Discrete Dynamics in Nature and Society

Volume 2016, Article ID 1045057, 11 pages

http://dx.doi.org/10.1155/2016/1045057

## An Improved Metabolism Grey Model for Predicting Small Samples with a Singular Datum and Its Application to Sulfur Dioxide Emissions in China

International Business School, Yunnan University of Finance and Economics, Kunming 650221, China

Received 15 December 2015; Revised 24 January 2016; Accepted 27 January 2016

Academic Editor: Filippo Cacace

Copyright © 2016 Wei Zhou and Demei Zhang. 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 study proposes an improved metabolism grey model [IMGM] to predict small samples with a singular datum, which is a common phenomenon in daily economic data. This new model combines the fitting advantage of the conventional GM in small samples and the additional advantages of the MGM in new real-time data, while overcoming the limitations of both the conventional GM and MGM when the predicted results are vulnerable at any singular datum. Thus, this model can be classified as an improved grey prediction model. Its improvements are illustrated through a case study of sulfur dioxide emissions in China from 2007 to 2013 with a singular datum in 2011. Some features of this model are presented based on the error analysis in the case study. Results suggest that if action is not taken immediately, sulfur dioxide emissions in 2016 will surpass the standard level required by the Twelfth Five-Year Plan proposed by the China State Council.

#### 1. Introduction

In response to technological advances and the progress of human society, many scholars have proposed and investigated various theories and methods that analyze uncertain information from different angles and perspectives. The grey system theory as proposed by Chinese scholar Julong Deng in 1982 is a novel uncertainty theory, which has received increasing attention and has recently become a popular subject of research. The grey system is an uncertainty system in which information is only partially known; it is also suitable to modeling small samples and poor information.

The grey prediction models, especially the GM, may be superior to other prediction techniques in the context of small samples and poor information. Thus, grey prediction models have received much attention since their introduction and successful application in the small sample prediction fields. They have been applied in many fields. Wang [1] predicted the stock price using a fuzzy grey prediction system. Sun et al. [2] proposed a small sample prediction technology for early warning of financial crises and recognition of crisis patterns of large-scale enterprises based on the GM. Vishnu and Syamala [3] used the basic GM to predict dynamic changes in water supply systems. Chang and Lu [4] used the GM predictor to forecast the flicker severity. Li et al. [5] used the adaptive GM to forecast short-term electricity consumption. Tang and Yin [6] forecasted the education expenditure and school enrollment by the GM. Yin and Tang [7] fitted and forecasted China’s labor formation based on the prediction GM. Li et al. [8] applied the GM and GRA to evaluate the financial burden of patients at hospitals in China by PPP model. Wu et al. [9] applied an improved GM to analyze the trends of the plum virus incidence in China. Tabaszewskin and Cempel [10] developed a grey prediction method for the fan vibration in a cogeneration plant (CHP) by combining the GM and the optimal estimated approaches. Pao et al. [11] forecasted CO_{2} emissions, energy consumption, and economic growth in China using a prediction entirely improved GM. Recently, Huang et al. [12] constructed a hybrid nonlinear grey prediction model to analyze the carbon reduction at dual levels in China. Wang et al. [13] provided an approach to forecast the coal mine water inflow based on the GM prediction theory. Ma and Fu [14] studied the accidents due to human factor in coal mine based on the GM. These applications of the grey prediction models show the feasibility and superiority of the grey system theory in dealing with small samples, which is also a reason why we choose this method to predict sulfur dioxide emissions.

With respect to the limitations of the basic grey prediction models, many scholars have proposed some optimization GM. Xie and Liu [15, 16] proposed and studied a discrete GM and verified its unbiasedness to fit an exponential sequence. Yao and Liu [17] and Wang et al. [18], respectively, developed an improved and optimal discrete GM based on Xie’s discrete GM. Shih et al. [19] improved the GM by changing the background and initial values. Li et al. [20, 21] provided an overall definition of the grey prediction model and proposed some basic GM. Truong and Ahn [22] proposed the SAGM to overcome the instability and improve the prediction accuracy of the GM. Cui et al. [23] provided the NGM based on the GM, which belongs to an entirely optimized GM. Dai and Huang [24] developed the GM by designing calculation program and selecting the initial value. Zhou and He [25] proposed the GGM which is generalized GM that includes the DGM, SAGM, and NGM. Lin and Lian [26] designed a grey prediction self-organizing model, which is an extended GM. Xiong et al. [27] proposed the nonequidistant GM based on the optimization initial condition. The main objectives of the above methods are to improve the accuracy of grey prediction models and, thereby, perfect them. However, these methods have a common data background referred to as small simples, which demonstrate an increasing or decreasing trend. This simple sequence does not include a small simple sequence with several singular data or a nonmonotone sequence, which generally constitute a practical phenomenon in the periodicity of economic and environmental uncertainty. For this reason, this study investigated the applicability of a new grey prediction model in a small simple sequence with a singular datum, which is a key point in the prediction of sulfur dioxide emissions in China.

Due to the recently increasing interest in environmental protection, many studies have focused on predicting energy consumption and pollution emissions. However, the prediction results of such studies remain relatively inaccurate due to changing environmental indicators and nonuniform standards. To solve this problem, some scholars have successfully applied the grey system theory and its prediction models to predict the environmental indicators with small simples. For example, Wang et al. [28] applied the GM to analyze and predict the atmospheric ozone levels in Asia. Lee and Tong [29] proposed the GPGM to forecast the energy consumption in China and achieved significant fitting results. Lin et al. [30] used the GM to predict the carbon dioxide emissions in China. Pan et al. [31] and Dai et al. [32] applied the GM to predict the degrees of air pollution in Tianjin and Shenzhen, respectively. These studies prove that the grey prediction model is suitable and useful in forecasting the energy consumption and pollution emission in China. However, the grey prediction model has yet to be investigated with small simples of singular data, which have become increasingly common in Chinese economic data to reflect the decline in the economic growth.

Based on the literature review above and practical trends in Chinese pollution indicators, this study proposes a new grey model to predict small simples with singular data and applies it to predict recent annual sulfur dioxide emissions in China. This paper is organized as follows: Section 2 describes the basic GM and MGM. Section 3 analyzes the limitations of the GM and MGM and then proposes the improved MGM called IMGM, based on which it provides the corresponding modeling steps. In Section 4, a practical example is provided to demonstrate the effectiveness and the practicability of the new grey prediction model. The paper ends in Section 5 with conclusions.

#### 2. The GM(1, 1) and MGM(1, 1)

##### 2.1. The Modeling Mechanism of the GM(1, 1)

Grey system theory, the name of which was derived from cybernetics and a clear degree of information, was proposed by Chinese scholar Julong Deng in 1982 [33]. In economics management processes and scientific research, incomplete information can arise in a general situation belonging to a grey system, as defined in grey system theory. As such, a grey prediction model has a comprehensive scope of applications in our society. However, a prerequisite for the application of grey prediction models, such as the GM, is that the original data must be monotonically increasing or decreasing. If this prerequisite is not met, the prediction results are likely to seriously deviate from the reality. This prerequisite and its influence are investigated through the following analysis of the modeling mechanism of the GM.

Let a nonnegative small sample sequence be the original data sequence, and . As a small sample sequence, is generally small at . Deng [33] proposed the GM and its modeling steps to fit and predict based on as follows.

*Step 1. *Use the one-time accumulated generating operation (1-AGO) to obtain , in which and

*Step 2. *Calculate the background value , in which , , and

*Step 3. *Construct the original form of GM, and then estimate the developing coefficient and the grey action quantity based on the ordinary least squares method presented as follows: where , , and

*Step 4. *Calculate the time response sequence through the following formula:

*Step 5. *The time response sequence is transformed into the restored time response sequence through the following formula:

The abovementioned steps provide the modeling process of the GM, which also is its modeling mechanism. Here, is the cumulative prediction value, is the restored prediction value, is the cumulative fitting value, and is the restored fitting value, where and .

With respect to the above process, the GM was found to have two advantages and two shortcomings. The* advantages* are as follows: (1) The model does not require many samples and is simpler and more suitable than general econometric models. (2) It can be used for recent or short-term prediction activities. The* shortcomings* are as follows: (1) The model is a monotonic exponential prediction approach, which is its modeling prerequisite. (2) It models and predicts all data when , but it ignores new information and it cannot accurately reflect the characteristics of the current situation.

Liu et al. [34] proposed the metabolism grey model, namely, MGM, by updating the modeling data and introducing new information to overcoming the second shortcoming, which is a significant improvement as explained in Section 2.2. In Section 3, we propose and analyze another new GM based on the above MGM to overcome the first shortcoming.

##### 2.2. Construction and Calculation of the MGM(1, 1)

As previously stated, the general grey prediction models, such as the GM, provide forecasting based on the original data without considering the impact of new information or dynamic predictions when information is subsequently supplied and updated. As such, Liu et al. [34] proposed the MGM. As an improved grey prediction model, the MGM can add new information and remove old information in what is also a metabolic process. The corresponding modeling steps of the MGM are presented as follows.

*Step 1. *Construct the GM for a small sample sequence also called original data sequence, and then calculate the time response sequence and the restored time response sequence based on (5) and (6), where the original data sequence is and .

*Step 2. *Eliminate and metabolize the fitting data. In particular, the new prediction data is used instead of the first data . Thus, we can obtain a metabolic small sample sequence called metabolic data sequence, which is similar to the original data sequence and includes a new datum.

*Step 3. *Use the MGM to calculate and predict, following the same modeling process of the GM. Then, a new prediction datum can be obtained.

If more prediction data is needed, then the above process is repeated to obtain the data. This calculation process demonstrates the modeling steps of the MGM.

Unlike the GM, the MGM uses the prediction data as the original data to predict other data. Therefore, there could be a propagation phenomenon of the prediction error from the MGM, which could further affect the metabolic modeling process. Thus, there are two error inspection methods (Step 4) to improve the MGM.

*Step 4. *The fitting errors are calculated based on the following equations.(1)The residual error and its corresponding form are as follows:(2)The relative error and its corresponding form are as follows:

According to the above calculation process of the MGM, this new grey prediction model is a small sample and updated information prediction model. Therefore, the MGM is more suitable than the GM in the prediction of changing small simple sequences.

#### 3. The IMGM(1, 1)

##### 3.1. Shortage Analysis of the GM(1, 1) and MGM(1, 1)

Based on the modeling processes of the GM and MGM and the analysis of their disadvantages in the previous section, the original data sequence, which is a monotonic exponential sequence for these two grey prediction models, is taken as a prerequisite. Therefore, if the monotonic exponential character is changed or if singular data exist in the original data, then the prediction results can be affected, and a larger error could result, which can be proved by (9) and the following analysis:

Equation (9) exhibits the following conclusions:(i)Given that the ratio between and in the GM is a constant and is related to the development coefficient , a monotonic exponential trend exists in this grey prediction model.(ii)This monotonic exponential trend also exists in the MGM, which leads to its prerequisite of the monotonic exponential sequence. In our opinion, this prerequisite is unsuitable.

We provide a simple example below to further show the conclusions and limitations discussed above.

*Example 1. *Let a small sample sequence with a singular datum be

The distribution and trend of this sequence are shown in Figure 1. This sequence is similar to a monotonic exponential sequence, but it has a singular datum “5.”