (2008) 1490–1498. [48] C.H. Wu, G.H. Tzeng, R.H. Lin, A Novel hybrid genetic algorithm for kernel function and parameter optimization in support vector regression, Expert Syst. Appl. 6 (3) (2009) 4725–4735. [49] L.Y. Chen, Application of SVR with chaotic GASA algorithm to forecast Taiwanese 3G mobile phone demand, Neurocomputing 127 (2014) 206–213. [50] W.C. Hong, Y.C. Dong, L.Y. Chen, S.Y. Wei, SVR with hybrid chaotic genetic algorithms for tourism demand forecasting, Appl. Soft Comput. 11 (2) (2011) 881–1890. [51] G.R. McGregor, A preliminary assessment of the spatial and temporal characteristics of human comfort in China, Int. J. Climatol. 3 (7) (1993) 707–725. [52] F. Lee, The impact of new holiday institution on the tourist flow during “Golden Week” and its significance-evidence from the key scenic areas monitored by National Holiday Office, Tour. Tribune 24 (6) (2009) 12–18. [53] E. Kyriakides, M. Polycarpou, Short term electric load forecasting: a tutorial, Stud. Comput. Intell. 35 (2007) 391–418. [54] J. Eugenio-Martin, M.T. Sinclair, I. Yeoman, Quantifying the effects of tourism crises: an application to Scotland, J. Travel Tour. Mark. 19 (2–3) (2006) 21–34. [55] J.C.H. Min, The effect of the SARS illness on tourism in Taiwan: an empirical study, Int. J. Manag. 22 (3) (2005) 497–506. [56] B. Prideaux, E. Laws, B. Faulkner, Events in Indonesia: exploring the limits to formal tourism trends forecasting methods in complex crisis situations, Tour. Manag. 24 (4) (2003) 475–487.