Modelling and Simulation in Engineering

Volume 2015 (2015), Article ID 126738, 10 pages

http://dx.doi.org/10.1155/2015/126738

## A New Approach to Improve Accuracy of Grey Model GMC in Time Series Prediction

^{1}Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand^{2}Department of Mathematics, Faculty of Science and Technology, Kamphaeng Phet Rajabhat University, Kamphaeng Phet 62000, Thailand

Received 2 September 2015; Accepted 17 November 2015

Academic Editor: Min-Chie Chiu

Copyright © 2015 Sompop Moonchai and Wanwisa Rakpuang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a modified grey model GMC for use in systems that involve one dependent system behavior and relative factors. The proposed model was developed from the conventional GMC 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 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 model was able to achieve higher fitting and prediction accuracy compared with the conventional GMC and D-GMC models.

#### 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 denotes a grey model which indicates that variables are employed in the model and that it is an -order differential equation. GM 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 to novel prediction types, such as GM [14], GMC [10, 15], D-GMC [14], DGDMC [16], and CAGM [17]. GM model is a grey multivariable model used for estimating the relationship between the system behavior and relative factors [14] and has been used for various applications [4, 11, 18–20]. However, there are some limitations existing in the GM which affect the prediction accuracy of GM [14, 15]. Then, the grey multivariable model with the convolution integral GMC, proposed by Tien [10], is developed from GM by adding the grey control parameter in the differential equation of GM to improve the forecasting accuracy of GM. The GMC model has successfully been applied to many real works [10, 15, 21, 22]. However, the prediction accuracy of the GMC 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 for an actual system [8, 16].

In this paper, we proposed a modified grey GMC model to improve the prediction accuracy of the conventional GMC 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 model in comparison with the conventional GMC and the discrete multivariate grey model D-GMC [5].

The paper is organized as follows: Section 2 describes the conventional GMC, Section 3 proposes a modified grey GMC model, Section 4 explains the statistical measure of the forecasting performance, and Section 5 presents the case study with the modified grey GMC. Finally, the conclusions are drawn in Section 6.

#### 2. GMC Model [8, 10]

Suppose that the original multivariate time series are a nonnegative series and available at an equispaced interval of time, where the main factor of the system behavior is , the relative factors are , and is the number of data. Then the first-order accumulative generation operation (1-AGO) of , is given by the following equation:where .

The prediction procedure of using the conventional GMC model is shown as follows.

The grey prediction model based on the 1-AGO data, , is given by the following differential equation:where and are model parameters to be estimated and is the number of entries to be predicted.

Taking the integral of both sides of (2) in the interval yieldswhere for , which are called the background values [9].

By using the trapezoidal rule, the values of for are approximated by the following equation:

Substituting all the data values in (3) gives the system of linear equations that can be written as a matrix equation in the formwhere

Applying the least square method to this system, we obtainSolving (2) with the initial condition , the predicted 1-AGO series is obtained as follows:where is the unit step function and and denotes the predicted value of .

Therefore, the predict series of is given by

#### 3. Modified GMC Model

The proposed model is modified from GMC by using two advanced improvements, as follows:

The equation of the predicted 1-AGO series is obtained accurately from the exactly integrated form of .

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 , satisfy the quasi-smooth and quasi-exponential condition. Then, the 1-AGO data , are fitted by the exponential function [9, 17], which can be written as

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 asDividing (12) by (11), we have

Let be the objective function. By using the least square method, can be made minimum using the parameter , which should satisfyBy solving this equation, we can obtainSubstituting (15) into (10) giveswhere .

Let be the objective function. The and should satisfySolving this equation yields

From (10), the background value can be written as follows:Substituting (20) into (3), we havewhere

According to the least squares estimation, parameters and of (22) are obtained aswhereBy solving (2) using the integrating factor method with the initial condition , the model prediction equation for the predicted 1-AGO series can be obtained as follows.

From (2), the integrating factor is . Multiplying (2) by obtainsTaking the integral of both sides of the above equation in the interval , we getSubstituting into (28) givesMultiplying both sides of (28) by , we getSubtracting (30) from (29), we have

Consequently, the predicted 1-AGO series of , is given bywhere , , , and for .

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 of (2) is zero, which is shown as follows.

Setting and taking the integral of both sides of (2) in the interval , we haveSubstituting into (35) givesSubtracting (36) from (35), we haveTherefore, (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:whereThe parameters and are determined using the least squares estimation, as follows.

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 equationwhere is Moore-Penrose pseudoinverse of matrix [25, 26].

Finally, the predicted series of 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, aswhere is the actual value at time , 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