FORECASTING ENERGY INTENSITY WITH FOURIER RESIDUAL MODIFIED GREY MODEL: AN EMPIRICAL STUDY IN TAIWAN

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1 ORECASTING ENERGY INTENSITY WITH OURIER RESIDUAL MODIIED GREY MODEL: AN EMPIRICAL STUDY IN TAIWAN Thah-Lam Nguye, Yig-ag Huag Natioal Kaohsiug Uiversity of Applied Scieces, Kaohsiug 80778, Taiwa ABSTRACT: Eergy itesity is defied as the eergy cosumptio for producig every uit of real GDP i a certai time frame. Studies i forecastig the eergy itesity have ot well positioed due to the difficulty i collectig relevat data o the determiats affectig the eergy cosumptio ad GDP. Therefore, i this study, it is proposed to use the Grey forecastig model GM(,) to predict the eergy cosumptio ad real GDP before the itesity is forecasted. To ehace the accuracy level of the forecastig models, their residuals are the modified with ourier series. I the case of Taiwa, the modified models resulted i very low values of mea of absolute percetage error (MAPE) of 0.33% ad 0.58%, respectively to the eergy cosumptio ad real GDP. Hece, the modified model is strogly suggested to forecast the eergy itesity i Taiwa from Keywords: GM(,), GM(,), Grey forecastig, ourier series, Eergy itesity. I. INTRODUCTION Eergy is the core of most ecoomic, evirometal ad developmetal issues aroud the globe. It has bee well proved that there is a close relatioship betwee the eergy cosumptio ad ecoomic developmet. As per the defiitio offered by the Departmet of Ecoomic ad Social Affairs of the Uited Natios Secretariat, eergy itesity is defied as the eergy cosumptio for producig every uit of real GDP i a certai time frame which meas that the lower the eergy itesity of a ecoomy is, the better the ecoomy performs. Eergy itesity idicates the total eergy used to support a wide rage of productio ad cosumptio (ecoomic ad social) activities []. Therefore, it is usually cosidered as oe of the measures of sustaiable developmet. A coutry with highly ecoomical productivity, pleasat weather, geographically well-allocated work places, fuel efficiet vehicles, mass trasportatio, etc., will have a far lower eergy itesity ad vice versa. May researches have bee coducted ad it has bee foud out that eergy cosumptio ad GDP are positively correlated though the correlatio coefficiets may be differet from coutry to coutry []. Chages i the ecoomy structure may result i ug less additioal eergy; however, total eergy cosumptio is still icreag [3]. I reducig the emissios through reducig the eergy cosumptio, it was poited out that the developed coutries ted to be more affected by such policy rather tha developig oes [4]. May differet researchers [5-5] have focused o the aalysis of the relatioship betwee the eergy cosumptio ad GDP. Nevertheless, the umber of studies i forecastig the eergy itesity is actually limited due to the fact that collectig relevat data o the determiats affectig the eergy cosumptio ad GDP rus ito a lot of difficulties. Therefore, i this study, it is proposed to use the covetioal Grey forecastig model GM(,), which has bee widely used i differet areas due to its ability to deal with the problems of ucertaity with few data poits ad/or partial kow, partial ukow iformatio, to predict the eergy cosumptio ad real GDP before the itesity is forecasted. To ehace the accuracy level of the forecastig model, its residuals are the modified with ourier series. A empirical study i Taiwa is ivestigated as a example for this improved model.

2 II. LITERATURE REVIEWS. Grey Model Grey theory offers a ew approach to deal maily with the problems of ucertaity with few data poits ad/or poor iformatio which is said to be partial kow, partial ukow [6]. The core of Grey theory is the Grey dyamic model which is usually called Grey model (GM). The Grey model is used to execute the short-term forecastig operatio with o strict hypothesis for the distributio of the origial data series [7]. The geeral GM model has the form of GM(d,v), d is the rak of differetial equatio ad v is the umber of variables appeared i the equatio. The basic model of Grey model is GM(,), a first-order differetial model with oe iput variable. The procedure to obtai GM(,) is as the followig: Step : Suppose a origial series with (0) etries is x : (0) (0) (0) (0) x x,, x ( k), x ( ) x (0) ( k ) is the value at time k k,. Step : rom the origial series x (0), a ew series x ca be geerated by oe time accumulated geeratig operatio (- AGO), which is x x,, x ( k),, x ( ) k (0) j x ( k) x ( j) Step 3: A first-order differetial equatio with oe variable is expressed as: ax b 3 dx dt a is called a developig coefficiet ad b is called a grey iput coefficiet. These two coefficiets ca be determied by the least square method as show below:, T T T a b B B B Y 4 x x () / x () x (3) / B x ( ) x ( ) / (0) (0) (0) Y x (), x (3),, x ( ) T Therefore, the solutio of equatio (3) is expressed as: b at b x t x e a 5 a Equatio (5) is also kow as time respose fuctio of the equatio (3). rom equatio (5), the time respose fuctio of the GM(,) is give by: (0) b a k ˆ b x k x e a a 6 k, Based o the operatio of oe time iverse accumulated geeratig operatio (0) (-IAGO), the predicted series ˆx ca be obtaied as the followig: (0) (0) (0) (0) xˆ xˆ,, xˆ k,, xˆ 7 (0) xˆ xˆ (0) xˆ k xˆ k xˆ k k,. ourier Residual Modificatio I order to improve the accuracy of forecastig models, the ourier series has bee widely ad successfully applied i modifyig the residuals i Grey forecastig model GM(,) which reduces the values of RMSE, MAE, MAPE, etc., [8-]. The overall procedure to obtai the modified model is as the followigs: Let x is the orgial series of etries ad ˆx is the predicted series obtaied from GM(,). Based o the predicted series ˆx, a residual series amed is defied as:, 3, 4,, k,, 8 ( k) x( k) xˆ ( k) k, Expressed i ourier series, k is rewritte as: k a0 ai D bi D 9 i D ik / ( ) k,

3 / called the rewritte as: miimum deploymet frequecy of ourier PC. 0 series [] ad oly take iteger umber [8-0]. Ad therefore, the residual series is P 3 C a, a, b, a, b,, a, b 0 The parameters a0, a, b, a, b,, a, b are obtaied by ug the ordiary least squares method (OLS) which results i the equatio of: T T T C P P P Oce the parameters are calculated, the modified residual series ˆ is the achieved based o the followig expressio: ˆ k a0 ai D bi D i rom the predicted series ˆx ad ˆ, the ourier modified series x of series ˆx is determied by: x x, x,, x k,, x x xˆ x k xˆ k ˆ k k, To evaluate the model accuracy, there are four importat idexes to be cosidered, such as: The mea absolute percetage error (MAPE) [9,, 3]: x( k) v( k) MAPE k, x( k) k v(k) is the forecasted value of kth etry from the model ( v( k) xˆ ( k) i T GM(,) or v( k) x( k) i GM(,)). The post-error ratio C [4, 5]: S C S S x ( k ) x x x ( k ) S k k k ( k) ( k) x( k) v( k) ( k) k The smaller the C value is, the higher accuracy the model has. The small error probability P [4, 5]: k P p S The higher the P value is, the higher accuracy the model has. The forecastig accuracy [5]: MAPE The above four idexes are used to classify the grades of forecastig accuracy as i Table. Table. our grades of forecastig accuracy Grade level MAPE C P I (Excellet) < 0.0 < 0.35 > 0.95 > 0.95 II (Good) < 0.05 < 0.50 > 0.80 > 0.90 III (Qualified) < 0.0 < 0.65 > 0.70 > 0.85 IV (Uqualified)

4 III. EMPIRICAL RESULTS The data of eergy cosumptio from i Taiwa are obtaied from the Bureau of Eergy of Miistry of Ecoomic Affairs of Taiwa [6]; as the data of the Taiwa GDP from the same period are collected from Iteratioal Moetary ud [7]. Oly data from are used to build relevat GM(,) ad GM(,) models. The data i 0 is used to compare with the forecasted value from the selected model to further affirm its forecastig power. 3.. orecastig model for the eergy cosumptio Based o the algorithm expressed i sectio., the fudametal Grey forecastig model for the eergy cosumptio amed GM(,) E is foud as the followig: ( k) xˆ ( k) e The residual series attaied from GM(,) E is the modified with ourier series, which results i the modified model GM(,) E as per the algorithm stated i sectio.. The evaluatio idexes of GM(,) E ad GM(,) E are summarized as i Table. Table clearly showed that betwee GM(,) E ad GM(,) E, GM(,) E is selected because it has a lower value of MAPE ad a better forecastig power. So, GM(,) E is used to forecast the eergy cosumptio i 0. The forecasted value is the compared with the actual cosumptio i order to further affirm its forecastig power as show i Table 3. The MAPE value of 5.55% idicates that GM(,) E ca be appropriately used to forecast the cosumptio i The forecasted values i this period are show i Table orecastig model for the GDP Similarly, the fudametal Grey forecastig model for the GDP amed GM(,) G is foud as the followig: ( ) ˆ ( ) k x k e GM(,) G is accordigly obtaied based o sectio.. It is also selected because it outperforms GM(,) G i term of low MAPE value as show i Table. Its forecasted value of GDP i 0 show i Table 3 has a MAPE value of.83% idicatig that it ca be used to forecast the GDP i Its relevat forecasted values are also show i Table 4. Table. Summary of evaluatio idexes of model accuracy Idex orecastig MAPE S S C P Model power GM(,) E Good GM(,) E Excellet GM(,) G Good GM(,) G Excellet Table 3. orecasted eergy cosumptio ad GDP i 0 Model Uit Actual value orecasted value MAPE GM(,) E 0 3 KLOE 3, GM(,) G 0 9 USD Table 4. orecasted eergy cosumptio ad GDP from 0-05 Item Uit Eergy 0 3 KLOE GDP 0 9 USD Eergy itesity KLOE/0 6 USD

5 Table 4 shows that there is a small decrease i the eergy itesity idex of Taiwa i the comig years. This could be explaied as a outcome of the past ad curret ivestmet i the ew productio techology as well as the moder facilities i the trasportatio, services ad residetial sectors. Besides, the movig of its maufacturig factories to other coutries icludig Chia, Vietam, Idoesia, Malaysia, Laos, Thailad, etc., as well as the ehacig of its service idustries make Taiwa ot oly cosume less eergy but also produce higher GDP, which sigificatly cotribute to the decrease of the eergy itesity idex of Taiwa. IV. CONCLUSION The accuracy level of the traditioal Grey forecastig model GM(,) ca be well improved if the model is modified with ourier series. I the case of eergy itesity of Taiwa, with the ourier modified Grey forecastig model GM(,), it was foud out that the eergy itesity of Taiwa becomes lower ad lower represetig a better & stable developmet of the coutry. This result plays as a excellet motivatio for the authorities to assert that they are o the right way to develop Taiwa i geeral ad its ecoomy i particular. Other coutries could refer to this as a good example to focus o research & developmet as well as ivest ad apply advaced techology i most of their activities. [] Departmet of Ecoomic ad Social Affairs of the Uited Natios Secretariat, Idicators of sustaiable developmet: Guidelies ad Methodologies, 3 rd Editio, pp.87, Uited Natios, New York, 007. [] R. Haesso, Eergy ad GDP growth, Iteratioal Joural of Eergy Sector Maagemet, Vol. 3, Issue, 009, pp [3] Europea Evirometal Agecy, Total primary eergy itesity (CSI 08/ENER 07), EEA, 006. [4] J. Chotaawat, L. C. Hut ad R. Pierse, Casuality betwee eergy cosumptio ad GDP: Evidece from 30 OECD ad 78 No-OECD coutries, Surrey Eergy Ecoomics Cetre (SEEC), Uiversity of Surrey, Uited Kigdom, 006, pp.4. [5] D. I. Ster, A multivariate Coitegratio Aalysis of the role of the eergy i US, Eergy Ecoomics, Vol., 000, pp [6]. Climet ad A. Pardo, Decouplig factors o the eergy-output likage: The Spaish case, Eergy Policy, Vol. 35, 007, pp REERENCES [7] K. Ghali ad M. El-Sakka, Eergy use ad output growth i Caada: A multivariate Coitegratio aalysis, Eergy Ecoomics, Vol. 6, 004, pp [8] U. Soytas ad R. Sari, The relatioship betwee eergy ad productio: Evidece from Turkish maufacturig idustry, Eergy Ecoomics, Vol. 9, pp [9] B. S. Warr ad R. U. Ayres, Evidece of causality betwee the quatity ad quality of eergy cosumptio ad ecoomic growth, Eergy, Vol. 35, 00, pp [0] C. Lee ad C. Chag, Structural breaks, eergy cosumptio, ad ecoomic growth revisited: Evidece from Taiwa, Eergy Ecoomics, Vol. 7, 005, pp [] J. E. Paye, US disaggregate fossil fuel cosumptio ad real GDP: A empirical ote, Eergy Sources, Part B: Ecoomics, Plaig ad Policy, Vol. 6, 0, pp [] J. Asafu-Adjaye, The relatioship betwee eergy cosumptio, eergy prices ad ecoomic growth: Time

6 series evidece from Asia developig coutries, Eergy Ecoomics, Vol., 000, pp [3] K. atai, L. Oxley ad. G. Scrimgeour, Modellig the causal relatioship betwee eergy cosumptio ad GDP i New Zealad, Australia, Idia, Idoesia, Philipies, ad Thailad, Mathematics ad Computers i Simulatio, Vol. 64, 004, pp [4]. Climet ad A. Pardo, Decouplig factors o the eergy-output likage: The Spaish case, Eergy Policy, Vol. 35, 007, pp [5] J. L. Hu ad C. H. Li, Disaggregated eergy cosumptio ad GDP i Taiwa: A threshold co-itegratio aalysis, Eergy Ecoomics, Vol. 30, 008, pp [6] S.. Liu, Advaces i Grey system theory ad its applicatios, Proceedigs of 007 IEEE Iteratioal Coferece o Grey Systems ad Itelliget Services, Chia, 007, pp. 6. [7] S. J. Wag, W. L. Wag, C. T. Huag ad S. C. Che, Improvig ivetory effectiveess i RID-eabled global supply chai with Grey forecastig model, Joural of Strategic Iformatio Systems, Vol. 0, 0, pp [8] M. Askari ad A. etaat, Log-term load forecastig i power system: Grey system predictio-based models, Joural of Applied Scieces, Vol., 0, pp [9] Z. Guo, X. Sog ad J. Ye, A Verhulst model o time series error corrected for port throughput forecastig, Joural of the Easter Asia Society for Trasportatio Studies, Vol. 6, 005, pp [0] L.C. Hsu, Applyig the grey predictio model to the global itegrated circuit idustry, Techological orecastig ad Social Chage, Vol. 70, 003, pp [] Y. L. Huag ad Y. H. Lee, Accurately forecastig model for the Stochastic Volatility data i tourism demad, Moder Ecoomy, Vol., 0, pp [] M.L. Ka, Y. B. Lee ad W. C. Che, Apply grey predictio i the umber of Tourist, The 4 th iteratioal coferece o Geetic ad Evolutioary Computig, 00, pp [3] C. C. Hsu ad C. Y. Che, Applicatios of improved grey predictio model for power demad forecastig, Eergy Coversio ad Maagemet, Vol. 44, 003, pp [4] J. Hua ad Q. Liag, Applicatio research of the Grey forecast i the Logistics demad forecast, irst iteratioal workshop o Educatio Techology ad Computer Sciece, 009, pp [5] H. Ma ad Z. Zhag, Grey predictio with Markov-Chai for Crude oil productio ad cosumptio i Chia, Advaces i Itelliget ad Soft Computig, Vol. 56, 009, pp [6] Bureau of Eergy of Miistry of Ecoomic Affairs of Taiwa, Eergy Statistics. Retrieved o April 5, 0 from Statistics/EStatistics.aspx [7] Iteratioal Moetary ud, Coutry iformatio. Retrieved o April 0, 0 from taiwa/gdp Correspodig author: Thah-Lam Nguye Graduate Istitute of Mechaical ad Precisio Egieerig, Natioal Kaohsiug Uiversity of Applied Scieces 45, Chie Kug Rd., Kaohsiug 80778, Taiwa, R.O.C. gree4rest.v@gmail.com

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