A New Approach to Improve Accuracy of Grey Model GMC ( 1 , n ) in Time Series Prediction

This paper presents a modified grey model GMC(1, n) for use in systems that involve one dependent system behavior and n − 1 relative factors. The proposed model was developed from the conventional GMC(1, n) model in order to improve its prediction accuracy by modifying the formula for calculating the background value, the system of parameter estimation, and the model prediction equation. The modified GMC(1, n)model was verified by two cases: the study of forecasting CO 2 emission in Thailand and forecasting electricity consumption in Thailand. The results demonstrated that the modified GMC(1, n) model was able to achieve higher fitting and prediction accuracy compared with the conventional GMC(1, n) and D-GMC(1, n)models.


Introduction
Grey model is a useful tool for modeling and forecasting future values of a system based on the information and knowledge obtained from the past and current data.Grey model was developed from the grey system theory introduced by Deng in the early 1980s [1].It can be used to predict behaviors of systems in the future value with high accuracy without knowing their mathematical models and used in the uncertain coefficients system with small nonnegative data.Grey model has been successfully applied to various systems [1][2][3][4][5][6][7][8].GM(, ) denotes a grey model which indicates that  variables are employed in the model and that it is an order differential equation.GM(1, 1) is a first-order onevariable grey differential equation, and it is the most widely used grey model in time series prediction [9].However, it does not find application in multivariable prediction models which are important for real applied works [4,5,[10][11][12][13].The grey prediction models have been expanded from the original GM(1, 1) to novel prediction types, such as GM(1, ) [14], GMC(1, ) [10,15], D-GMC(1, ) [14], DGDMC(1, ) [16], and CAGM(1, ) [17].GM(1, ) model is a grey multivariable model used for estimating the relationship between the system behavior and −1 relative factors [14] and has been used for various applications [4,11,[18][19][20].However, there are some limitations existing in the GM(1, ) which affect the prediction accuracy of GM(1, ) [14,15].Then, the grey multivariable model with the convolution integral GMC(1, ), proposed by Tien [10], is developed from GM(1, ) by adding the grey control parameter in the differential equation of GM(1, ) to improve the forecasting accuracy of GM(1, ).The GMC(1, ) model has successfully been applied to many real works [10,15,21,22].However, the prediction accuracy of the GMC(1, ) model depends on many factors such as smooth condition of the raw data, background value calculation, and model prediction equation.Moreover, there exists a contradiction between discrete equations for parameter estimation and continuous equations for model predictions [5,23].Therefore, a high accuracy of prediction cannot be expected of GMC(1, ) for an actual system [8,16].
In this paper, we proposed a modified grey GMC(1, ) model to improve the prediction accuracy of the conventional GMC(1, ) by modification of the formula for calculating the background value, the system of the parameter estimation, and the model prediction equation.In addition, we presented the case studies with the numerical results for prediction 2 Modelling and Simulation in Engineering accuracy for the modified GMC(1, ) model in comparison with the conventional GMC(1, ) and the discrete multivariate grey model D-GMC(1, ) [5].
The paper is organized as follows: Section 2 describes the conventional GMC(1, ), Section 3 proposes a modified grey GMC(1, ) model, Section 4 explains the statistical measure of the forecasting performance, and Section 5 presents the case study with the modified grey GMC(1, ).Finally, the conclusions are drawn in Section 6.
Substituting all the data values in (3) gives the system of linear equations that can be written as a matrix equation in the form where . . .

Modified GMC(1, 𝑛) Model
The proposed model is modified from GMC(1, ) by using two advanced improvements, as follows: (1) The equation of the predicted 1-AGO series is obtained accurately from the exactly integrated form of  (1)   .
(2) The system of parameter estimation is derived by using the model prediction equation in order to eliminate the problem of a contradiction between the system of discrete equations for parameter estimation and the system of continuous equations for model predictions.
Next, parameters   ,   , and   are determined using the least square method, which is as follows: From ( 9), the raw data can be expressed by the 1-AGO data as ( − 1) =  (1)   ( − 1) −  (1)   ( − 2) Dividing ( 12) by ( 11), we have where Let ) 2 be the objective function.By using the least square method,  can be made minimum using the parameter   , which should satisfy By solving this equation, we can obtain Substituting ( 15) into (10) gives where be the objective function.The   and   should satisfy Solving this equation yields From ( 10), the background value  (1)   () can be written as follows: Substituting ( 20) into (3), we have where According to the least squares estimation, parameters  * 2 ,  * 3 , . . .,  *  and  * of ( 22) are obtained as where . . .
By solving (2) using the integrating factor method with the initial condition X(1) 1 (1) =  (1)  1 (1), the model prediction equation for the predicted 1-AGO series can be obtained as follows.
From (2), the integrating factor is Taking the integral of both sides of the above equation in the interval [1, ], we get Substituting  =  + 1 into (28) gives Multiplying both sides of (28) by  − 1 , we get Modelling and Simulation in Engineering 5 Subtracting (30) from ( 29), we have Consequently, the predicted 1-AGO series of  (1)  1 (),  = 2, 3, . . ., , is given by (1)   ( + 1) where In the traditional grey model, the parameters are evaluated by (23), whereas the model predictions are given by (32).However, a paradox between these two equations would lead to high levels of error of prediction [5].We can infer that (23) for parameter estimation is equivalent to (32) for model prediction if parameter  1 of ( 2) is zero, which is shown as follows.
The different forms of these two equations affect the prediction accuracy of the model.Therefore, in this study, (32) is used for parameter estimation and model prediction in order to overcome this problem.
If    is a nonsingular matrix, then the solution of (38) can be obtained using the following equation: However, if    is a singular matrix, then the solution of ( 28) can be determined using the equation where  + is Moore-Penrose pseudoinverse of matrix  [25,26].Finally, the predicted series of  (0) 1 (),  = 2, 3, . . .,  is calculated by using ( 32) and (9).

Statistical Measure of the Forecasting Performance
To evaluate the performance of model simulation and prediction, two criteria, namely, the mean absolute percentage error (MAPE) and the root mean square percentage error (RMSPE), are applied for this study.MAPE and RMSPE are the most commonly used accuracy measures in prediction models [27][28][29][30].Generally, MAPE and RMSPE are defined, respectively, as where  (0) () is the actual value at time , X(0) () is its model value, and  is the number of data used for prediction.The values of MAPE and RMSPE sufficiently describe the goodness of prediction effect.The lower the values of MAPE and RMSPE, the more accurate the prediction.The criteria of MAPE and RMSPE are presented in Table 1 [29,30].In addition, absolute percentage error (APE) is used to evaluate the accuracy of the model for each data point, which is defined as

Application of Modified GMC(1, 𝑛) Model
In order to verify the performance of the modified GMC(1, ) prediction model, two different real cases of prediction problems are considered.In addition, we compare prediction accuracy with the conventional GMC(1, ) and the discrete multivariate grey model (D-GMC(1, )).
Case 1 (forecasting CO 2 emission levels in Thailand).From a previous study [31], it has been established that the main factors that affect the amount of CO 2 released into the atmosphere are natural gas, solid fuel, liquid fuel, population, and GDP.Therefore, these factors are used to predict the CO 2 emission levels in Thailand for this study.All the data, from 1994 to 2013, were obtained from the Energy Policy and Planning Office (EPPO) of Thailand (data source: http://www.eppo.go.th/info/index-statistics.html).These factors along with the factor of CO 2 emission are defined as the variables in the modified grey model GMC (1,6), which are written as follows: 1 () is the time series of CO 2 emission (1,000 tons).
3 () is the time series of solid fuel consumption (1,000 tons).
4 () is the time series of liquid fuel consumption (barrels/day).
5 () is the time series of population (persons).
is the time series of GDP (billion bahts).
Case 2 (forecasting electricity consumption in Thailand).From a previous study [32], it has been established that population, gross domestic product (GDP), stock index (SET index), and total revenue from exporting industrial products (export) are the main factors that influence the electricity consumption (measured in GWh) of Thailand.Therefore, we used these factors to construct the grey model.All data were obtained from the Energy Policy and Planning Office and Stock Exchange of Thailand.The data set was collected annually from 2002 to 2014 and is as presented in Table 4.These factors and electricity consumption are defined as the variables in the modified grey model GMC (1,5), which are written as follows: 1 () is the time series of electricity consumption (GWh).
2 () is the time series of population (persons).
is the time series of gross domestic product (billion bahts).
4 () is the time series of stock index (point).
(0) 5 () is the time series of the total revenue from exporting industrial products (billion bahts).
is an order of the time series:  = 1 refers to the year 2002 and  = 13 refers to the year 2014.

Conclusion
This study presents a modified GMC(1, ) model for improving the prediction accuracy of conventional GMC(1, ) by modification of the formula for calculating the background value, the system of parameter estimation, and the model prediction equation.The prediction accuracy can be compared between the conventional GMC(1, ), D-GMC(1, ), and the modified GMC(1, ) models in terms of the MAPE and RMSPE values.The empirical results obtained using the modified GMC(1, ) reveal the least error for MAPE and RMSPE in comparison with conventional GMC(1, ) and D-GMC(1, ) for two real cases of prediction problems.This success indicates that the modified GMC(1, ) improves the accuracy of the simulation and prediction of the conventional GMC(1, ) models.In addition, using the criteria in Table 1, it can be substantiated that the modified GMC(1, )  model demonstrates highly accurate forecasting.However, the modified GMC(1, ) model needs to be validated with more real problems and cannot be directly written in the form of the differential equation which is given as (2).For practical applications, the proposed model can be applied to many real systems, especially the systems which have the similarity in trend between the predicted variables and their influencing factors [8].In addition, the proposed techniques can be applied to the real studies [10,15,21,22] which have successfully used the conventional GMC(1, ).

Table 1 :
Criteria of MAPE and RMSPE.