Forecasting energy intensity with fourier residual modified grey model: an empirical study in Taiwan

FORECASTING ENERGY INTENSITY WITH FOURIER RESIDUAL MODIFIED GREY MODEL: AN EMPIRICAL STUDY IN TAIWAN Thanh-Lam Nguyen, Ying-Fang Huang National Kaohsiung University of Applied Sciences, Kaohsiung 80778, Taiwan ABSTRACT: Energy intensity is defined as the energy consumption for producing every unit of real GDP in a certain time frame. Studies in forecasting the energy intensity have not well positioned due to the difficulty in collecting relevant data on the determinants affecting the energy consumption and GDP. Therefore, in this study, it is proposed to use the Grey forecasting model GM(1,1) to predict the energy consumption and real GDP before the intensity is forecasted. To enhance the accuracy level of the forecasting models, their residuals are then modified with Fourier series. In the case of Taiwan, the modified models resulted in very low values of mean of absolute percentage error (MAPE) of 0.33% and 0.58%, respectively to the energy consumption and real GDP. Hence, the modified model is strongly suggested to forecast the energy intensity in Taiwan from 2012-2015. Keywords: GM(1,1), FGM(1,1), Grey forecasting, Fourier series, Energy intensity. I. INTRODUCTION Energy is the core of most economic, environmental and developmental issues around the globe. It has been well proved that there is a close relationship between the energy consumption and economic development. As per the definition offered by the Department of Economic and Social Affairs of the United Nations Secretariat, energy intensity is defined as the energy consumption for producing every unit of real GDP in a certain time frame which means that the lower the energy intensity of an economy is, the better the economy performs. Energy intensity indicates the total energy used to support a wide range of production and consumption (economic and social) activities [1]. Therefore, it is usually considered as one of the measures of sustainable development. A country with highly economical productivity, pleasant weather, geographically well-allocated work places, fuel efficient vehicles, mass transportation, etc., will have a far lower energy intensity and vice versa. Many researches have been conducted and it has been found out that energy consumption and GDP are positively correlated though the correlation coefficients may be different from country to country [2]. Changes in the economy structure may result in using less additional energy; however, total energy consumption is still increasing [3]. In reducing the emissions through reducing the energy consumption, it was pointed out that the developed countries tend to be more affected by such policy rather than developing ones [4]. Many different researchers [5-15] have focused on the analysis of the relationship between the energy consumption and GDP. Nevertheless, the number of studies in forecasting the energy intensity is actually limited due to the fact that collecting relevant data on the determinants affecting the energy consumption and GDP runs into a lot of difficulties. Therefore, in this study, it is proposed to use the conventional Grey forecasting model GM(1,1), which has been widely used in different areas due to its ability to deal with the problems of uncertainty with few data points and/or “partial known, partial unknown” information, to predict the energy consumption and real GDP before the intensity is forecasted. To enhance the accuracy level of the forecasting model, its residuals are then modified with Fourier series. An empirical study in Taiwan is investigated as an example for this improved model. II. LITERATURE REVIEWS 2.1 Grey Model Grey theory offers a new approach to deal mainly with the problems of uncertainty with few data points and/or poor information which is said to be “partial known, partial unknown” [16]. The core of Grey theory is the Grey dynamic model which is usually called Grey model (GM). The Grey model is used to execute the short-term forecasting operation with no strict hypothesis for the distribution of the original data series [17]. The general GM model has the form of GM(d,v), where d is the rank of differential equation and v is the number of variables appeared in the equation. The basic model of Grey model is GM(1,1), a first-order differential model with one input variable. The procedure to obtain GM(1,1) is as the following: Step 1: Suppose an original series with n entries is (0)x :  (0) (0) (0) (0)(1), , ( ), ( )x x x k x n  1 where (0) ( )x k is the value at time k  1,k n . Step 2: From the original series (0)x , a new series (1)x can be generated by one time accumulated generating operation (1- AGO), which is  (1) (1) (1) (1)(1), , ( ), , ( )x x x k x n  2 where (1) (0) 1 ( ) ( ) k j x k x j   Step 3: A first-order differential equation with one variable is expressed as: (1) (1)dx ax b dt    3 where a is called a developing coefficient and b is called a grey input coefficient. These two coefficients can be determined by the least square method as shown below:     1 , T T Ta b B B B Y    4 where       (1) (1) (1) (1) (1) (1) (1) (2) / 2 1 (2) (3) / 2 1 1( 1) ( ) / 2 x x x x B x n x n                   (0) (0) (0)(2), (3), , ( ) T Y x x x n    Therefore, the solution of equation (3) is expressed as:    (1) (1) 1 at b b x t x e a a         5 Equation (5) is also known as time response function of the equation (3). From equation (5), the time response function of the GM(1,1) is given by:         1(1) (0)ˆ 1 1, a kb bx k x e a a k n           6 Based on the operation of one time inverse accumulated generating operation (1-IAGO), the predicted series (0)xˆ can be obtained as the following:       (0) (0) (0) (0)ˆ ˆ ˆ ˆ1 , , , ,x x x k x n  7 where         (0) (1) (0) (1) (1) ˆ ˆ(1) (1) ˆ ˆ ˆ 1 2, x x x k x k x k k n        2.2 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 RMSE, MAE, MAPE, etc., [18-22]. The overall procedure to obtain the modified model is as the followings: Let x is the orginal series of n entries and xˆ is the predicted series obtained from GM(1,1). Based on the predicted series xˆ , a residual series named  is defined as:           2 , 3 , 4 , , , ,k n       8 where  ˆ( ) ( ) ( ) 2,k x k x k k n    Expressed in Fourier series,  k is rewritten as:       0 1 1 cos sin 2 F i i i k a a D b D      9 where 2 / ( 1)D ik n   2,k n where   1 / 2 1F n   called the minimum deployment frequency of Fourier series [21] and only take integer number [18-20]. And therefore, the residual series is rewritten as: .PC   10 where 1 2 1 2 1 2 2 cos 2 sin 2 cos 2 sin 2 2 1 1 1 1 1 2 1 2 1 2 2 cos 3 sin 3 cos 3 sin 2 1 1 1 1 2 1 2 1 2 cos sin cos 2 1 1 1 F F n n n n F n n nP F n n n n n n                                                                                          3 1 2 sin 1 F n F n n                             0 1 1 2 2, , , , , , , T F FC a a b a b a b The parameters 0 1 1 2 2, , , , , , ,F Fa a b a b a b are obtained by using the ordinary least squares method (OLS) which results in the equation of:     1 TT TC P P P    Once the parameters are calculated, the modified residual series ˆ is then achieved based on the following expression:       0 1 1 ˆ cos sin 2 F i i i k a a D b D      11 From the predicted series xˆ and ˆ , the Fourier modified series x of series xˆ is determined by:         1 , 2 , , , ,x x x x k x n  12 where             ˆ1 1 ˆˆ 2, x x x k x k k k n      To evaluate the model accuracy, there are four important indexes to be considered, such as:  The mean absolute percentage error (MAPE) [19, 22, 23]:   1 1 ( ) ( ) 1, ( ) n k x k v k MAPE k n n x k    where v(k) is the forecasted value of kth entry from the model ( ˆ( ) ( )v k x k in GM(1,1) or ( ) ( )v k x k in FGM(1,1)).  The post-error ratio C [24, 25]: 2 1 S C S  where  21 1 1 1 1 ( ) ( ) n n k k S x k x x x k n n       22 1 1 1 ( ) 1 ( ) ( ) ( ) ( ) n k n k S k n k x k v k k n               The smaller the C value is, the higher accuracy the model has.  The small error probability P [24, 25]:    1 0.6745 k P p S     The higher the P value is, the higher accuracy the model has.  The forecasting accuracy  [25]: 1 MAPE   The above four indexes are used to classify the grades of forecasting accuracy as in Table 1. Table 1. Four grades of forecasting accuracy Grade level MAPE C P  I (Excellent) 0.95 > 0.95 II (Good) 0.80 > 0.90 III (Qualified) 0.70 > 0.85 IV (Unqualified) ≥ 0.10 ≥ 0.65 ≤ 0.70 ≤ 0.85 III. EMPIRICAL RESULTS The data of energy consumption from 1999 – 2011 in Taiwan are obtained from the Bureau of Energy of Ministry of Economic Affairs of Taiwan [26]; whereas the data of the Taiwan GDP from the same period are collected from International Monetary Fund [27]. Only data from 1999- 2010 are used to build relevant GM(1,1) and FGM(1,1) models. The data in 2011 is used to compare with the forecasted value from the selected model to further affirm its forecasting power. 3.1. Forecasting model for the energy consumption Based on the algorithm expressed in section 2.1, the fundamental Grey forecasting model for the energy consumption named GM(1,1)E is found as the following: (1) 0.030273( 1)ˆ ( ) 3377369.76 3285397.26kx k e   The residual series attained from GM(1,1)E is then modified with Fourier series, which results in the modified model FGM(1,1)E as per the algorithm stated in section 2.2. The evaluation indexes of GM(1,1)E and FGM(1,1)E are summarized as in Table 2. Table 2 clearly showed that between GM(1,1)E and FGM(1,1)E, FGM(1,1)E is selected because it has a lower value of MAPE and a better forecasting power. So, FGM(1,1)E is used to forecast the energy consumption in 2011. The forecasted value is then compared with the actual consumption in order to further affirm its forecasting power as shown in Table 3. The MAPE value of 5.55% indicates that FGM(1,1)E can be appropriately used to forecast the consumption in 2012 – 2015. The forecasted values in this period are shown in Table 4. 3.2. Forecasting model for the GDP Similarly, the fundamental Grey forecasting model for the GDP named GM(1,1)G is found as the following: (1) 0.032091( 1)ˆ ( ) 8943.01 8667.89kx k e   FGM(1,1)G is accordingly obtained based on section 2.2. It is also selected because it outperforms GM(1,1)G in term of low MAPE value as shown in Table 2. Its forecasted value of GDP in 2011 shown in Table 3 has an MAPE value of 1.83% indicating that it can be used to forecast the GDP in 2012 – 2015. Its relevant forecasted values are also shown in Table 4. Table 2. Summary of evaluation indexes of model accuracy Index Model MAPE S1 S2 C P  Forecasting power GM(1,1)E 0.0352 14781.86 5032.84 0.34 1.00 0.9648 Good FGM(1,1)E 0.0033 14781.86 475.76 0.03 1.00 0.9967 Excellent GM(1,1)G 0.0414 42.22 15.94 0.38 1.00 0.9586 Good FGM(1,1)G 0.0058 42.22 2.31 0.05 1.00 0.9942 Excellent Table 3. Forecasted energy consumption and GDP in 2011 Model Unit Actual value Forecasted value MAPE FGM(1,1)E 10 3 KLOE 131,832.50 139148.40 0.0555 FGM(1,1)G 10 9 USD 430.58 422.71 0.0183 Table 4. Forecasted energy consumption and GDP from 2012-2015 Item Unit 2012 2013 2014 2015 Energy 10 3 KLOE 148519.70 147592.90 152744.00 157530.00 GDP 10 9 USD 447.45 454.54 465.26 479.98 Energy intensity KLOE/10 6 USD 331.92 324.71 328.30 328.20 Table 4 shows that there is a small decrease in the energy intensity index of Taiwan in the coming years. This could be explained as an outcome of the past and current investment in the new production technology as well as the modern facilities in the transportation, services and residential sectors. Besides, the moving of its manufacturing factories to other countries including China, Vietnam, Indonesia, Malaysia, Laos, Thailand, etc., as well as the enhancing of its service industries make Taiwan not only consume less energy but also produce higher GDP, which significantly contribute to the decrease of the energy intensity index of Taiwan. IV. CONCLUSION The accuracy level of the traditional Grey forecasting model GM(1,1) can be well improved if the model is modified with Fourier series. In the case of energy intensity of Taiwan, with the Fourier modified Grey forecasting model FGM(1,1), it was found out that the energy intensity of Taiwan becomes lower and lower representing a better & stable development of the country. This result plays as an excellent motivation for the authorities to assert that they are on the right way to develop Taiwan in general and its economy in particular. Other countries could refer to this as a good example to focus on research & development as well as invest and apply advanced technology in most of their activities. REFERENCES [1] Department of Economic and Social Affairs of the United Nations Secretariat, Indicators of sustainable development: Guidelines and Methodologies, 3 rd Edition, pp.87, United Nations, New York, 2007. [2] R. Hannesson, Energy and GDP growth, International Journal of Energy Sector Management, Vol. 3, Issue 2, 2009, pp.157 – 170. [3] European Environmental Agency, Total primary energy intensity (CSI 028/ENER 017), EEA, 2006. [4] J. Chontanawat, L. C. Hunt and R. Pierse, Casuality between energy consumption and GDP: Evidence from 30 OECD and 78 Non-OECD countries, Surrey Energy Economics Centre (SEEC), University of Surrey, United Kingdom, 2006, pp.4. [5] D. I. Stern, A multivariate Cointegration Analysis of the role of the energy in US, Energy Economics, Vol. 22, 2000, pp.267 – 283. [6] F. Climent and A. Pardo, Decoupling factors on the energy-output linkage: The Spanish case, Energy Policy, Vol. 35, 2007, pp.522 – 528. [7] K. Ghali and M. El-Sakka, Energy use and output growth in Canada: A multivariate Cointegration analysis, Energy Economics, Vol. 26, 2004, pp.225 – 238. [8] U. Soytas and R. Sari, The relationship between energy and production: Evidence from Turkish manufacturing industry, Energy Economics, Vol. 29, pp.1151 – 1165. [9] B. S. Warr and R. U. Ayres, Evidence of causality between the quantity and quality of energy consumption and economic growth, Energy, Vol. 35, 2010, pp.1688 – 1693. [10] C. Lee and C. Chang, Structural breaks, energy consumption, and economic growth revisited: Evidence from Taiwan, Energy Economics, Vol. 27, 2005, pp.857 – 872. [11] J. E. Payne, US disaggregate fossil fuel consumption and real GDP: An empirical note, Energy Sources, Part B: Economics, Planning and Policy, Vol. 6, 2011, pp.63 – 68. [12] J. Asafu-Adjaye, The relationship between energy consumption, energy prices and economic growth: Time series evidence from Asian developing countries, Energy Economics, Vol. 22, 2000, pp.615 – 625. [13] K. Fatai, L. Oxley and F. G. Scrimgeour, Modelling the causal relationship between energy consumption and GDP in New Zealand, Australia, India, Indonesia, Philipines, and Thailand, Mathematics and Computers in Simulation, Vol. 64, 2004, pp.431 – 445. [14] F. Climent and A. Pardo, Decoupling factors on the energy-output linkage: The Spanish case, Energy Policy, Vol. 35, 2007, pp.522 – 528. [15] J. L. Hu and C. H. Lin, Disaggregated energy consumption and GDP in Taiwan: A threshold co-integration analysis, Energy Economics, Vol. 30, 2008, pp.2342 – 2358. [16] S. F. Liu, Advances in Grey system theory and its applications, Proceedings of 2007 IEEE International Conference on Grey Systems and Intelligent Services, China, 2007, pp.1 – 6. [17] S. J. Wang, W. L. Wang, C. T. Huang and S. C. Chen, Improving inventory effectiveness in RFID-enabled global supply chain with Grey forecasting model, Journal of Strategic Information Systems, Vol. 20, 2011, pp.307 – 322. [18] M. Askari and A. Fetanat, Long-term load forecasting in power system: Grey system prediction-based models, Journal of Applied Sciences, Vol. 11, 2011, pp.3034 – 3038. [19] Z. Guo, X. Song and J. Ye, A Verhulst model on time series error corrected for port throughput forecasting, Journal of the Eastern Asia Society for Transportation Studies, Vol. 6, 2005, pp.881 – 891. [20] L.C. Hsu, Applying the grey prediction model to the global integrated circuit industry, Technological Forecasting and Social Change, Vol. 70, 2003, pp.563 – 574. [21] Y. L. Huang and Y. H. Lee, Accurately forecasting model for the Stochastic Volatility data in tourism demand, Modern Economy, Vol. 2, 2011, pp.823 – 829. [22] M.L. Kan, Y. B. Lee and W. C. Chen, Apply grey prediction in the number of Tourist, The 4 th international conference on Genetic and Evolutionary Computing, 2010, pp.481 – 484. [23] C. C. Hsu and C. Y. Chen, Applications of improved grey prediction model for power demand forecasting, Energy Conversion and Management, Vol. 44, 2003, pp.2241 – 2249. [24] J. Hua and Q. Liang, Application research of the Grey forecast in the Logistics demand forecast, First international workshop on Education Technology and Computer Science, 2009, pp. 361 – 363. [25] H. Ma and Z. Zhang, Grey prediction with Markov-Chain for Crude oil production and consumption in China, Advances in Intelligent and Soft Computing, Vol. 56, 2009, pp.551 – 561. [26] Bureau of Energy of Ministry of Economic Affairs of Taiwan, Energy Statistics. Retrieved on April 15, 2012 from www.moeaboe.gov.tw/English/ Statistics/EnStatistics.aspx [27] International Monetary Fund, Country information. Retrieved on April 20, 2012 from www.tradingeconomics.com/ taiwan/gdp Corresponding author: Thanh-Lam Nguyen Graduate Institute of Mechanical and Precision Engineering, National Kaohsiung University of Applied Sciences 415, Chien Kung Rd., Kaohsiung 80778, Taiwan, R.O.C. Email: green4rest.vn@gmail.com