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

Authors: Sompop  Moonchai; Wanwisa  Rakpuang
Journal: Modelling and Simulation in Engineering
 (2015)

---

## Abstract

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 CO2 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.


---

## Body

## 1. 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 [1–8]. GM(m,n) denotes a grey model which indicates that n variables are employed in the model and that it is an m-order differential equation. GM(1,1) is a first-order one-variable 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, 10–13]. The grey prediction models have been expanded from the original GM(1,1) to novel prediction types, such as GM(1,n) [14], GMC(1,n) [10, 15], D-GMC(1,n) [14], DGDMC(1,n) [16], and CAGM(1,n) [17]. GM(1,n) model is a grey multivariable model used for estimating the relationship between the system behavior and n-1 relative factors [14] and has been used for various applications [4, 11, 18–20]. However, there are some limitations existing in the GM(1,n) which affect the prediction accuracy of GM(1,n) [14, 15]. Then, the grey multivariable model with the convolution integral GMC(1,n), proposed by Tien [10], is developed from GM(1,n) by adding the grey control parameter in the differential equation of GM(1,n) to improve the forecasting accuracy of GM(1,n). The GMC(1,n) model has successfully been applied to many real works [10, 15, 21, 22]. However, the prediction accuracy of the GMC(1,n) 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,n) for an actual system [8, 16].In this paper, we proposed a modified grey GMC(1,n) model to improve the prediction accuracy of the conventional GMC(1,n) 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 accuracy for the modified GMC(1,n) model in comparison with the conventional GMC(1,n) and the discrete multivariate grey model D-GMC(1,n) [5].The paper is organized as follows: Section2 describes the conventional GMC(1,n), Section 3 proposes a modified grey GMC(1,n) model, Section 4 explains the statistical measure of the forecasting performance, and Section 5 presents the case study with the modified grey GMC(1,n). Finally, the conclusions are drawn in Section 6.


## 2. GMC(1,n) Model [8, 10]

Suppose that the original multivariate time seriesXi(0)=Xi(0)(1),Xi(0)(2),…,Xi(0)(r),i=1,2,…,n are a nonnegative series and available at an equispaced interval of time, where the main factor of the system behavior is X1(0), the relative factors are Xi(0),i=2,3,…,n, and r is the number of data. Then the first-order accumulative generation operation (1-AGO) of Xi(0),i=1,2,…,n, is given by the following equation:(1)Xi1=Xi11,Xi12,…,Xi1r,i=1,2,…,n,where Xi1k=∑j=1kXi0j,i=1,2,…,n,k=1,2,…,r.The prediction procedure of using the conventional GMC(1,n) model is shown as follows.The grey prediction model based on the 1-AGO data,Xi(1),i=1,2,…,n, is given by the following differential equation:(2)dX11tdt+b1X11t=b2X21t+b3X31t+⋯+bnXn1t+u,t=1,2,…,rf,where b1,b2,…,bn and u are model parameters to be estimated and rf is the number of entries to be predicted.Taking the integral of both sides of (2) in the interval [k-1,k] yields(3)X10k+b1z11k=b2z21k+b3z31k+⋯+bnzn1k+u,where zi(1)(k)=∫k-1kXi(1)(t)dt for i=1,2,…,n, which are called the background values [9].By using the trapezoidal rule, the values of∫k-1kXi(1)(t)dt for i=1,2,…,n are approximated by the following equation:(4)zi1k≈Xi1k+Xi1k-12t,i=1,2,…,n.Substituting all the data values in (3) gives the system of linear equations that can be written as a matrix equation in the form(5)Ab=Y,where(6)A=-z112z212⋯zn121-z113z213⋯zn131⋮⋮⋱⋮⋮-z11rz21r⋯zn1r1,Y=X102X103⋮X10r,b=b1b2b3⋯bnuT.Applying the least square method to this system, we obtain(7)b=ATA-1ATY.Solving (2) with the initial condition X^1(1)(1)=X1(1)(1), the predicted 1-AGO series is obtained as follows:(8)X^11k+1=X101e-b1k+θk-1∑i=2k+1e-b1k+3/2-i12fi+fi-1,fi=b2X21i+b3X31i+⋯+bnXn1i+u,where θ(k-1) is the unit step function and k=2,3,…,rf and X^1(1)(k) denotes the predicted value of X1(1)(k).Therefore, the predict series ofX1(0)(k),k=2,3,…,rf is given by(9)X^10k=X^11k-X^11k-1.


## 3. Modified GMC(1,n) Model

The proposed model is modified from GMC(1,n) 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 Xi(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.The grey model can perform well as regards predictions if the raw data satisfies the quasi-smooth and quasi-exponential conditions [17, 24]. However, if the raw data does not satisfy these conditions, function transformation methods are applied in original data sequences [17]. Suppose that the original data Xi(0),i=1,2,…,n, satisfy the quasi-smooth and quasi-exponential condition. Then, the 1-AGO data Xi(1),i=1,2,3,…,n, are fitted by the exponential function [9, 17], which can be written as(10)Xi1k=Eieaik+Fi,i=1,2,3,…,n,k=1,2,3,…,r.Next, parametersEi, ai, and Fi 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(11)Xi0k-1=Xi1k-1-Xi1k-2=Eieaik-2eai-1,(12)Xi0k=Xi1k-Xi1k-1=Eieaik-1eai-1.Dividing (12) by (11), we have(13)Xi0k=eaiXi0k-1=MiXi0k-1,whereMi=eai.LetH(Mi)=∑k=1r1/2(Xi0(k)-MiXi0(k-1))2 be the objective function. By using the least square method, H can be made minimum using the parameter Mi, which should satisfy(14)∂H∂Mi=-∑k=1rXi0k-MiXi0k-1Xi0k-1=0.By solving this equation, we can obtain(15)Mi=eai=∑k=1rXi0kXi0k-1∑k=1rXi0k-12,(16)ai=ln⁡∑k=1rXi0kXi0k-1∑k=1rXi0k-12.Substituting (15) into (10) gives(17)Xi1k=EiWik+Fi,where Wi(k)=Mik=eaik.LetQ(Ei,Fi)=∑k=1r1/2(Xi(1)(k)-EiWi(k)-Fi)2 be the objective function. The Ei and Fi should satisfy(18)∂Q∂Ei=-∑k=1rXi1k-EiWik-FiWik=0,∂Q∂Fi=-∑k=1rXi1k-EiWik-Fi=0.Solving this equation yields(19)Ei=r∑k=1rWikXi1k-∑k=1rWik∑k=1rXi1kr∑k=1rWik2-∑k=1rWik2=r∑k=1reaikXi1k-∑k=1reaik∑k=1rXi1kr∑k=1reaik2-∑k=1reaik2,Fi=∑k=1rXi1kr-Ei∑k=1reaikr.From (10), the background value zi(1)(k) can be written as follows:(20)zi1k=∫k-1kXi1tdt=∫k-1kEieait+Fidt=Eieaitai+Fitk-1k=Eieaikai-Eieaik-1ai+Fik-k-1=1aiEieaik-Eieaik-1+Fik-k-1=1aiXi1k-Xi1k-1+Fi=Xi0kai+Fi,i=1,2,3,…,n.Substituting (20) into (3), we have(21)X10k+b1X10ka1+F1=∑i=2nbiXi0kai+Fi+u,a1+b1a1X10k=∑i=2nbiXi0kai+Fi+u-b1F1,X10k=∑i=2na1bia1+b1Xi0kai+Fi+a1u-b1F1a1+b1,(22)X10k=∑i=2nbi∗Xi0kai+Fi+u∗,where(23)bi∗=a1bia1+b1,u∗=a1u-b1F1a1+b1.According to the least squares estimation, parametersb2∗,b3∗,…,bn∗ and u∗ of (22) are obtained as(24)b2∗b3∗b4∗⋯bn∗u∗T=BTB-1BTZ,where(25)B=X201a2+F1X301a3+F3⋯Xn01an+Fn1X202a2+F2X302a3+F3⋯Xn02an+Fn1⋮⋮⋱⋮⋮X20ra2+F2X30ra3+F3⋯Xn0ran+Fn1,Z=X101X102⋮X10r.By solving (2) using the integrating factor method with the initial condition X^1(1)(1)=X1(1)(1), the model prediction equation for the predicted 1-AGO series can be obtained as follows.From (2), the integrating factor is e∫b1dt=eb1t. Multiplying (2) by eb1t obtains(26)ddteb1tX11t=∑i=2neb1tbiXi1tdt+eb1tu.Taking the integral of both sides of the above equation in the interval [1,k], we get(27)∫1kddteb1tX11tdt=∫1k∑i=2neb1tbiXi1tdt+∫1keb1tudt,X11keb1k-X111eb1=∫1k∑i=2neb1tbiEieait+Fidt+∫1keb1tudt=∫1k∑i=2nbiEieb1+ait+eb1tFidt+∫1keb1tudt,X11keb1k=X111eb1+∑i=2nbiEieb1+aik-eb1+aib1+ai+Fieb1k-eb1b1+ueb1kb1-ueb1b1,(28)X11k=X111e-b1k-1+∑i=2nbiEieaikb1+ai+Fib1-∑i=2nbiEie-b1k-1+aib1+ai+Fie-b1k-1b1+ub1-ue-b1k-1b1.Substituting k=k+1 into (28) gives(29)X11k+1=X111e-b1k+∑i=2nbiEieaik+1b1+ai+Fib1-∑i=2nbiEie-b1k+aib1+ai+Fie-b1kb1+ub1-ue-b1kb1.Multiplying both sides of (28) by e-b1, we get(30)e-b1X11k=X111e-b1k+∑i=2nbiEjeaik-b1b1+ai+Fie-b1b1-∑i=2nbiEje-b1k+aib1+ai+Fie-b1kb1+ue-b1b1-ue-b1kb1.Subtracting (30) from (29), we have(31)X11k+1-e-b1X^11k=X111e-b1k+∑i=2nbiEieaik+1b1+ai+Fib1-∑i=2nbiEie-b1k+aib1+ai+Fie-b1kb1+ub1-ue-b1kb1-X111e-b1k-∑i=2nbiEieaik-b1b1+ai+Fie-b1b1+∑i=2nbiEie-b1k+aib1+ai+Fie-b1kb1-ue-b1b1+ue-b1kb1=∑i=2nbiEieaik+1b1+ai+Fib1-∑i=2nbiEieaik-b1b1+ai+Fie-b1b1+ub1-ue-b1b1=∑i=2nbiEieaik+1+Fib1+ai-Fib1+ai+Fib1-∑i=2nbiEieaik-b1+Fib1+ai-Fib1+ai+Fie-b1b1+ub1-ue-b1b1=∑i=2nbiXi1k+1b1+ai+Fib1-∑i=2nbiXi1ke-b1b1+ai+Fie-b1b1+ub1-ue-b1b1=∑i=2nbiXi1k+1b1+ai-∑i=2nbiXi1ke-b1b1+ai+∑i=2nbiFib1-Fie-b1b1+ub1-ue-b1b1=e-b1X^11k+∑i=2nbiXi1k+1b1+ai-∑i=2nbiXi1ke-b1b1+ai+∑i=2nbiFib1-Fie-b1b1+ub1-ue-b1b1.Consequently, the predicted 1-AGO series ofX1(1)(k),k=2,3,…,rf, is given by(32)X^11k+1=c1X^11k+∑i=2nciXi1k+1+∑i=2ndiXi1k+v,(33)X^10k+1=X^11k+1-X^11k,where c1=e-b1, ci=bi/b1+ai, di=bic1/b1+ai, and v=∑i=2nbiFi/b1-Fic1/b1+u/b1-uc1/b1 for i=2,3,…,n.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 b1 of (2) is zero, which is shown as follows.Settingb1=0 and taking the integral of both sides of (2) in the interval [1,k], we have(34)∫1kddtX11tdt=∫1k∑i=2nbiXi1tdt+∫1kudt,X11k-X111=∑i=2nbi∫1kEieait+Fidt+∫1kudt=∑i=2nbiEiaieaik-eai+Fik-1+uk-1=∑i=2nbiXi1k-Xi11ai+Fik-1+uk-1=∑i=2nbiaiXi1k-Xi11+∑i=2nbiFik-1+uk-1,(35)X11k-1-X111=∑i=2nbiaiXi1k-1-Xi11+∑i=2nbiFik-2+uk-2.Substituting k=k+1 into (35) gives(36)X11k-X111=∑i=2nbiaiXi1k-Xi11+∑i=2nbiFik-1+uk-1.Subtracting (36) from (35), we have(37)X11k-X11k-1=∑i=2nbiaiXi1k-Xi1k-1+∑i=2nbiFi+u,X10k=∑i=2nbiXi0kai+F+u.Therefore, (32) is the same as (23).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.Firstly, we substitute all the data values into (32). Then, the linear system is derived as follows:(38)Dc=P,where(39)D=X111X212X312⋯Xn12X211X311⋯Xn111X112X213X313⋯Xn13X212X312⋯Xn121⋮⋮⋮⋱⋮⋮⋮⋱⋮⋮X11r-1X21rX31r⋯Xn1rX21r-1X31r-1⋯Xn1r-11,P=X112X113⋮X11r,c=c1c2c3⋯cnd2d3⋯dnvT.The parameters c1,c2,c3,…,cn,d2,d3,…,dn and v are determined using the least squares estimation, as follows.IfDTD is a nonsingular matrix, then the solution of (38) can be obtained using the following equation:(40)c=c1c2c3⋯cnd2d3⋯dnvT=DTD-1DTP.However, if DTD is a singular matrix, then the solution of (28) can be determined using the equation(41)c=c1c2c3⋯cnd2d3⋯dnvT=D+P,where D+ is Moore-Penrose pseudoinverse of matrix D [25, 26].Finally, the predicted series ofX1(0)(k),k=2,3,…,rf is calculated by using (32) and (9).


## 4. 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 [27–30]. Generally, MAPE and RMSPE are defined, respectively, as(42)MAPE=1r-1∑k=2rX0k-X^0kX0k×100%,(43)RMSPE=∑k=2rX0k-X^0k/X0k2r-1×100%,where X(0)(k) is the actual value at time k, X^(0)(k) is its model value, and r 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(44)APE=X0-X^0X0×100%.Table 1
Criteria of MAPE and RMSPE.
MAPE and RMSPE (%)
Forecasting power
<10
Highly accurate forecasting
10–20
Good forecasting
20–50
Reasonable forecasting
>50
Inaccurate forecasting


## 5. Application of Modified GMC(1,n) Model

In order to verify the performance of the modified GMC(1,n) prediction model, two different real cases of prediction problems are considered. In addition, we compare prediction accuracy with the conventional GMC(1,n) and the discrete multivariate grey model (D-GMC(1,n)).Case 1 (forecasting CO2 emission levels in Thailand).
From a previous study [31], it has been established that the main factors that affect the amount of CO2 released into the atmosphere are natural gas, solid fuel, liquid fuel, population, and GDP. Therefore, these factors are used to predict the CO2 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 ofCO2 emission are defined as the variables in the modified grey model GMC(1,6), which are written as follows:X1(0)(t) is the time series of CO2 emission (1,000 tons).X2(0)(t) is the time series of natural gas consumption (MMSCFD).X3(0)(t) is the time series of solid fuel consumption (1,000 tons).X4(0)(t) is the time series of liquid fuel consumption (barrels/day).X5(0)(t) is the time series of population (persons).X6(0)(t) is the time series of GDP (billion bahts).tis an order of the time series: t=1 refers to the year 1994 and t=20 refers to the year 2013.
The actual data are separated into two parts: the first part (1994–2010) is used to construct the models, while the second part (2011–2013) is employed for model prediction. All the data satisfy the quasi-smooth and quasi-exponential conditions. The result of the modified GMC(1,6) model, obtained from (32), has the form(45)X^11k+1=0.248X^11k-1.133X21k+1-1.359X21k+0.326X31k+1+0.227X31k+0.487X41k+1+0.300X41k+0.031X51k+1+0.015X51k+0.064X61k+1-0.095X61k+1.285,k=1,2,…,19.
The actual values, the model values, the predictive values, and the APE of D-GMC(1,6) and the modified GMC(1,6) models are presented in Table 2. Table 3 shows the performance of the model simulation and prediction of D-GMC(1,6) as well as the modified GMC(1,6). As demonstrated in these tables, both the models have highly accurate forecasting abilities. However, the modified GMC(1,6) has lower MAPE and RMSPE than D-GMC(1,6), which indicates that the modified GMC(1,6) has higher accuracy of prediction than D-GMC(1,6). As far as the conventional GMC(1,6) is concerned, large errors were obtained for the model simulation and prediction (results not shown).Table 2
Comparison of actual values, model values, and APEs of D-GMC(1,6) and modified GMC(1,6) models.
Years
t
CO2 emission data
D-GMC(1,6)
Modified GMC(1,6)
Model value
APE
Model value
APE
1994
1
122,876.13
122,876.13
0.00
122,876.13
0.00
1995
2
137,691.67
142,056.72
3.17
138,089.13
0.29
1996
3
155,553.15
148,750.34
4.37
154,325.20
0.79
1997
4
161,057.54
154,933.62
3.80
162,115.22
0.66
1998
5
145,451.52
161,292.65
10.89
146,188.49
0.51
1999
6
149,881.53
140,012.17
6.58
148,098.26
1.19
2000
7
150,832.00
150,968.83
0.09
150,578.46
0.17
2001
8
153,545.22
157,968.41
2.88
156,157.34
1.70
2002
9
163,558.08
164,645.66
0.66
161,506.52
1.25
2003
10
172,053.83
176,232.82
2.43
172,557.16
0.29
2004
11
187,418.22
176,180.65
6.00
189,201.76
0.95
2005
12
192,574.90
193,410.05
0.43
188,058.52
2.35
2006
13
193,233.99
197,621.97
2.27
196,939.39
1.92
2007
14
200,494.35
194,241.16
3.12
199,340.27
0.58
2008
15
203,261.89
208,243.34
2.45
204,008.47
0.37
2009
16
208,267.86
211,387.03
1.50
207,577.79
0.33
2010
17
220,467.16
217,488.33
1.35
220,488.97
0.01
2011∗
18
224,465.07
236,377.49
5.31
222,859.77
0.72
2012∗
19
240,583.97
234,631.65
2.47
240,026.56
0.23
2013∗
20
240,802.07
248,902.18
3.36
241,530.87
0.30
∗Prediction.Table 3
Performance evaluation for fitting and Prediction accuracies of D-GMC(1,6) and modified GMC(1,6) models.
Criteria
D-GMC(1,6)
Modified GMC(1,6)
Model value
Predictive value
Model value
Predictive value
MAPE (%)
3.25
3.71
0.83
0.42
RMSPE (%)
4.19
3.90
1.06
0.47Case 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:X1(0)(t) is the time series of electricity consumption (GWh).X2(0)(t) is the time series of population (persons).X3(0)(t) is the time series of gross domestic product (billion bahts).X4(0)(t) is the time series of stock index (point).X5(0)(t) is the time series of the total revenue from exporting industrial products (billion bahts).t is an order of the time series: t=1 refers to the year 2002 and t=13 refers to the year 2014.Table 4
Comparison of actual values, model values, and APE of D-GMC(1,5) and modified GMC(1,5) models.
Years
t
Electricity consumption data
D-GMC(1,5)
Modified GMC(1,5)
Model value
APE
Model value
APE
2002
1
100,091
100,091.00
0.000
100091.00
0
2003
2
106,987
105,968.82
0.952
106987.17
4.11 × 10−12
2004
3
115,101
117,863.18
2.400
115101.15
1.29 × 10−11
2005
4
121,240
119,605.49
1.348
121240.03
1.28 × 10−11
2006
5
127,879
127,053.81
0.645
127879.26
1.25 × 10−11
2007
6
133,113
132,870.66
0.182
133112.82
1.24 × 10−11
2008
7
135,520
137,355.47
1.354
135520.31
1.24 × 10−11
2009
8
135,181
134,748.54
0.320
135180.80
1.22 × 10−11
2010
9
149,301
148,048.60
0.839
149301.14
1.15 × 10−11
2011
10
148,855
149,991.01
0.763
148855.54
1.28 × 10−11
2012∗
11
161,779
169,923.11
5.034
159807.95
1.22
2013∗
12
164,341
161,959.15
1.449
164233.19
0.07
2014∗
13
168,620
165,305.30
1.966
163396.00
3.10
∗Prediction.We will use the steps of the modified grey GMC(1,5) model for finding the parameters from 2002 to 2011 for the prediction of electricity consumption (measured in GWh) from 2012 to 2014. All the data satisfy the quasi-smooth and quasi-exponential conditions. The result of the modified GMC(1,5) model, obtained from (32), has the form(46)X^11k+1=0.417X^11k+0.398X21k+1-0.025X21k+0.910X31k+1-0.491X31k+0.006X41k+1-0.013X41k-0.139X51k+1-0.018X51k-0.226,k=1,2,…,12.The actual values, the model values, the predictive values, and the APE of D-GMC(1,5) and the modified GMC(1,5) models are presented in Table 4. Table 5 shows the MAPE and RMSPE values of D-GMC(1,5) and the modified GMC(1,5) models, which are used to measure the accuracy of prediction of the models. These results indicate that the modified GMC(1,5) has lower MAPE and RMSPE values than D-GMC(1,5). For the conventional GMC(1,5), large errors were obtained for the model simulation and prediction (results not shown). Therefore, it can be safely concluded that the modified GMC(1,5) has a higher degree of accuracy of prediction than the conventional GMC(1,5) and D-GMC(1,5).Table 5
Performance evaluation for fitting and prediction accuracies of D-GMC(1,5), and modified GMC(1,5) models.
Error
D-GMC(1,5)
Modified GMC(1,5)
Model value
Predictive value
Model value
Predictive value
MAPE (%)
0.98
2.82
1.15 × 10−11
1.46
RMSPE (%)
1.16
3.23
1.17 × 10−11
1.92


## 6. Conclusion

This study presents a modified GMC(1,n) model for improving the prediction accuracy of conventional GMC(1,n) 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,n), D-GMC(1,n), and the modified GMC(1,n) models in terms of the MAPE and RMSPE values. The empirical results obtained using the modified GMC(1,n) reveal the least error for MAPE and RMSPE in comparison with conventional GMC(1,n) and D-GMC(1,n) for two real cases of prediction problems. This success indicates that the modified GMC(1,n) improves the accuracy of the simulation and prediction of the conventional GMC(1,n) models. In addition, using the criteria in Table 1, it can be substantiated that the modified GMC(1,n) model demonstrates highly accurate forecasting. However, the modified GMC(1,n) 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,n).



---

*Source: 126738-2015-12-07.xml*