Enhanced prediction accuracy of grey forecasting model: a case by tourism industry in Vietnam

Hue University Journal of Science ISSN 1859-1388 Vol. 113, No. 14, 2015, pp. 157-167 *Corresponding: thanhkem2710@gmail.com March 22, 2016; Revised: April 01, 2016; Accepted: February 25, 2016. ENHANCED PREDICTION ACCURACY OF GREY FORECASTING MODEL: A CASE BY TOURISM INDUSTRY IN VIETNAM Phan Van Thanh* Quang Binh Univerisity Abstract: grey forecasting based on the grey system theory is a diverse forecasting model and has been successfully applied in various fields. In recent years, many scholars have proposed new procedures or new models with different ways to improve the prediction accuracy of grey forecasting for fluctuating data sets. However, the prediction accuracy of the existing grey forecasting models may not be always satisfactory in different scenarios. For example, the data not only consist of trend, seasons but highly fluc- tuate with lots of noise as well. To overcome this drawback, this paper proposed two effective combined grey models, namely Fourier grey Model (1, 1) (abbreviated as F-GM (1, 1)) and Fourier Nonlinear grey Bernoulli Model (abbreviated as F-NGBM (1, 1). Two proposed models were built by using Fourier series to modify their residual values. To verify their performance and effectiveness, these proposed models were used to forecast the international tourism demand in Vietnam from Jan. 2006 to Mar. 2016. The em- pirical results demonstrated that the accuracy of both GM (1, 1) and NGBM (1, 1) forecasting models after using Fourier series to revise their residual error provided more accuracy than original ones in terms of in- sample and out-of-sample cases. Further more, this paper also indicated that the F-GM (1, 1) is the better model than other forecasting models in forecasting the international tourist arrival to Vietnam with aver- age MAPE of in-sample and out-of-sample of 0.013% and 5.19 %, respectively. Keywords: grey system theory, GM (1, 1), NGBM (1, 1), Fourier series, International tourism demand 1 Introduction Grey system theory established during the 1980s by Deng [12] is a quantitative method dealing with grey systems that are characterized by both partially known and partially unknown in- formation [13-16]. The main purpose of the grey system theory focuses on the relationship be- tween the analysis model constructions, for circumstances such as: uncertainty, multi-data in- put, discrete data, and insufficient data through prediction and decision- making. Nowadays, the field of grey system can be summarized in six main parts, namely grey generating, grey relational analysis, grey model, grey prediction, grey decision making and grey control. With its advantages and various approaches, grey system theory has been widely used in industry, so- cial systems, ecological systems, economy, geography, traffic, management, agriculture, envi- ronment, education, etc. Grey prediction models are one of most important parts of grey system theory. They have been popularized in the time series prediction due to their simplicity and ability to characterize an unknown system with high accuracy, using as little as four data points [17, 18]. During the past two decades, the grey prediction models have been successfully applied to various fields, Phan Van Thanh Vol. 113, No. 14, 2015 158 such as tourism [19], energy [20, 21], financial and economics [22-24], IC industry [25], marine and sea port industry [26-28]. Although grey models have been successfully adopted in various fields and they have provided us with promising results, in some cases, the classical GM (1, 1) model exhibits certain limitations that directly affect the model applicability as well as prediction accuracy. Therefore, it is necessary to improve the prediction performance as well as to overcome the restriction ex- isting in the traditional grey model (abbreviated as GM (1, 1)) and Nonlinear grey Bernoulli Model (abbreviated as NGBM (1, 1). To upgrade the prediction accuracy of GM (1, 1) and NGBM (1, 1), this paper used Fourier series to modify the residual errors in GM (1, 1) and NGBM (1, 1) models, and these models are finally compared based on their accuracy and the better one is selected to forecast the inbound tourism demand in Vietnam. The remaining of this paper was organized as follows. In section 2, the concept of classical GM (1, 1) and NGBM (1, 1), and Fourier Residual Modification of these models were presented. Based on the fundamental functions of these models, the empirical results were shown in section 3. Finally, section 4 summarized the findings. 2 Modeling and Methodology 2.1 Classical GM (1, 1) model GM (1, 1) is the basic model of grey forecasting modeling, a first order differential model with one input variable which has been successfully applied in many different researches. It is ob- tained as the following procedure. Step1: Let raw matrix )0(X stands for the non-negative original historical time series data  )()0()0( itxX  , ni ,...,2,1 (1.1) Where )( )0( itx is the value at time ti, and n is the total number of modeling data Step 2: Construct )1(X by one time accumulated generating operation (1-AGO), which is  )()1()1( itxX  , ni ,...,2,1 (1.2) Where nktxtx k i ik ,...,2,1,)()( 1 )0(1   (1.3) Step 3: )1(X is a monotonic increasing sequence which is modeled by the first order linear differential equation (1.4) baX dt dX  )1( )1( (1.4) Where the parameter “a” is the developing coefficient and “b” is the grey input. Step 4: In order to estimate the parameter “a” and “b”, Eq. (1.4) is approximated as btaX dt tX k k k   )( )( )1( )1( (1.5) Jos.hueuni.edu.vn Vol. 113, No. 14, 2015 159 Where )()()()( )0(1 )1()1()1( kkkk txtxtxtX   (1.6) 1 kkk ttt (1.7) If the sampling time interval is units, then let 1 kt , using )()1()()( 1 )1()1()1(  kkk txptpxtz , nk ,..,3,2 (1.8) to replace )()1( ktX in Eq. (1.5), we obtain btaztx kk  )()( )1()0( , nk ,...,3,2 (1.9) Where )()1( ktz in Eq. (1.8) is termed background value, and p is production coefficient of the background value in the range of (0, 1), which is traditionally set to 0.5. Step 5: From Eq. (1.9), the value of parameter “a” and “b” can be estimated using least square method. That is n TT YBBB b a 1)(       (1.10) where                     1 ..... )( ...... 1)( 1)( )1( 3 )1( 2 )1( ntz tz tz B (1.11) and  Tnn txtxtxY ))(),...,(),( )0(3)0(2)0( (1.12) Step 6: The solution of Eq. (1.4) can be obtained after the parameter “a” and “b” have been estimated. That is               a b e a b txtx tta k k )( 1 )0()1( 1)()(ˆ , ,...3,2,1k (1.13) Step 7: Applying inverse accumulated generating operation (IAGO) to )(ˆ )1( ktx , the pre- dicted datum of )()0( ktx can be estimated as        )(ˆ)(ˆ)(ˆ )()(ˆ 1 )1()1()0( 1 )0( 1 )0( kkk txtxtx txtx , ,...3,2k 2.2 Nonlinear-grey Bernoulli model “NGBM (1, 1)” The procedures of deriving NGBM are as follows: Step1: Let raw matrix )0(X stands for the non-negative original historical time series data  )()0()0( itxX  , ni ,...,2,1 (2.1) (1.14) (1.15) Phan Van Thanh Vol. 113, No. 14, 2015 160 Where )()0( itx corresponds to the system output at time ti, and n is the total number of modeling data. Step 2: Construct )1(X by one time accumulated generating operation (1-AGO), which is  )()1()1( itxX  , ni ,...,2,1 (2.2) where nktxtx k i ik ,...,2,1,)()( 1 )0(1   (2.3) Step 3: )1(X is a monotonic increasing sequence which is modeled by the Bernoulli differential equation  rXbaX dt dX )1()1( )1(  (2.4) Where the parameter “a” is called the developing coefficient and “b” is the named the grey in- put and “r” is any real number excluding r = 1. Step 4: In order to estimate the parameter “a” and “b”, Eq. (2.4) is approximated as  rkk k k tXbtaX dt tX )()( )( )1()1( )1(   (2.5) where )()()()( )0(1 )1()1()1( kkkk txtxtxtX   (2.6) 1 kkk ttt (2.7) If the sampling time interval is units, then let 1 kt , using )()1()()( 1 )1()1()1(  kkk txptpxtz , nk ,..,3,2 (2.8) To replace )( )1( ktX in Eq. (2.5), we obtain  rkkk tzbtaztx )()()( )1()1()0(  , nk ,...,3,2 (2.9) Where )()1( ktz in Eq. (2.8) is termed background value, and p is production coefficient of the background value in the range of (0, 1), which is traditionally set to 0.5. Step 5: From Eq. (2.9), the value of parameter “a” and “b” can be estimated using least square method. That is n TT YBBB b a 1)(       (2.10) Where                   r nn r r tztz tztz tztz B ))(()( ...... ))(()( ))(()( )1()1( 3 )1( 3 )1( 2 )1( 2 )1( (2.11) And  Tnn txtxtxY ))(),...,(),( )0(3)0(2)0( (2.12) Jos.hueuni.edu.vn Vol. 113, No. 14, 2015 161 Step 6: The solution of Eq. (2.4) can be obtained after the parameter “a” and “b” have been estimated. That is r ttrar k a b e a b txtx k               1 1 ))(1()1( 1 )0()1( 1)()(ˆ , 1r , ,...3,2,1k (2.13) Step 7: Applying inverse accumulated generating operation (IAGO) to )(ˆ )1( ktx , the pre- dicted datum of )()0( ktx can be estimated as        )(ˆ)(ˆ)(ˆ )()(ˆ 1 )1()1()0( 1 )0( 1 )0( kkk txtxtx txtx , ,...3,2k )15.2( )14.2( 2.3 Fourier Residual Modification In order to improve the accuracy of forecasting models, the Fourier series has been widely and successfully applied in modifying the residuals in grey forecasting model GM (1,1) which reduces the values of MAPE. So, this good methodology should also be considered in this case. The overall procedure to obtain the modified model is as the followings: Let x is the original series of n entries and v is the predicted series (obtained from GM (1, 1) or NGBM (1, 1). Based on the predicted series v , a residual series named is defined as:  )(k  , nk ,...3,2 (2.16) where )()()( kvkxk  , nk ,...3,2 (2.17) Expressed in Fourier series, )(k is rewritten as                        Z i ii k n i bk n i aak 1 )0( )( 1 2 sin)( 1 2 cos 2 1 )(ˆ   , nk ,.,3,2,1 (2.18) where 1) 2 1 (    n Z is the minimum deployment frequency of Fourier series [19] and only takes integer number. Therefore, the residual series is rewritten as PC (2.19) where                                                                                                                                  n n Z n n Z n n n n n Z n Z nn n Z n Z nn P 1 2 sin 1 2 cos.... 1 12 sin 1 12 cos 2 1 ............ 3 1 2 sin3 1 2 cos...3 1 12 sin3 1 12 cos 2 1 2 1 2 sin2 1 2 cos...2 1 12 sin2 1 12 cos 2 1    (2.20) and  ZZ bababaaC ,,...,,,,, 22110 (2.21) The parameter a0, a1, b1, a2, b2 aZ, bZ are obtained by using the ordinary least squares method (OLS) which results in the equation of Phan Van Thanh Vol. 113, No. 14, 2015 162   TTT PPPC 1 (2.22) Once the parameters are calculated, the modified residual series is then achieved based on the following expression                        Z i ii k n i bk n i aak 1 )0( )( 1 2 sin)( 1 2 cos 2 1 )(ˆ   (2.23) From the predicted series v and ˆ , the Fourier modified series vˆ of series v is determined by  nk vvvvvv ˆ,...,ˆ,....,ˆ,ˆ,ˆˆ 321 (2.24) where       ),...,3,2(ˆˆ ˆ ˆ 11 nkvv vv v kkk  (2.25) 2.4 Evaluative precision of forecasting models In order to evaluate the forecast capability of the model, Means Absolute Percentage Error (MAPE) index was used to evaluate the performance and reliability of forecasting technique [29]. It is expressed as follows: %100 2 )( )0( )( )0(ˆ)( )0( 1     n k kx kxkx n MAPE (2.26) where )( )0( kx and )( )0( ˆ kx are actual and forecasting values in time period k, respectively, and n is the total number of predictions. Lewis [30] interprets the MAPE results as a method to judge the accuracy of forecasts, where more than 50% is an inaccurate forecast, 20%-50% is a reasonable forecast, 10%-20% is a good forecast, and less than 10% is an excellent forecast. 3 Data and Empirical Results Tourism is one of the world’s most important and fastest growing economic sectors, generating quality jobs and substantial wealth for economies around the globe. According to the data col- lected from World Travel & Tourism Council (WTTC) [1], tourism directly and indirectly con- tributes US $7 trillion to the global economy (9.5 percent of global GDP), not only outpacing the wider economy, but also growing faster than other significant sectors such as financial and business services, transport and manufacturing. It also supports directly nearly 266 million jobs - 1 in 11 of all jobs in the world. The sustained demand for tourism, together with its ability to generate high levels of employment continues to prove the importance and value of the sector as a tool for economic development and job creation. In Vietnam, tourism industry has witnessed significant development in the last 20 years. According to Vietnam National Administration of Tourism (VNAT) [2], there were only about 25,000 inbound arrivals in 1990, but in 2013, more than 7 million arrivals were recorded in total. Jos.hueuni.edu.vn Vol. 113, No. 14, 2015 163 The contribution of tourism industry to GDP was VND 311,117 billion (9.6% of total GDP) in 2013, along with more than 4,071,500 directly and indirectly jobs created, equivalent to 7.9 of total employment [1]. These figures indicate that tourism is an important industry in Vietnam. In order for the authorities to make proper plans for the development of the tourism industry, it is critical to forecast the tourism demand accurately. However, “tourism demand” is a vague concept which is not easily measured by a cer- tain standard. It was suggested that inbound tourism demand be measured in terms of the number of tourist arrivals, tourist expenditure (tourist receipts) or the number of nights tourists spent [3, 4]. However, due to their complexities in collecting the data of tourist expenditure as well as the number of nights tourists spent, the number of tourist arrivals has been widely used as an appropriate indicator of inbound tourism demand in many researches [5-11]. Therefore, in this study, the monthly arrivals of international tourists to Vietnam are used to denote the in- bound tourism demand in Vietnam. 3.1 Data collection The historical data of the inbound tourism demand in Vietnam are obtained from the monthly statistical data published on the website of Vietnam National Administration of Tourism from January 2006 to March 2015. The inbound tourism demand is referred to the number of monthly arrivals of international tourists to Vietnam. There are totally 122 observations available as stat- ed in Table 1 [2]. Because we want to know where the in-sample fit predicts the out-of-sample forecasting performance of the model, we separate our data into two parts: the in-sample data set (Jan. 2006 - Dec. 2015) and out-of-data set (Jan., Feb., and Mar. of 2016). Table 1. Monthly arrivals of international tourists to Vietnam (Unit: Tourist arrivals) Month Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Jan 349000 369017 420000 370000 416249 506424 630000 651812 776174 700692 805072 Feb 336000 380000 441000 342913 446323 542671 681849 570476 842026 756000 833098 Mar 307081 362336 424954 303489 473509 475733 561877 587366 709725 617895 820480 Apr 309000 350878 411000 305430 432608 460000 620000 613919 745980 690440 May 282500 304848 382000 292842 350982 480886 456749 558751 674204 576868 Jun 274070 335000 210333 279150 375707 446966 417429 567291 539776 529445 Jul 303000 343000 330000 277998 410000 460000 466000 658325 564736 593566 Aug 288148 356000 339000 314915 427935 490000 525292 676719 618588 664985 Sep 277000 358000 315000 294000 383463 286618 460238 614827 590881 626324 Oct 276000 332762 296742 227859 440071 518477 495576 628695 559002 649099 Nov 305577 340000 279904 387871 428295 611864 655701 731034 608617 732740 Dec 324625 354000 375995 376400 449570 593408 614673 722349 657304 760798 Source: Website of Vietnam National Administration of Tourism [2] To solve the GM (1, 1) and NGBM (1, 1) model, and modified model with Fourier residu- al modification, Microsoft Excel is used in this study. Excel offers two useful functions, namely Mmult (array 1, array 2) to return the matrix product of two relevant arrays and Minverse (ar- Phan Van Thanh Vol. 113, No. 14, 2015 164 ray) to return the inverse matrix. These two functions are of great help to find out the values of parameters in GM (1, 1), NGBM (1, 1) and Fourier residual modification. To find out the parameters in GM (1, 1) and NGBM (1, 1) model as well as modified model of their models, Microsoft Excel is used. Besides a basic function in excel, Excel software also offers two useful functions named Mmult (array 1, array 2) to return the matrix product of two relevant arrays and Minverse (array) to return the inverse matrix. These two functions are of great help to find out the values of parameters in GM (1, 1), NGBM (1, 1) and Fourier residual modification. 3.2 GM (1, 1) model for the international tourist arrivals to Vietnam From historical data in Table 1 and based on the algorithm expressed in section 2.1, the coeffi- cient parameters a and b in GM (1, 1) for the inbound tourism in Vietnam are calculated as a = - 0.00805, b = 279746.5047, and the GM (1, 1) model for inbound tourism in Vietnam is then 31.34748803 )1(00805.0 31.35097803)( )1( ˆ    k ekx The evaluation indexes of GM (1, 1) model are also listed in Table 2. The residual series of GM (1, 1) is modified with Fourier series as illustrated in section 3.3 3.3 Modified GM (1, 1) model by Fourier series “F-GM (1, 1) model” The residual series of GM (1, 1) obtained in section 3.2 is now modified with Fourier series as per the algorithm stated in section 2.3. With this modified series, the forecasted values of the international arrivals to Vietnam based on Fourier residual modified GM (1, 1) model F-GM (1, 1) are calculated based on the equations (2.23) and (2.24). The evaluation index of F-GM (1, 1) is summarized in Table 2. 3.4 NGBM (1, 1) model for the international tourism arrivals to Vietnam Using the data in Table 1 and based on the mathematical algorithm expressed in section 2.2, the coefficient parameters a, b and the power of r in NGBM (1, 1) for the international arrivals to Vietnam are calculated as a = -0.0086, b = 524990.85934, and r = -0.041, and the NGBM (1, 1) model for the international visitors to Vietnam is then   041.01 1 23.60616388 )1)(041.01(0086.0 11.61206321)( )1(ˆ         k ekx The evaluation index of NGBM (1, 1) model is also listed in Table 2. The residual series of NGBM (1, 1) is modified with Fourier series as illustrated in section 3.5 3.5 Modified NGBM (1, 1) model by Fourier series “F-NGBM (1, 1) model” The residual series of NGBM (1, 1) obtained in section 3.4 is now modified with Fourier series as per the algorithm stated in section 2.3. With this modified series, the forecasted values of the Jos.hueuni.edu.vn Vol. 113, No. 14, 2015 165 international arrivals to Vietnam based on Fourier residual modified NGBM (1, 1) model are calculated based on the equations (2.23) and (2.24). The evaluation index of F-NGBM (1, 1) is summarized in Table 2. Table 2. Summary of evaluation indexes of model accuracy Models MAPE (%) In sample Performance Out -of -sample Performance GM (1,1) 12.84 Good 9.04 Excellent NGBM(1, 1) 12.68 Good 8.09 Excellent FGM (1, 1) 0.013 Excellent 5.19 Excellent FNGBM (1 ,1) 0.014 Excellent 6.90 Excellent Table 2 shows the evaluation indexes of each model of GM (1, 1), F-GM (1, 1), NGBM (1, 1) and F-NGBM (1, 1) with its performance in forecasting the international tourist arrivals to Vietnam. Both GM (1, 1) and NGBM (1, 1) forecasting models after using Fourier series revised their residual error provided more accuracy than original ones in terms of in-sample and out-of- sample cases. Furthermore, the results also indicate that the forecast performance of F-GM (1, 1) is the better model compared with to other models with the average MAPE for in-sample and out-of-sample of 0.013 and 5.19 %, respectively. Therefore, F-GM (1, 1) is strongly suggested in this situation. The forecasted values of the international tourism demand from Apr. of 2016 to Jun. 2016 in Vietnam are illustrated in Table 3. Table 3. Forecasted value of international visitors from Apr. 2016 –Jun. 2016 by F-GM (1, 1) Time Number of international tourist arrival Apr. 2016 749,404 May. 2016 744,748 Jun. 2016 777,483 Source: Author’s estimate Table 3 shows that the demand of international visitors in Jun. 2016 will be 777,483 arri- vals increasing 30.9% compared with Jun. 2015. With the rapid growth of the demand, provid- ing enough facilities (accommodations, transportations), having different well-organized events, providing enough skillful human resources and upgrading the local services are of ex- treme importance. Social evils, poor behaviours and corrupt customs need to be removed total- ly and replaced by the warm, kindhearted, helpful and constructive actions which will gradual- ly build up a good image of Vietnam worldwide. In recent years, people more and more con- Phan Van Thanh Vol. 113, No. 14, 2015 166 cern about the living environment (climate change, air/water pollution, etc.) and eco-tourism is of the trend. Therefore, the authorities of Vietnam tourism industry should make good plans in protecting the natural forests, waterfalls, rivers, beaches, etc. to turn these into green regions for the sake of stable development of Vietnam tourism industry. 4 Conclusions The prediction performances in Table 2 clearly show that the modified GM (1, 1) and NGBM (1, 1) models by Fourier series gain much higher accuracy than traditional models. Both F-GM (1, 1) and F-NGBM (1, 1) are good models, but F-GM (1, 1) is the better model with lowest MAPE. Therefore, F-GM (1, 1) is strongly suggested to forecast the inbound tourism demand in Vi- etnam. Highly precise forecasting result will help the policy makers and related organizations in the tourism industry of Vietnam to arrange enough facilities and human resources for high seasons and also make regular maintenance and training in low seasons just for a stable growth of the industry. Fourier residual modification has been successfully applied to the fundamental form of GM (1, 1) and the NGBM (1, 1) models to enhance their accuracy; therefore, it is thought to work well with other forecasting models as well. Further researches on this application should be conducted before the suggestion is firmly verified. References 1. Website of World Travel & Tourism Council, online available: /media/files/reports/economic%20impact%20research/country%20reports/vietnam2014.pdf 2. Website of Vietnam National Administration of Tourism: online available: 3. Ouerfelli, C. Co-integration analysis of quarterly European tourism demand in Tunisia. Tourism Man- agement. 2008, 29, 127-137. 4. 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