Application of a novel second-order differential equation grey model to forecast NOx emissions in China

Nitrogen oxide (NOx) contains two harmful air pollutants: nitric oxide (NO) and nitrogen dioxide (NO2). The reasonable prediction of China’s NOx emissions is of positive significance for the government to formulate environmental protection policies. To this end, a new grey prediction model with second-order differential equation is proposed in this paper, which has more reasonable model structure and better modeling performance than the traditional grey model. Secondly, according to the data characteristics of NOx emissions of China in recent years, a smoothing algorithm and weakening buffer operator are employed to process the original data to solve the rationality of the prediction results of the new model. Thirdly, the model for predicting China’s NOx emissions has been constructed by the new proposed model. The results show that the mean comprehensive error of the new model is only 0.0692%, and its performance is much better than that of several other mainstream grey prediction models. Finally, the new model is applied to China’s carbon dioxide prediction in the next 5 years, and the rationality of the prediction results is analyzed. Based on the prediction results, relevant countermeasures and suggestions are put forward.


Introduction
In recent years, China has been experiencing rapid economic growth. As a country with a large population, environmental, ecology, and health issues make the government pay more attention to environmental protection. In terms of environmental protection, the emission regulations of toxic or harmful gases are becoming more and more strict. Two very important air pollutants, nitric oxide (NO) and nitrogen dioxide (NO 2 ), are collectively referred to as nitrogen oxides (NOx) (Stamenković et al. 2017). NOx is emitted into the atmosphere from artificial emissions and many natural resource utilizations. NOx emissions from manufacturing resources in China include fossil fuel power plants, industrial production, road transportation, and motor vehicles. The natural resource emissions mainly have soil, lightning, wildfire, etc. (Zhang et al. 2020). The emission of NOx into the air causes haze, acid rain, and tropospheric ozone. The depletion of the ozone layer and, ultimately, global warming (Kaya et al. 2018) can easily lead to respiratory diseases and harm humans.
China has been committed to vigorously developing renewable clean energy to reduce NOx emissions, such as wind power generation, solar power generation, and electric vehicles. Research shows that renewable energy can reduce NOx emissions by 20-80% (Safdarnejad et al. 2019). China has also made great achievements in this regard. According to the data collected from the National Bureau of Statistics of China (Table 1) Currently, China is in the most critical period of air quality management (Ding et al. 2017a, b). Therefore, the main purpose of this research is to predict the NOx emissions in the next 5 years according to the existing sequence. The results will assist the Chinese government in formulating effective controlling policies to break the plight of air pollution, to further reduce pollution during the 14th Five-Year Plan period, and to achieve the national energy conservation and emission reduction goals.
Many factors produce NOx, and its prediction is complex. NOx emission is supervised by environmental protection organizations around the world, and its harm to health makes people subjectively control it, NOx emission future development trend is of great application value to study, and it has become an active research field. The research contents and methods of NOx emissions in various open research literature differ. Geographically, there are those taking the world as the research object (Wild et al. 2000. Stamenković et al. 2017. Some take the sub-regional or regional emissions as the research aim (Van aardenne et al. 1999;Frost et al. 2006;Ding et al. 2017a, b;Zhang et al. 2020). Comprehensive analyses of multi-factor emissions (Stamenković et al., 2017, Ding et al. 2017a, human factors (Kaya et al. 2018, Safdarnejad et al. 2019, Zhang et al. 2020, Chandrasekaran et al. 2020, and natural resources (Yan et al. 2005;Nault et al. 2017) are also analyzed. The prediction models used in the literature mainly include the air quality models (Pan et al., 2014, Zhang et al. 2020Nault et al. 2017), statistical models (Yan et al. 2005, Chandrasekaran et al. 2020, dynamic models (Safdarnejad et al. 2019), and intelligent algorithm (neural network models) (Obodeh and Ajuwa 2009, Smrekar et al. 2013, Kaya et al. 2018. The accuracy and practicability of the systems are systematically analyzed and studied. This study mainly explores the prediction of China's NOx emissions. Limited by the statistical caliber and statistical methods, the amount of relevant data available is small. Therefore, using the existing big data processing model to predict China's NOx emissions effectively is difficult. It is a good choice to use the grey model, which has the advantage of processing "small data" to predict it. Deng Julong put forward the grey system theory in 1982 (Deng 1982). A grey system contains some available information and some unknown information. Grey prediction is an important branch of grey theory. In the past decades, the grey prediction model has made a series of academic achievements. The model structure ranges from homogeneous index series (Zhan and Shi 2013) and non-homogeneous index series (Ma et al. 2017;Dang et al. 2017) to intelligent, flexible structure TDGM (1, 1, r) Li 2018, Wang andLi 2019). In terms of parameter optimization, some models have been upgraded in terms of optimization initial value , background value (Ding et al. 2017a, b), and accumulation sequence (Ma et al. 2020). These studies enrich and perfect the grey prediction model system, promote the wide application of the grey prediction model, and lay a theoretical foundation for solving various practical problems.
The grey prediction model is widely used in the prediction of clean energy output and consumption Li 2016, Ma et al. 2019a, b) and pollutant emission (Ding et al. 2017a, b, Wang andLi, 2019). All these research results provide a useful reference for the government to formulate energy policies. Although the research results of the existing grey prediction models have achieved effective results in many fields, not every model is suitable for prediction in all areas because of the different digital characteristics for diverse research contents. To accurately predict the emission of NOx, it is necessary to establish a more suitable grey prediction model according to its data characteristics. NOx emissions are affected by many human and natural factors. Its data composition is complex, and the available data are limited. Therefore, it is necessary to make qualitative analysis and quantitative treatment of China's NOx emissions. Then, a new intelligent grey prediction model suitable for medium-and long-term prediction of China's NOx emissions is reconstructed from the initial data. The model we proposed has three advantages. First, its development trend is nonlinear and can predict complex initial data more accurately. Second, we have established a second-order differential whitening equation, which is more flexible and easier to understand. Finally, the r-order accumulation operator can be flexibly adjusted to meet the optimal prediction results of NOx emissions. Therefore, the new model can provide an effective and feasible research method for the prediction of China's NOx emissions and provide reference data for the government to formulate reasonable policies.
The remainder of this paper is organized as follows. The "Related models and their defects" section briefly introduces the relevant models and analyzes their shortcomings. In the "The new second-order differential equation grey prediction model TWGM (2,1,r)" section, a second-order differential grey prediction model is established, and its basic form is derived. The initial data are smoothed and analyzed, the parameters are estimated, and its time response function and final expression are derived. The section also introduces the error definition of the proposed model. In the "Predicting China's NOx emissions using the TWGM (2,1,r) model" section, the effectiveness of the new model is verified based on China's NOx emissions, and the results are compared with other relevant models. Finally, a 5-year forecast of China's NOx emissions is made. The "Conclusion" section summarizes the full text and makes some prospects for the future research direction.

Related models and their defects
where Z (1) is the nearest neighbor mean series of X (1) , that is, where then The basic form of the three-parameter discrete grey prediction model is called TDGM (1,1) for short (Zeng et al. 2020a, b). A three-parameter differential whitening equation is established to obtain the time response function, and the functional relationships x (0) (k) and k are derived by solving the differential equation.

Definition 2.2
Assuming that X (0) and X (1) are as shown in Definition 2.1, then.
is the whitening differential equation of the TWGM (1,1) model.
The TWGM (1,1) differential model has a good simulation of all homogeneous, non-homogeneous exponential, and linear function sequences. However, the following three main shortcomings limit the prediction ability of the model and its application.

Structural defects
The TWGM (1,1) model uses the linear function 0.5(2k − 1)b of the grey information action quantity, and c is an arbitrary disturbance term. It can have a high-precision prediction for linear series, but in practical application, most of the series are nonlinear, so the model cannot achieve the effective prediction.

Lack of data preprocessing
The original sequence is generally non-smooth, and the increase or decrease speed is too fast, which is not in line with reality. It is difficult to obtain a reasonable solution using the grey prediction model directly. Therefore, the raw sequence should be smoothed, weakened, and then predicted by the grey prediction model.

Mechanism defect
The TWGM (1,1) model only uses the first-order accumulation operator to act on the original sequence. However, with the change in the raw data, only the first-order accumulation operator cannot reach the best prediction effect. The fractional r-order accumulation operator can improve the accuracy of the grey prediction model.

The new second-order differential equation grey prediction model TWGM (2, 1, r)
Given the deficiencies of the TWGM (1,1) model, a second-order differential equation grey prediction model is established in this study. The increment of the increase and decrease speed of the sequence is no longer a constant but a linear function of time t ; that is, the second-order derivative of the sequence is a linear function. The time response function can be solved by solving the differential equation. Secondly, the original series is processed by the smoothing operator and weakening operator due to data characteristics and development trend. Finally, the optimal r-order accumulation operator is used to act on the model.
Data processing method of TWGM (2, 1, r) The reduction of the original sequence that is too fast or too slow may greatly affect the change law of the system. A reduction that is too fast results in a high reduction for the local system, while a reduction that is too slow results in a low reduction. It may lead to inconsistencies between the predicted results and the actual situation. To solve this problem, Professor Liu Sifeng proposed a series of grey buffer operators in 1997. Essentially, a grey buffer operator is a mathematical transformation method. It applies the qualitative analysis results of the system's future development toward the original modeling order. The grey weakening buffer operator is one of the grey buffer operators. It eliminates the contradiction between the low-reduction and high-reduction systems and achieves ideal modeling results by suppressing the high-reduction system.
This section analyzes the data characteristics of NOx emissions in China to support establishing a reasonable grey prediction model. The NOx emissions in China from 2011 to 2020 provided by the National Bureau of Statistics of China were used as experimental data, as shown in Table 1. It is easy to see that China's NOx emissions show a decreasing trend from 2011 to 2020, and the decreasing trend is increasing. The annual decrease in 2020 is 17.34%, much higher than 4.37% in 2019, and is also much higher than this decade. The yearly decrease rate was 5.7%. But this high reduction is unlikely to be sustained over the long term, and if the raw data is used directly for modeling and forecasting, the results will be unsatisfactory. To establish a more realistic prediction model, this paper intends to use grey smoothing and weakening buffer operators to slow down the high decreasing trend of the original data to achieve a balance between qualitative analysis and quantitative data.
where Definition 3.2 Let the sequence X (0) and weight vector XD1 be as shown in Definition 3.1., It can be seen that D is a weakening operator. And the sequence x (0) is a monotonically increasing, monotonically decreasing, or vibrating series. The D is called a weighted average weakening buffer operator. A grey buffer operator is essentially a qualitative analysis and quantification computing. The weight selection is based on the principle of prioritizing new sequences by grey system theory while considering future changes.

D e f i n i t i o n 3 . 3 A s s u m i n g t h a t
is called a sequence of r-order of XD . r(n) is a gamma function, and.

X (−r)
R is called the r-order inverse sequence of XD , where.
And Z (r) is the adjacent neighbor average sequence of X (r) R , that is. and TWGM (2, 1, r) model and its basic form Definition 3.4 Assume that the sequences X (0) , XD1 , and XD are as shown in Definitions 3.1 and 3.3, and X (r) R is as shown in Definition 3.3, then.
x (−r) Equation (10) is called three-parameter second-order whitening differential equation grey prediction TWGM (2, 1, r) model for real number field grey generation operators. By integrating the two ends of Eq. (10) with respect to t , we can obtain Calculate the definite integral in the interval [k − 1, k] on the left and right sides of Eq. (11) as follows: by therefore The same as the TWGM (1, 1) model, the TWGM (2, 1, r) model background value can be approximately expressed by the following formula: By substituting Eq. (13) into Eq. (12), we can get. that is Equation (14) is called the basic form of the TWGM (2, 1, r) model.
The basic form of the TWGM (2, 1, r) model includes four parameters a, b, c, and d . a is called the development coefficient, and its value reflects the development trend of model eigenvalues. b and c are grey action quantities, which reflect all grey information affecting the system, and 3t 2 − 3t + 1 ∕6 b + (t − 1∕2)c indicates that the grey action quantity is nonlinearly related to time. d is the error disturbance term, which represents the information with accidental or weak factors other than the main variable. According to the modeling idea of the TWGM (2, 1, r) model, the model parameters are estimated by Eq. (14), and the time response function of the model is obtained by solving the differential Eq. (11).

Parameter estimation of TWGM (2, 1, r) model
This model's basic idea of parameter estimation is similar to that of the TWGM (1, 1) model. The parameters are estimated by the least square method on the condition that the sum of squares of simulation errors of data preprocessed sequences is the minimum. where That is, p = (a, b, c, d) T can be estimated by the least square method, satisfies.

Time response function of the TWGM 2, 1, r model
Parameter estimation is the first step in structuring a grey prediction model. The development trend of a variable system cannot be predicted according to Eq. (14), and parameter estimation is also necessary to establish the time response function of the model. In this way, the functional x (0) R (t) relationship with time variables t can be obtained. The time response function is expressed as the relation of x (0) R (t) and t , which can be obtained by establishing the whitening differential equation and then solving the differential equation. According to Eq. (11), its corresponding homogeneous differential equation is It is easy to obtain the general solution of the homogeneous differential Eq. (15) as Next, we need to find a special solution to the non-homogeneous Eq. (11). Using the constant-coefficient variation method, let On the derivation of the above formula about t , we can get By substituting it into Eq. (11) then Therefore, the general solution of the differential Eq. (11) can be expressed as Equation (17) can be expressed as.

Let
Equation (18) can be simply written as The simulation values x (0) R (t) can be obtained by using Eq. (8), Equation 21 is the time response function of the TWGM (2, 1, r) model. When t ≤ n , x (0) R (t) is the simulation value, and when t > n , x (0) R (t) is the prediction value.

Modeling steps of the TWGM (2, 1, r) model
According to the proposed TWGM (2, 1, r) model, the fundamental modeling steps are summarized as follows: Step 1. Data preprocessing. First, collect the original data. Then, according to Definition 3.1, do a threeterm moving average to get the smooth sequence XD1 . According to the sequence development trend, weaken the XD1 sequence to find the most suitable grey weakening operator and get the new sequence XD.
Step 2. Calculating the optimal r-order accumulation sequence of the model. According to Definition 3.3, the particle swarm optimization algorithm calculates the optimal r-order r*, and the adjacent neighbor means sequence Z (r) R is calculated.
Step 3. Parameter estimating. According to Theorem 3.1, construct matrices B and Y, and then calculate the parameters a, b and c.
Step 4. The time response function building. Calculate the parameters according to Eq. (19). Substituting into recipes Eqs. (20) and (21), the time response function of the model TWGM (2, 1, r) is established.
Step 5. Simulation error test. According to the TWGM (2, 1, r) model, when t ≤ n , calculating the simulation value, the mean comprehensive percentage error ) can be calculated. If the accuracy of MRPE meets the requirements, go to Step 6; if not, return to Step 1.
Step 6. Predict the future. The TWGM (2, 1, r) model is used to predict the future trend. The new data replace the old data, x (0) R (n + 1) is added to the model sequence, while x (0) R (1) was removed. The dimension of the sequence is unchanged, and the new sequence can be expressed as follows: The modeling flow chart is shown in Fig. 1.

Error discrimination method
Error test of the model is an important prerequisite for model prediction. To evaluate the effectiveness of the TWGM (2, 1, r) model, we use the mean relative simulation percentage error (MRPE) Δ s to test the model's overall performance. It is calculated according to the following formula. where Predicting China's NOx emissions using the TWGM (2, 1, r) model .51,15.68,13.8,12.9,11.81,10.20). Next, we consider two ways of weakening the operators (b) and taking the weight coefficient W1=[0.05,0.05，0.1,0.1,0.15,0.2] to drain the smoothing sequence (c). It is considered to use the following method to weight W2= [1,2,3,4,5,6]/21; their experimental results are shown in Table 2. The mean relative percentage errors of the three data processing modes are 0.2494, 0.1713, and 0.0129, respectively.Next, we consider two ways of weakening the operators In Table 2, symbols x di (t) , x (r) Ri (t) , Δ i (t) , r i * , and Δ si are described as follows.
• x di (t) : pretreated sequence of (a), (b), and (c), respectively, (i = 1, 2, 3); : residual error, (i = 1, 2, 3); • r i * : optimal r-order accumulation calculated by PSO algorithm, (i = 1, 2, 3); The line graph of the processed and simulated data corresponding to the three data processing methods is shown in Fig. 2. The simulated performance of the processed data cannot be seen directly, so the simulated data is inversed, and (b) and (c) are compared with (a). Their line chart is shown in Fig. 3.
It can be seen from Fig. 3 that the (b) and (c) preprocess the inverse operation sequence of the data obtained from the above two data preprocessing methods, the method (c) is the best for the TWGM (2, 1, r) model for prediction of China's NOx emissions, and the following prediction chooses (c), that is, XD = [12.55, 12.26, 11.84, 11.38, 10.85, 10.2] . The corresponding steps of method (c), the scatter diagram of the original sequence X (0) , the three-moving average smoothing sequence, and the weakening operator sequence are shown in Fig. 4. The new series smoothed the curve, weakening the original data's reduction speed and conforming to the actual reduction trend to ensure a more reasonable prediction. (2, 1, r) According to the modeling steps of the method (c), the particle swarm optimization algorithm is used to calculate with MATLAB. The optimal r-order data of the TWGM (2, 1, r) model for the prediction of China's NOx emissions and the values of all parameters are shown in Table 3.

China's NOx emissions prediction model with TWGM
By substituting the data in Table 3 into Eq. (20), the r-order accumulation function of China's NOx emissions can be obtained.

Model comparisons and performance analysis
To verify the prediction performance of the TWGM (2, 1, r) model for China's NOx emissions, this study selects the relevant grey prediction models TDGM (1, 1, r) model, TWGM (1, 1) model, the optimal r-order TWGM (1, 1, r)  Table 4. The MRPE of the TWGM (2, 1, r) model is much smaller than the other models. The TWGM (1, 1) model, DGM (1,1) model, and GM (1,1) model are used to predict the data because the first-order generating sequence is used, so the error is relatively large. To intuitively compare the simulation effects of the different models, we draw the simulation diagram, as shown in Fig. 5. The error analysis of the above four methods is shown in Fig. 6. The mean simulation percentage error of the TWGM (2, 1, r) prediction model is 0.0692%. According to the inspection grade in the accuracy reference table, it belongs to class I (≥ 99%), and the complete accuracy is 99.9308%.

Prediction of NOx emissions in China
The TWGM (2, 1, r) model is used to predict China's NOx emissions from 2021 to 2025. The 2021 data was expected to be 9.26 million tons, then, the 2015 data was discarded, and the 2021 data was added and modeled again. It was repeated until the prediction data were completed from 2021 to 2025. The prediction results are shown in Table 5.
The initial data of China's NOx emissions from 2015 to 2020 and the forecast data from 2021 to 2025 are put together to form a series, as shown in Fig. 7, and we can visually see the reduction trend of China's NOx emissions. According to the forecasting results, China's NOx emissions will continue to decline in the next 5 years, but the decline rate will slow down. The prediction results have a reference value for the Chinese government to formulate relevant Fig. 4 Scatter diagram of the raw data, three-term moving average, and weakening operator data    policies. According to the forecast value, China's NOx emissions will drop to 8.14 million tons in 2025, a cumulative decrease of 20.2% during the 14th Five-Year Plan period. Therefore, the Chinese government has two main aspects in controlling NOx emissions. On the one hand, it is to reduce artificial emissions: for example, the use of diesel engines, gas turbines, coal-fired boilers, etc. is restricted in the industry, and clean energy (wind power generation, solar energy, electric vehicles) is strongly encouraged and supported. On the other hand, we should reduce the emission of NOx from natural resources. For example, because the use of nitrogen fertilizers produces NOx emissions, so does the burning of crop straw. When the government formulates policies to reduce NOx emissions, more publicity is needed, so that every citizen can participate in protecting the environment and contribute to reducing NOx emissions in China.

Conclusion
China has a large population and a wide range of regions.
To protect the environment, it is necessary to reasonably predict China's future NOx emissions, which can provide reference data for the Chinese government to formulate energy conservation and emission reduction policies. In this paper, we propose a new differential equation grey model TWGM (2, 1, r) , which solves the time response function of the grey model by solving the differential equation. Moreover, we preprocessed the initial series to accurately predict the emission of NOx in China in the past 10 years to improve the prediction rationality. It has higher simulation accuracy than the TWGM (1, 1, r) model and a more stable model performance. In general, this paper not only puts forward an effective grey prediction model but also makes a scientific prediction of China's NOx emissions. The prediction results can be used as a reference for the government to formulate relevant policies. The model is a perfect development of the grey system theory methodology. In the next step, we can consider a nonlinear function that is more suitable for the prediction sequence in grey action or consider the fractional order differential equation grey model, so that it can achieve the ideal prediction effect for any prediction sequence in an adaptive way.