A Multivariate Grey Prediction Model Using Neural Networks with Application to Carbon Dioxide Emissions Forecasting

Chiu, Yu-Jing; Hu, Yi-Chung; Jiang, Peng; Xie, Jingci; Ken, Yen-Wei

doi:https://doi.org/10.1155/2020/8829948

Mathematical Problems in Engineering

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Abstract Introduction Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Computational Intelligence in Complex Decision Making

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Research Article | Open Access

Volume 2020 | Article ID 8829948 | https://doi.org/10.1155/2020/8829948

A Multivariate Grey Prediction Model Using Neural Networks with Application to Carbon Dioxide Emissions Forecasting

Yu-Jing Chiu,¹Yi-Chung Hu,¹Peng Jiang,²Jingci Xie,³and Yen-Wei Ken¹

Academic Editor: Tomas Balezentis

Received08 Sept 2020

Revised10 Oct 2020

Accepted12 Oct 2020

Published29 Oct 2020

Abstract

The forecast of carbon dioxide (CO₂) emissions has played a significant role in drawing up energy development policies for individual countries. Since data about CO₂ emissions are often limited and do not conform to the usual statistical assumptions, this study attempts to develop a novel multivariate grey prediction model (MGPM) for CO₂ emissions. Compared with other MGPMs, the proposed model has several distinctive features. First, both feature selection and residual modification are considered to improve prediction accuracy. For the former, grey relational analysis is used to filter out the irrelevant features that have weaker relevance with CO₂ emissions. For the latter, predicted values obtained from the proposed MGPM are further adjusted by establishing a neural-network-based residual model. Prediction accuracies of the proposed MGPM were verified using real CO₂ emission cases. Experimental results demonstrated that the proposed MGPM performed well compared with other MGPMs considered.

1. Introduction

Carbon dioxide (CO₂) is mainly produced from fossil fuel combustion [1], and reducing the impact that energy consumption and economic growth have on CO₂ emissions has become a global challenge [2]. According to the International Energy Agency (IEA) [3], total emissions of greenhouse gas in 2018 were a record 33.1 billion tons, along with a global economic growth rate that increased by 3.2%. Despite a CO₂ emission plateau from 2014 to 2016, the IEA reported that China and USA were the highest energy-using and carbon-emitting countries, and CO₂ emissions went up in each country by 2.5% and 3.1%, respectively, mainly arising from an increased use of fossil fuel to meet the energy demand. In fact, CO₂ emissions can significantly give rise to climate change and have a negative impact on economic growth. Therefore, to keep a green economic growth, the national authorities make an effort to devise energy development policies that reduce the impact of CO₂ emissions. An accurate forecast of CO₂ emissions becomes a remarkable issue when public sectors set up policies.

From the viewpoint of the grey system theory [4–6], the prediction of CO₂ emissions can be viewed as a grey system problem because although the relevant features, such as energy consumption, population, and gross domestic product (GDP) [7–9], influence CO₂ emissions, the precise relationship between these features and emissions is not clear. Furthermore, it is possible that emissions data do not conform to any statistical assumptions [7]. Compared with the prediction models implemented by artificial intelligence techniques [10–15], statistical models including logistic models [16], multivariate regression [17, 18], and time series analysis [19, 20], MGPMs have the advantage of characterizing an unknown system using limited samples [6], without requiring conformance with statistical assumptions. Despite huge amount of data we can collect, only a few sample data points are required to achieve reliable and acceptable prediction accuracy [21, 22]. Therefore, it is interesting to apply the multivariate grey prediction models (MGPMs) to CO₂ emissions.

In contrast to the frequently used GM (1, 1), the first-order grey differential equation with one variable, which does not consider the influence of relevant factors on the system [23], the GM (1, N), which consists of a system’s characteristic sequence and several relevant factor sequences, is fundamental to MGPMs and has been widely applied to time series forecasting, such as the forecasting of traffic flow [24], number of motor vehicles [25], production in high-tech industries [26, 27], energy consumption [28, 29], integrated circuit output [30, 31], OFDI analysis [32], and pattern classification [33, 34]. Many GM (1, N) variants have been introduced to improve the prediction accuracy of the traditional GM (1, N), such as a rolling multivariable model [8], background value optimization [27], the optimization of GM (1, N) (OGM (1, N)) using grey differential equations with linear correction and grey quantity terms [25, 35], nonlinear grey models (NGM (1, N)) using grey differential equations with power exponents, and transformed NGM (1, N) (TNGM (1, N)) [28]. These MGPMs have shown their superior prediction performance when used with time series problems.

Grey prediction has demonstrated its effectiveness on CO₂ emissions forecasting, such as univariate grey prediction models by Pao et al. [36], Wu et al. [2], and Xu et al. [37], prediction models based on trends of driving coefficients (TDVGM (1, N)) by Ding et al. [7], TNGM (1, N) by Wang and Ye [28], multikernel nonlinear multivariable grey model by Duan et al. [23], a forecasting method for the traffic-related emissions by Xie et al. [38], a nonlinear grey power model (DGPM (1, N)) by Ding et al. [39], and a nonequigap grey Verhulst model by Wang and Li [40]. This study contributed to developing a distinctive MGPM with feature selection and residual modification performed to improve prediction accuracy. For the former, since system performance can be improved by feature selection [41], the proposed MGPM performs grey relational analysis (GRA) [4–6, 42] to estimate relevance between independent variables and CO₂ emissions. For the latter, it has been suggested that the prediction accuracy of the GM (1, 1) can be improved by residual modification [4]. However, it is very interesting to extend residual modification to the GM (1, N). A neural-network-based residual model is thus created for the proposed MGPM to adjust predicted values from the GM (1, N). To sum up, the proposed MGPM with feature selection and residual modification can be treated as a residual modification model. Genetic algorithms (GAs) are employed to determine required parameters of the GRA and GM (1, N) to construct the proposed MGPM with high prediction accuracy. Experimental results have indicated that the proposed MGPM performs well compared with the other MGPMs considered. It is noted that, to estimate the correlation between the system behavior variable and the influential factors, the proposed prediction model used GA to determine a threshold value that is not easily prespecified. In contrast, Xie et al. [38] applied GRA to identify relevant factors from passenger cars per 1000 inhabitants, stock of vehicles, volume of freight transport relative to GDP (VFTRG), and volume of passenger transport relative to GDP, but only VFTRG (with time lags four) with maximum correlation was considered. Meanwhile Ding et al. [39] used a prespecified threshold value to evaluate the relationship among factors.

The remainder of the paper is organized as follows. Section 2 introduces the traditional GM (1, N), grey residual modification model, and Section 3 introduces the proposed MGPM with residual modification on the basis of a neural network. Section 4 examines prediction performances of the proposed MGPM using two real cases of CO₂ emissions. Section 5 discusses the outcomes and presents conclusions.

2. The GM (1, N) Model

The GM (1, N) is a grey prediction model with N variables, including a dependent variable (system characteristic), x₁, and N − 1 explanatory variables (relevant factors), x₂, x₃,…, x_N [5]. Then, an original sequence or a time series is associated with x_i, where i = 1, 2, …, N, and m is the number of samples. is usually a sequence taken at successive equally spaced points in time. Step 1. Present an original and nonnegative sequence . Step 2. Perform the accumulated generating operation (AGO). A new sequence can be generated from by the AGO as follows: The AGO can help identify potential hidden regularities in data sequences [4, 43]. can be approximated by an exponential function, which is a first-order whitening equation, where a and b_j are the development and the driving coefficients, respectively. The time response expression of (k) (k = 2, 3, …, m) can be obtained by solving the differential equation with the initial condition (1) = (1): Step 3. Determine the development and the driving coefficients. A grey differential equation of the GM (1, N) is as follows: where the background value with the generating coefficient α (0 ≤ α ≤ 1) being usually set to 0.5 is formulated as Thus, a linear regression model consisting of grey differential equations is used to estimate a, b₂,…, and b_N through the ordinary least squares (OLS) method: where Step 4. Perform the inverse accumulated generating operation (IAGO). When varies slightly, can be generated by means of the IAGO: where = .

To improve the prediction performance of the traditional GM (1, N), several improved versions of MGPMs have been proposed by deriving new whitening and grey differential equations:(1)The transformed model of nonlinear GM (1, N) (TNGM (1, N)) [28]: the TNGM (1, N) has been applied to forecast Chinese carbon emissions. The whitening equation of the TNGM (1, N) is defined as The solution of the whitening equation is thus given by a, b₂,…, and b_N can be further derived by OLS as where Any optimization technique such as a GA can be used to derive the optimal γ₂, γ₃,…, and γ_N.(2)The optimization of the GM (1, N) (OGM (1, N)) [35]: the grey difference equation of the OGM (1, N) is defined as The time response expression of is thus given by where a, b₂,…, b_N, h₁, and h₂ can be obtained by OLS as where Zeng et al. [25] applied a variant of the OGM (1, N) to forecast the number of motor vehicles in Beijing.(3)The MGPM based on trends of driving coefficients (TDVGM (1, N)) [7]: the development of the TDVGM (1, N) addressed an issue of forecasting Chinese CO₂ emissions. This prediction model initially divided data from the first year by data from each year. Then, it predicted trends of driving coefficients by defining the grey differential equation for b_j (j = 2, 3, …, N) as The time response expression of is formulated as a_j and b_j were obtained by OLS as where Then, a time response expression of can be derived as where

3. The Proposed Multivariate Grey Prediction Model

For the proposed MGPM, GRA was first used to keep explanatory variables that are more relevant to CO₂ emissions. Then, the proposed MGPM can be constructed by GAs. Subsequently, a residual GM (1, 1) model can be embedded into the proposed MGPM by establishing a functional-link net to adjust .

3.1. Feature Selection by Grey Relational Analysis

In contrast to statistical correlation analysis that measures the relationship between any two random variables, GRA can effectively measure the relationships between one reference sequence and the other comparative sequences by viewing the reference sequence as the desired goal [44]. The grey relational coefficient, ξ_jk, for the time period k (1 ≤ j ≤ N − 1, 1 ≤ k ≤ m) is addressed by the discriminative coefficient ρ (0 ≤ ρ ≤ 1) to indicate the relationship between and :where ρ is usually specified as 0.5 and

It is seen that ξ_jk lies in [0, 1] and approaches 1 if approaches .

To measure the degree of proximity between x_j and x₁, the grey relational grade (GRG) can be calculated as follows:where ranges from 0 to 1. The greater the value of is, the more relevant x_j is to x₁. In other words, to construct the proposed model, x_j can be retained for constructing the proposed MGPM when surpasses a threshold value λ; otherwise, x_j can no longer be considered for the proposed MGPM. That means, for equation (3), both b_j and associated with x_j can be retained when is above λ; otherwise, they can be removed directly. However, λ may not be prespecified easily beforehand.

3.2. Construction of Multivariate Grey Prediction Models Using Genetic Algorithms

This study aims to find the optimal solution to construct the proposed MGPM with high prediction accuracy. This problem can be thus formulated as the following single objective optimization problem by minimizing the mean absolute percentage error (MAPE):

Instead of OLS, a real-valued GA using MAPE as its fitness function can be developed to automatically determine the optimal values of development coefficient (a), the driving coefficients (b_2,b_3, …, b_N), and the cut value (λ).

For the GA, a set of strings making up a population is generated initially. All parameters for each string are randomly generated as real numbers. Using the Genetic Algorithm and the Direct Search Toolbox in MATLAB, a real-valued GA is easily developed to automatically determine all parameters. The best chromosome with the maximum fitness value of all successive generations is the desired solution for examining the generalizability of the proposed MGPM.

3.3. Residual Modification Using a Functional-Link Net

produced by the proposed MGPM can be further adjusted by residual modification to improve prediction accuracy. Let denote the sequence of absolute residual values, where

Because no dependent variables are considered for ε₁, a residual GM (1, 1) model can be established for , where the predicted value of iswhere a_ɛ and b_ɛ are the developing coefficient and the control variable, respectively. = . The problem of constructing a residual GM (1, 1) model can be formulated as the following single objective optimization problem by minimizing the MAPE:

MAPE can be used as the fitness function of a real-valued GA that is used to determine optimal a_ɛ and b_ɛ instead of OLS.

Then, following a mechanism of residual modification recommended by Wang and Hu [41], the final predicted value of the proposed MGPM is produced by means of :where y_k ranges from −1 to 1 and can be computed by presenting (t_k, sin (πt_k), cos (πt_k), sin (2πt_k), and cos (2πt_k)) to a single-layer perceptron, namely, the functional-link net, with effective function approximation capability [45–48]:where tanh represents a hyperbolic tangent function, and are connection weights. This means that the amount of adjusting could be as much as 3 if y_k = 1. In contrast, the adjustable amount can be −3 if y_k = −1.

4. Empirical Results

Empirical studies were conducted using real datasets to compare the CO₂ emission forecasting ability of the proposed MGPM with the other MGPMs considered. As mentioned above, urban population (UP), GDP, and energy consumption have a dominant influence on CO₂ emissions. In addition to the traditional GM (1, N), the Autoregressive Integrated Moving Average model (ARIMA) and the aforementioned improved MGPMs with comprehensible distinctive features were considered.

4.1. Case I

Statistics from the IEA [3] revealed that, in 2015, the total amount of CO₂ emissions in China was 9,040 million tons, reaching the highest level worldwide. To devise energy plans that would effectively reduce CO₂ emissions while promoting green economic growth, the ability to predict CO₂ emissions has played a very significant role in China. Therefore, we were intrigued to examine the prediction performance of MGPMs that consider CO₂ emissions in China. The data on urban population (million persons) and GDP (million USD dollars) were collected from the World Bank (http://data.worldbank.org.cn), and energy consumption (million tons of oil equivalent) and CO₂ emissions (million tons) were collected from the IEA (http://www.iea.org).

As shown in Table 1, the historical annual data were collected from 2005 to 2015, data from 2005 to 2012 were used for the model-fitting, and data from 2013 to 2015 were used for ex-post testing. Results shown in Table 2 are summarized as follows:(1)The MAPE of the traditional GM (1, N), the TNGM (1, N), the OGM (1, N), the TDVGM (1, N), the ARIMA, and the proposed MGPM for model-fitting were 3.34%, 0.81%, 0.02%, 3.14%, 1.98%, and 1.71%, respectively. The OGM (1, N) thus demonstrated its superiority in model-fitting.(2)For ex-post testing, the MAPE of the traditional GM (1, N), the TNGM (1, N), the OGM (1, N), the TDVGM (1, N), the ARIMA, and the proposed MGPM were 2.06%, 2.44%, 2.62%, 4.54%, 2.43%, and 1.87%, respectively.(3)Although the proposed MGPM is inferior to the TNGM (1, N) and the OGM (1, N) for model-fitting, it is superior to other MGPMs considered for ex-post testing.

It should be noted that, when evaluating a prediction model, more emphasis should be placed on generalization rather than on model-fitting [49]. Figure 1 also demonstrates the superiority of the generalization ability of the proposed MGPM over the other prediction models considered.

4.2. Case II

From statistics reported by the IEA [47], the total and average amounts of CO₂ emissions in Taiwan in 2015 were the 21st and the 19th highest in the world, respectively. This means that Taiwan still has room to reduce CO₂ emissions. The second real case involved the historical annual CO₂ emission data collected in Taiwan from 2005 to 2015. The data on urban population (million persons) and GDP (million USD dollars) were collected from the United Nations Conference on Trade and Development (UNCTAD) (http://unctad.org/en/Pages/statistics.aspx), and energy consumption (million tons of oil equivalent) and CO₂ emissions (million tons) were collected from the IEA.

As shown in Table 3, data collected from 2005 to 2012 were used for the model-fitting, and data from 2013 to 2015 were used for ex-post testing. The results obtained from the different prediction models are shown in Table 4. The results are summarized as follows:(1)The MAPE of the traditional GM (1, N), the TNGM (1, N), the OGM (1, N), the TDVGM (1, N), the ARIMA, and the proposed MGPM for model-fitting were 3.35%, 0.23%, 0.21%, 2.59%, 2.10%, and 1.86%, respectively. The TNGM (1, N) and the OGM (1, N) demonstrated their superiority in model-fitting.(2)The MAPE of the traditional GM (1, N), the TNGM (1, N), the OGM (1, N), the TDVGM (1, N), the ARIMA, and the proposed MGPM for ex-post testing were 1.28%, 4.75%, 8.43%, 4.54%, 0.86%, and 1.09%, respectively. The proposed MGPM is slightly inferior to the TDVGM (1, N).(3)Although the proposed MGPM is inferior to the TNGM (1, N) and the OGM (1, N) for model-fitting, it is superior to those two MGPMs for ex-post testing.

In Figure 2, we can see that the proposed MGPM performs well compared with other prediction models considered.

5. Discussion

What this study focuses on is the forecast rather than the projection. Compared with the projection, the forecast places value on estimating the amount of CO₂ emissions based on an established model such as a MGPM for which MAPE is a useful metric for measuring the prediction performance [50]. The projection can answer “what-if ” kind of questions to extrapolate the development trend. In other words, it is concerned about what would happen to CO₂ emission based on some future scenarios, for instance, how CO₂ is produced and how it is influenced by what kind of factors.

As mentioned above, the forecast of CO₂ emissions can be regarded as a grey system problem. Furthermore, CO₂ emission data might well not conform to statistical assumptions. Therefore, it is reasonable to apply MGPMs to forecast the amount of CO₂ emissions. Compared with the other MGPMs, feature selection and residual modification are taken into account in the proposed MGPM to improve prediction accuracy. In particular, GRA and a functional-link net are employed to implement feature selection and residual modification, respectively. The empirical results reveal that feature selection and residual modification can boost the prediction performance of the proposed MGPM. It is noted that although the OGM (1, N) variant [25] also applied GRA to filter out irrelevant features, the cut value (λ) was arbitrarily assigned by a prespecified value. However, λ can be optimized by the GA to improve the prediction accuracy of the proposed MGPM.

With real-world datasets, from Tables 2 and 4, we can see that the generalization ability of the proposed MGPM for CO₂ emissions was quite encouraging. The outcomes verified that the results obtained by the proposed MGPM are comparable to other prediction models considered. It is interesting to note that the TNGM (1, N) and the OGM (1, N) are superior to the traditional GM (1, N) and the proposed MGPM for model-fitting but inferior for ex-post testing. In other words, both the TNGM (1, N) and the OGM (1, N) appear to be overfitting. Experimental results show that the fitting and generalization abilities of the proposed MGPM are superior to the traditional GM (1, N). Thus, the prediction ability of the traditional GM (1, N) could be effectively improved by feature selection and residual modification indeed.

6. Conclusions

Undoubtedly, the reduction of greenhouse gas emissions is critical to environmental protection. For many countries, CO₂ is mainly produced from fuel combustion, which forms the majority of greenhouse gases. How to reduce the impact that energy consumption and economic growth have on CO₂ emissions has gained increasing global attention. Revelations from the IEA [3] showed that, along with global economic growth, global CO₂ emissions plateaued from 2014 to 2016 due to the growth of renewable electricity, the replacement of coal with natural gas, and changes to the economic structure. There appeared to be a decoupling of economic growth and environmental degradation. However, this does not mean that CO₂ emissions have reached a summit. To continuously inhibit carbon emissions and remain competitive, it is necessary for the authorities to make use of prediction models on carbon emissions, set up comprehensive policies on the development of new energy technologies, and increase demand for renewable energies (e.g., solar, wind, and natural gas) and environmental protection.

In addition to CO₂ emissions, there are several multivariate prediction problems, such as energy demand forecasting, which need to be resolved. In fact, energy demand prediction has become increasingly important when devising development plans for a country, particularly for developing countries [51]. Meanwhile, energy demand forecasting can be regarded as a grey system problem [52] because a few factors, such as income and population, have an influence on energy demand, but how exactly these factors affect energy demand is not clear. On the basis of the conspicuous forecasting performance of the proposed MGPM for CO₂ emissions, it would be interesting to explore its applicability to energy demand forecasting.

Data Availability

Statistics used in this paper are from the IEA (International Energy Agency).

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

This research was supported by the Ministry of Science and Technology, Taiwan, under Grant MOST 108-2410-H-033-038-MY2, the National Natural Science Foundation of China (no. 71772106), Humanities and Social Sciences Foundation of the Ministry of Education of China (no. 17YJCZH198), Shandong Provincial Natural Science Foundation, China (No. ZR2017MG012).

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Copyright © 2020 Yu-Jing Chiu et al. 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.

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