This study proposes an improved metabolism grey model [IMGM
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
With respect to the limitations of the basic grey prediction models, many scholars have proposed some optimization GM
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. [
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
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 [
Let a nonnegative small sample sequence
Use the one-time accumulated generating operation (1-AGO) to obtain
Calculate the background value
Construct the original form of GM
Calculate the time response sequence
The time response sequence
The abovementioned steps provide the modeling process of the GM
With respect to the above process, the GM
Liu et al. [
As previously stated, the general grey prediction models, such as the GM
Construct the GM
Eliminate and metabolize the fitting data. In particular, the new prediction data
Use the MGM
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 fitting errors are calculated based on the following equations. The residual error The relative error
According to the above calculation process of the MGM
Based on the modeling processes of the GM
Equation ( Given that the ratio between This monotonic exponential trend also exists in the MGM
We provide a simple example below to further show the conclusions and limitations discussed above.
Let a small sample sequence with a singular datum be
The distribution and trend of this sequence are shown in Figure
Original data sequence of Example
Let the first five numbers be the original data sequence, that is,
Use the 1-AGO operator to obtain
Calculate the background values as follows:
Estimate the developing coefficient
Calculate the time response sequence
Transform the time response sequence
Thus, the modeling steps of the GM
Fitting and prediction results of the GM
Original data sequence | Modeling data sequence | Fitting and prediction values | Residual errors | Relative errors (%) |
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4 | 4 | 4 | 2.6539 | 0 |
6 | 6 | 3.3460 | 3.5454 | 44.2330 |
9 | 9 | 5.4545 | 3.8919 | 39.3935 |
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5 | 5 | 8.8919 |
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20.25 | 20.25 | 14.495 | 6.7450 | 28.4180 |
30.375 | — | 23.6299 | 7.0416 | 22.2059 |
45.5625 | — | 38.5208 | 5.5480 | 15.4548 |
68.34375 | — | 62.7957 | 2.6539 | 8.1178 |
We also provide the following fitting and prediction results using the MGM
Fitting and prediction results of the MGM
Original data sequence | Fitted and predictive value | Residual errors | Relative errors |
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4 | 4 | 2.6539 | 0 |
6 | 3.3460 | 3.5454 | 44.2330 |
9 | 5.4545 | 3.8919 | 39.3935 |
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5 | 8.8919 |
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20.25 | 14.4953 | 6.7450 | 28.4180 |
30.375 | 23.6299 | 7.0416 | 22.2059 |
45.5625 | 36.2875 | 9.2749 | 20.3565 |
68.34375 | 57.3535 | 10.9902 | 16.0808 |
In Figure
Comparison of the three sequences.
Based on Tables A clear monotonic exponential trend is observed in the results of the GM The fourth data point is a singular datum in the original data, and it cannot be shown in the fitting results of the GM The GM The fitting results after the singular datum and the prediction results are inappropriate for the above second conclusion. The later prediction results are worse than the initial results.
Therefore, the GM
In this subsection, the IMGM
Let a small sample sequence with a singular datum be
Construct and apply the GM
Estimate the singular datum, and then update it by substituting
Construct and apply the GM
Derive the metabolism process for the original sequence which can be referenced in the modeling process of the MGM
Calculate the fitting errors based on the above error equations.
Thus, the modeling process of the IMGM
Therefore, we think that the IMGM
Sulfur dioxide is an environmental pollutant that is a major component of acid rain. It originates from the combustion of sulfur-containing fuels, metal smelting, and petroleum refining, which produce sulfuric acid and silicate products. Given the rapid development of its economy, China has seen its highest concentrations of sulfur dioxide emissions in recent years. Three years ago, China officially promulgated several policies to measure and control sulfur dioxide pollution. As such, we believe that the prediction of sulfur dioxide emissions is important to effectively measure and control such emissions. Although many prediction techniques can be used, these approaches are relatively inaccurate because of changing environmental indicators and nonuniform standards. Thus, small simple prediction approaches are more suitable than other prediction models. In this section, three grey models are applied to predict sulfur dioxide emissions based on the data from 2007 to 2013. Table
Sulfur dioxide emissions of China from 2007 to 2013.
Year | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 |
---|---|---|---|---|---|---|---|
Emission data (unit: 104 tons) | 2468.1 | 2321.2 | 2214.4 | 2185.1 | 2217.9 | 2117.6 | 2044 |
Table
The first five numbers are considered the original data, that is, 2007-to-2011 data, and the data from 2012 to 2013 are considered the unknown data. This demarcation is the reason why the real fitting and prediction errors can be calculated and obtained and why the more suitable models can be selected for further predictions. Based on the above set, we can obtain the original data sequence; namely,
Use the 1-AGO operator, and then we have
Calculate the background values, and then we can obtain
Estimate the developing coefficient
Calculate the time response sequence
Transform
Based on the time response sequence
Fitting and prediction results based on the GM
Year | Actual data | Modeling data | Fitting and prediction values | Residual errors | Relative error (%) |
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2007 | 2468.1 | 2468.1 | 2468.1 | 0 | 0 |
2008 | 2321.2 | 2321.2 | 2286.3 | −34.82 | −1.500 |
2009 | 2214.4 | 2214.4 | 2251.5 | 37.10 | 1.6756 |
2010 | 2185.1 | 2185.1 | 2217.1 | 32.06 | 1.4674 |
2011 | 2217.9 | 2217.9 | 2183.3 | −34.55 | −1.5578 |
2012 | 2117.6 | — | 2150.0 | 32.44 | 1.5322 |
2013 | 2044.0 | — | 2117.2 | 73.25 | 3.5838 |
Based on the above results and modeling steps of the MGM
Fitting and prediction results based on the MGM
Year | Actual data | Modeling data | Fitting and prediction value | Residuals | Relative errors (%) |
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2007 | 2468.1 | 2468.1 | 2468.1 | 0 | 0 |
2008 | 2321.2 | 2321.2 | 2286.3 | −34.82 | −1.5001 |
2009 | 2214.4 | 2214.4 | 2251.5 | 37.10 | 1.6756 |
2010 | 2185.1 | 2185.1 | 2217.1 | 32.06 | 1.4674 |
2011 | 2217.9 | 2217.9 | 2183.3 | −34.55 | −1.5578 |
2012 | 2117.6 | — | 2150.0 | 32.44 | 1.5322 |
2013 | 2044.0 | — | 2152.2 | 108.21 | 5.2945 |
Table
Construct and apply the GM
Thus, we have
Eliminate the singular datum
Here, the new original data sequence exhibits a decreasing trend, which is different from the original data sequence. Thus, this sequence is more suitable to be used in the grey prediction models.
Construct and apply the GM
Perform the metabolism process by eliminating the fitting data
Fitting and prediction results based on the IMGM
Year | Actual data | Modeling data | Fitting and prediction values | Residual errors | Relative errors (%) |
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2007 | 2468.1 | 2468.1 | 2468.1 | 0 | 0 |
2008 | 2321.2 | 2321.2 | 2293.4 | −27.77 | −1.1966 |
2009 | 2214.4 | 2214.4 | 2247.8 | 33.42 | 1.5095 |
2010 | 2185.1 | 2185.1 | 2203.1 | 18.03 | 0.8254 |
2011 | 2217.9 | 2183.348 | 2159.3 | −24.01 | −1.0998 |
2012 | 2117.6 | — | 2116.4 | −1.19 | −0.0565 |
2013 | 2044.0 | — | 2074.3 | 30.32 | 1.4835 |
Compare the fitting and prediction errors, and then check the results which are shown and analyzed in Section
The original data and the fitting and prediction results previously provided through three grey prediction models are displayed in Figure
Original data and the fitting and prediction results.
Figure If a singular datum exists in the original data sequence, the GM If the singular datum does not suitably fit, other fitting data after the singular datum will be inaccurate, their corresponding errors could be amplified, and the prediction results are deflected. The IMGM
The residuals and relative errors of the three grey prediction models are compared, which are shown in Figures
Comparisons of the residuals based on three grey prediction models.
Comparisons of the relative errors based on three grey prediction models.
If the residuals and relative errors are near zero, then the prediction model has high forecast accuracy. Based on this principle and Figures The singular datum exerts an inappropriate influence on the GM The IMGM The IMGM
By distinguishing the fitting and prediction results, we divide the residuals and relative errors into four parts: fitting average residuals, prediction average residuals, fitting average relative errors, and prediction relative errors; their calculated results for this case study are shown in Table
Comparisons of four types of errors based on three grey prediction models.
Models | Fitting average residuals | Prediction average residuals | Fitting average relative errors | Prediction average relative errors |
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GM |
27.7088 | 52.8512 | 0.00017 | 0.0255 |
MGM |
27.7088 | 70.3336 | 0.00017 | 0.0341 |
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Table
Based on the results from Section
Prediction results of the sulfur dioxide emissions in China from 2014 to 2018.
Year | 2014 | 2015 | 2016 | 2017 | 2018 |
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Predictive value (unit: 104 tons) | 2013.3621 | 1969.4262 | 1926.4491 | 1884.4098 | 1843.2879 |
Table
Recently, the
Actual data on 2014 sulfur dioxide emission in China was
This study proposed an improved grey prediction model, the IMGM
A practical case study on the sulfur dioxide emissions in China from 2007 to 2013, as well as the development trend, was used to illustrate the improvements and advantages of the proposed model. The results from applying the three grey prediction models lead to the following conclusions.
(1) The singular data can seriously affect the prediction effect of the GM
(2) The IMGM
(3) The level of sulfur dioxide emissions in China will not decrease to the requisite level proposed in the Twelfth Five-Year Plan by the China State Council in 2015 if effective action is not taken immediately.
The IMGM
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Natural Science Foundation of China (nos. 71301141 and 71561026), Humanity and Social Science Youth Foundation of Ministry of Education of China (no. 13YJC630247), Science Foundation and Major Project of Educational Committee of Yunnan Province (no. 2014Z100), Applied Basic Research Programs of Science and Technology Commission of Yunnan Province (no. 2013FD029), Social Science Fund of Yunnan Province (no. YB2015087), and China Postdoctoral Science Foundation (no. 2015M570792).