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

The forecast of carbon dioxide (CO 2 ) emissions has played a signiﬁcant role in drawing up energy development policies for individual countries. Since data about CO 2 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 CO 2 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 CO 2 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 CO 2 emission cases. Experimental results demonstrated that the proposed MGPM performed well compared with other MGPMs considered.


Introduction
Carbon dioxide (CO 2 ) is mainly produced from fossil fuel combustion [1], and reducing the impact that energy consumption and economic growth have on CO 2 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 CO 2 emission plateau from 2014 to 2016, the IEA reported that China and USA were the highest energy-using and carbon-emitting countries, and CO 2 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, CO 2 emissions can significantly give rise to climate change and have a negative impact on economic growth. erefore, to keep a green economic growth, the national authorities make an effort to devise energy development policies that reduce the impact of CO 2 emissions.
An accurate forecast of CO 2 emissions becomes a remarkable issue when public sectors set up policies.
From the viewpoint of the grey system theory [4][5][6], the prediction of CO 2 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][8][9], influence CO 2 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][11][12][13][14][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]. erefore, it is interesting to apply the multivariate grey prediction models (MGPMs) to CO 2 emissions. In contrast to the frequently used GM (1,1), the firstorder 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].
ese MGPMs have shown their superior prediction performance when used with time series problems.
is 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][5][6]42] to estimate relevance between independent variables and CO 2 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. e 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 CO 2 emissions. Section 5 discusses the outcomes and presents conclusions. N) is a grey prediction model with N variables, including a dependent variable (system characteristic), x 1 , and N − 1 explanatory variables (relevant factors), x 2 , x 3 ,. . ., x N [5]. en, an original sequence or a time series

The GM (1, N) Model
i is usually a sequence taken at successive equally spaced points in time.
Step 1. Present an original and nonnegative sequence i by the AGO as follows: e AGO can help identify potential hidden regularities in data sequences [4,43]. (x (1) i (1), x (1) i (2), ..., x (1) i (m)) 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. e time response expression of x (1) k (k) (k � 2, 3, . . ., m) can be obtained by solving the differential equation with the initial condition 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 z (1) 1 (k) with the generating coefficient α (0 ≤ α ≤ 1) being usually set to 0.5 is formulated as us, a linear regression model consisting of grey differential equations is used to estimate a, b 2 ,. . ., and b N through the ordinary least squares (OLS) method: Step 4. Perform the inverse accumulated generating operation (IAGO). When x (1) i varies slightly, x (0) k can be generated by means of the IAGO: e solution of the whitening equation x (0) 1 (k) is thus given by a, b 2 ,. . ., 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 c 2 , c 3 ,. . ., and c N . is defined as e time response expression of x (1) 1 (k) is thus given by where a, b 2 ,. . ., b N , h 1 , and h 2 can be obtained by OLS as Mathematical Problems in Engineering where Zeng et al. [25] applied a variant of the OGM (1, N) to forecast the number of motor vehicles in Beijing.  N) addressed an issue of forecasting Chinese CO 2 emissions. is prediction model initially divided data from the first year by data from each year. en, it predicted trends of driving coefficients by defining the grey differential equation a j and b j were obtained by OLS as where en, a time response expression of x (1) 1 (k) can be derived as where

The Proposed Multivariate Grey Prediction Model
For the proposed MGPM, GRA was first used to keep explanatory variables that are more relevant to CO 2 emissions. en, 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 x (0) 1 (k).

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]. e 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 x (0) j (k) and where ρ is usually specified as 0.5 and It is seen that ξ jk lies in [0, 1] and approaches 1 if Δ jk approaches Δ min .
To measure the degree of proximity between x j and x 1 , the grey relational grade (GRG) c j can be calculated as follows: where c j ranges from 0 to 1. e greater the value of c i is, the more relevant x j is to x 1 . In other words, to construct the proposed model, x j can be retained for constructing the proposed MGPM when c j surpasses a threshold value λ; otherwise, x j can no longer be considered for the proposed MGPM. at means, for equation (3), both b j and x (1) j (k + 1) associated with x j can be retained when c j is above λ; otherwise, they can be removed directly. However, λ may not be prespecified easily beforehand.

Construction of Multivariate Grey Prediction Models
Using Genetic Algorithms.
is study aims to find the optimal solution to construct the proposed MGPM with high prediction accuracy.
is 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 realvalued GA is easily developed to automatically determine all parameters. e best chromosome with the maximum fitness value of all successive generations is the desired solution for examining the generalizability of the proposed MGPM.

Residual Modification Using a Functional-Link Net.
x (0) 1 (k) produced by the proposed MGPM can be further adjusted by residual modification to improve prediction accuracy. Let ε (0) 1 � (ε (0) 1 (2), ε (0) 1 (3), ..., ε (0) 1 (m)) denote the sequence of absolute residual values, where Because no dependent variables are considered for ε 1 , a residual GM (1, 1) model can be established for ε (0) 1 , where the predicted value of ε (0) 1 (k) is where a ε and b ε are the developing coefficient and the control variable, respectively. ε (1) 1 (2) � ε (0) 1 (2). e 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 ε and b ε instead of OLS. en, following a mechanism of residual modification recommended by Wang and Hu [41], the final predicted value x (0) k 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][46][47][48]: where tanh represents a hyperbolic tangent function, and w 1 , . . . , w 5 are connection weights. is means that the amount of adjusting x (0) k could be as much as 3ε (0) k if y k � 1. In contrast, the adjustable amount can be −3ε (0) k if y k � −1.

Empirical Results
Empirical studies were conducted using real datasets to compare the CO 2 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 CO 2 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.

Case I.
Statistics from the IEA [3] revealed that, in 2015, the total amount of CO 2 emissions in China was 9,040 million tons, reaching the highest level worldwide. To devise energy plans that would effectively reduce CO 2 emissions while promoting green economic growth, the ability to predict CO 2 emissions has played a very significant role in China. erefore, we were intrigued to examine the prediction performance of MGPMs that consider CO 2 emissions in China. e 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 CO 2 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  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.

Case II.
From statistics reported by the IEA [47], the total and average amounts of CO 2 emissions in Taiwan in 2015 were the 21st and the 19th highest in the world, respectively. is means that Taiwan still has room to reduce CO 2 emissions. e second real case involved the historical annual CO 2 emission data collected in Taiwan from 2005 to 2015. e 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 CO 2 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. e results obtained from the different prediction models are shown in Table 4. e results are summarized as follows:  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.

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 CO 2 emissions based on an established model such as a MGPM for which MAPE is a useful metric for measuring the prediction performance [50]. e projection can answer "what-if " kind of questions to extrapolate the development trend. In other words, it is concerned about what would happen to CO 2 emission based on some future scenarios, for instance, how CO 2 is produced and how it is influenced by what kind of factors.
As mentioned above, the forecast of CO 2 emissions can be regarded as a grey system problem. Furthermore, CO 2 emission data might well not conform to statistical assumptions. erefore, it is reasonable to apply MGPMs to forecast the amount of CO 2 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 functionallink net are employed to implement feature selection and residual modification, respectively. e 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 CO 2 N). us, the prediction ability of the traditional GM (1, N) could be effectively improved by feature selection and residual modification indeed.

Conclusions
Undoubtedly, the reduction of greenhouse gas emissions is critical to environmental protection. For many countries, CO 2 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 CO 2 emissions has gained increasing global attention. Revelations from the IEA [3] showed that, along with global economic growth, global CO 2 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. ere appeared to be a decoupling of economic growth and environmental degradation. However, this does not mean that CO 2 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 CO 2 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 CO 2 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
e authors declare that there are no conflicts of interest.