Research on price forecasting method of China's carbon trading market based on PSO-RBF algorithm

ABSTRACT The forecasting of carbon emissions trading market price is the basis for improving risk management in the carbon trading market and strengthening the enthusiasm of market participants. This paper will apply machine learning methods to forecast the price of China's carbon trading market. First, the daily average transaction prices of the carbon trading market in Hubei and Shenzhen are collected, and these data are preprocessed by PCAF approach to choose the input variables. Second, a prediction model based on Radical Basis Function (RBF) neural network is established and the parameters of the neural network are optimized by Particle Swarm Optimization (PSO). Finally, the PSO-RBF model is validated by the actual data and proved that the PSO-RBF model has better prediction effect than BP and RBF neural network in China's carbon prices prediction. It is indicated that the prediction model has more significant applicability and deserves further popularization.


Introduction
The establishment of a carbon emission trading system provides a market-oriented means for countries around the world to implement low-carbon development (Yan & Tang, 2017). China, as the world's largest carbon emission country, is curbing the greenhouse gas emissions with pragmatic actions. The carbon emissions trading market is an effective policy tool for the Chinese government to control carbon emissions. China launched the carbon emissions trading pilot markets in seven provinces and cities, including Beijing, Shanghai, Guangdong, Shenzhen, Tianjin, Chongqing and Hubei from 2011, and launched all online transactions in 2014. In the end of 2017, China has fulfilled its emission reduction commitments to the international community and completed the overall design of the country's carbon emissions trading system (Lo & Francesch-Huidobro, 2017;Yue & Cheng, 2017). National unified carbon emissions trading market will gradually mature in the next few years and will play an increasingly important role as a national policy adjustment platform in the area of emission reduction in the next decade.
The key to building a carbon trading market is a scientific and reasonable carbon price mechanism. However, Chinese carbon emissions trading market has just been formed, resulting in unstable carbon price, irregular changes and more prominent market risk. Therefore, CONTACT Shijian Liu liushijian1992@163.com research on the carbon price forecasting model will help reduce the risk of carbon price fluctuations and promote the construction of China's carbon trading market. Machine learning technology is increasingly being cited in the nonlinear and non-stationary carbon price predictions. Zhu and Ming (2011) constructed a carbon price forecasting based on integration of group method of data handing (GMDH), PSO and least squares support vector machines (LSSVM). Zhu (2012) also proposed a multi-scale ensemble model, which is composed of empirical mode decomposition (EMD), genetic algorithm (GA) and artificial neural network (ANN) to forecast carbon price in ECX. Gao and Jian (2014) set up an international carbon financial market price error correction prediction model based on EMD, PSO and LSSVM. Fan, Li, and Tian (2015) established a multi-layer perceptron neural network (MLP) forecasting model to study the carbon price of EU EST from the perspective of chaos. Jiang and Wu (2015) built a model based on Support Vector Regression (SVR) algorithm to predict international carbon market price. Sun et al. (2016) found that using variational mode decomposition (VMD) and spiking neural networks (SNNS) to forecast the ICE carbon price can obtain a better empirical test effect. Zhang, Zhang, Xiong, and Su (2017) advanced a hybrid approach integrating PSO and multi-output support vector regression (MSVR) to forecast carbon prices. Jiang and Peng (2018) presented a carbon prices prediction model based on back propagation (BP) neural network optimized by Chaos Particle Swarm Optimization (CPSO) algorithm.
In summary, the neural network theory has shown good prediction ability in carbon price forecasting. RBF is a kind of neural network which is better than traditional neural network in time series prediction in terms of approximation ability, classification ability and learning speed. The structure of RBF is simpler, and its training success rate is higher. Qing-Wen, Chen, Zhu, and WU (2006) trained historical electricity price by RBF neural network and hierarchical genetic algorithm (HGA), and the test results are satisfactory. Zhang et al. (2008) established a model to forecast short-term load by combining the RBF neural network with the adaptive neural fuzzy inference system (ANFIS). Coelho and Santos (2011) found RBF neural network model with the generalized autoregressive conditional heteroskedasticity (GARCH) errors can show good results in application to electricity price forecasting. Shen, Guo, Wu, and Wu (2011) chose RBF neural network optimized by the artificial fish swarm algorithm (AFSA) to forecast the stock indices. Cecati, Kolbusz, Rzycki, Siano, and Wilamowski (2015) designed a new algorithm called ErrCor in machine learning to train RBF for 24h electric load prediction.
RBF neural network has been successfully applied into the time series prediction and analysis. However, due to the inherent characteristics of RBF neural network can easily cause itself to fall into a local optimum. Furthermore, the centre number of RBF neural network and the weights extremely need to rely on experience. If the optimization algorithm is adopted to overcome these defects, the network performance will be further improved. Therefore, this paper uses PSO to optimize RBF neural network and applies it into the China's carbon prices prediction for the first time. Compared with the traditional forecasting model, the PSO-RBF model is more suitable for China's carbon prices prediction, thus providing a practical and accurate price prediction method for China's carbon trading market.
The rest of this paper is organized as follows: Section 2 describes the fundamentals of RBF and PSO, then elaborates on the PSO-RBF model; Section 3 presents the used dataset and obtained results; Section 4 concludes the study.

RBF neural network
Radical Basis Function neural network (RBFNN) is a threelayer feedforward neural network, which is proposed by Moody and Darken (2014) in the 1980s. The input layer consists of the signal source nodes. The second layer is the hidden layer, and the number of hidden units depends on the needs of the described problem. The third layer is the output layer, which responds to the role of the input mode. The general structure of an RBFNN is shown in Figure 1 is Gaussian function selected as activation function, where n means the number of neurons in hidden layer. In g i (X) = g i (||X − C i ||), C i is the centre of ith activation function, and || * || is Euclid norm.
The output of the ith neuron in hidden layer of RBFNN can be represented in the form.
where σ i is the width of the receptive field. The activation of the output layer is linear combination of units on the hidden layer, which can be expressed as where w i is the connecting weights from hidden layer to output layer. When the centre point of the RBF is determined, the mapping relationship is determined. The transformation from the input space to the hidden layer space is nonlinear, and the mapping from the hidden layer to the output layer is linear. The output of the network is the linear weighted sum of the output of the hidden layer, where the weight here is the tunable parameter of network.
The theoretical basis of RBFNN is that radial basis exists as a hidden base to form the hidden layer space. So that the input vector can be directly mapped to the hidden space, does not need to connect through the weight. According to the Cover Theorem, the inseparable data in the low-dimensional space is more likely to become separable in high-dimensional space. In other words, the function of the hidden layer of the RBFNN is to map the input of the low-dimensional space to a high-dimensional space through a non-linear function, and then to fit the curve in this high-dimensional space. It is equivalent to finding a surface that best fits the training data in an implied high-dimensional space, so that the transition from the low dimension to the high dimension can be achieved. It can be easily solved problems which cannot be solved in the low dimension in the high-dimensional space.

PSO algorithm
PSO is an evolutionary optimization algorithm, introduced by Kennedy and Eberhart (2002) in 1995. Similar to GA, PSO also belongs to the population iteration. However, the particles of PSO follow its own optimal particle in the entire swarm to search the global optimum solution. Each particle keeps track of its own best position and velocity in the problem space, and its initial position and velocity are generated randomly. So PSO is an easy stochastic optimization technique because of a few parameters to adjust. For a complex nonlinear system, PSO is a better global optimization capability and high searching speed.
Let the position and velocity of the ith particle in the n-dimensional search space be respectively assumed as According to a specific fitness function, the local best of the ith particle could be P l i = p l i,1 , p l i,2 , . . . , p l i,n , and P g = p g 1 , p g 2 , . . . , p g n is the global best found. The new positions and velocities of the particles will be updated at each iteration. This process follows the following two formulas: where i = 1, 2, . . . , m, m is the number of particles in a population. P i (k) and P l i (k) are the position and the local best of ith particle at iteration k, respectively, P g is the global best of all particles, V i (k) is the velocity of ith particle at iteration k. c 1 and c 2 are both acceleration coefficient, but one is the cognitive parameter and other is the social parameter. r 1 and r 2 are the random numbers between 0 and 1.

RBF optimized by PSO
Predicting accuracy of the RBFNN lies on three parameters: output weights w i , the hidden layer nodes widths σ i , the data centre of basis function C i . The three parameters are updated by PSO, taking the place of the Gradient Descent method. In this paper, the number of hidden layer nodes of the RBFNN is 10, and the input data is normalization. The training error of RBFNN is taken as the fitness function of PSO, as shown in Equation (5). The minimum value of the fitness function is get through the optimal particle positions calculated by the PSO algorithm, and the optimal value of each parameter of the RBFNN is obtained.
The specific steps of adopting PSO to search the optimal values of the RBF network parameters are as followed and the flow chart of RBF optimized by PSO is demonstrated in Figure 2.
Step 1 Define the number of particles, initialize their position and velocity.
Step 2 Calculate the fitness function to obtain the optimal value of each particle and the global optimal value.
Step 3 Update each particle's velocity and position.
Step 4 Recalculate its fitness and acquire the optimal value of each particle and the global optimum value again.
Step 5 Determine whether the fitness reaches the minimum value. If not, loop to step 3 until a criterion is met.
Finally, the optimal w i , σ i and C i are get to form a trained RBFNN to predict the data of the testing set.

Data preprocessing
This article mainly forecasts China's carbon market transaction prices. As shown in Figure 3, Hubei's carbon trading volume is the largest among seven carbon emission trading pilot markets in China (we do not consider carbon market in Fujian), and its turnover is also the largest. The carbon trading pilot market in Shenzhen, as the earliest carbon markets in China, has a certain significance in research. Therefore, the daily average transaction prices (Data does not include holidays and no trading days.) of the carbon trading pilot market in Hubei and Shenzhen are selected as sample data, and details of samples are reported in Table 1. All the data comes from the Wind economic database (win, 2018).
In order to confirm the format of training data, PACF is selected to derive the autocorrelogram of data.    partial autocorrelation at lag k is out of the 95% confidence interval, x(i − k) can be one of the input variables. Table 2 shows input variables of carbon prices data in Hubei and Shenzhen obviously.
As can be seen from Table 2, the carbon price of Hubei on the ith day is predicted by the carbon prices of the (i − 1)th and (i − 2)th days as the input variables of the model. Similarly, the carbon prices of the (i − 1)th, (i − 2)th and (i − 3)th days are used as input variables to output the ith day's carbon price of Shenzhen.

The obtained results of forecasting carbon prices by PSO-RBF
This paper uses MATLAB software to Implement the prediction model. Set up RBF parameters by PSO algorithm and use samples of carbon prices data to train the RBF network. In this research, Gaussian function is selected as the radial basis function, and the number of hidden layer nodes is set as 15. The input values are given by the data after dealing with by PACF. And the parameters of the PSO algorithm are as follows: the number of particles is 1000, and the learning factor is c1 = c2 = 2. At the same time, the PSO-RBF model proposed in this article is compared with using RBF and BPNN (there are 20 hidden nodes in BPNN) to predict the carbon prices in Hubei and Shenzhen. The carbon price forecasting results for Hubei and Shenzhen are shown in Figures 6 and 7.  The predicted value error of different prediction models is presented in Table 3, we also can be the same conclusion by comparing the size of the Mean Absolute Percentage Error (MAPE) and the Mean Absolute Error (MAE). From Table 3, the MAPE and the MAE of PSO-RBF model for the Hubei's carbon price forecasting are respectively 1.580% and 0.233, and the Shenzhen's are separately 6.942% and 1.863. These values are both smaller than the MAPE and MAE of RBF neural network and BP neural network. It is proved that the PSO-RBF model is superior to the  BPNN model and RBFNN model in terms of prediction effect, prediction accuracy, etc. It also indicates the validity of the PSO-RBF model in Chinese carbon price prediction. Table 4 shows the time consumption of different prediction models. As seen in Table 4, the time consumption of the PSO-RBF model is longer than the other models. It is noted that the pro of our suggested method is to improve the forecasting performance. Specifically, the parameters for the RBF neural network are optimized by PSO while the single RBF neural network employs the random initial value for the parameters. However, the con of our suggested method is the PSO-RBF model can spend more time.

Conclusion
Price forecasting of carbon market is of great significant, especially after China launched the nation's unified carbon emissions trading market in 2017, either for the government or companies. The PSO-RBF model is presented and applied into the price prediction in this study. Set up a China's carbon prices forecasting model by comprehensively utilizing the self-learning ability of RBF network and the optimized advantages of PSO. As shown in this research by the results of an example of factual forecasting which is the prices of carbon market in Hubei and Shenzhen, this forecasting model can work effectively and enhance the predicting precision. Compared with RBF and BP neural network, the simulation results evidence that the PSO-RBF model has stronger approximation ability, faster convergence rate and higher forecasting accuracy. As one of the largest suppliers of emission reduction markets, China's annual carbon trading volume will exceed 200 million tons in the next 5 years, and it is expected to become the world's largest market for carbon emissions trading. Chinese government has vigorously developed non-fossil energy, promoted the clean development of fossil energy, accelerated industrial restructuring and controlled industrial emissions. By establishing and running the nationwide carbon emissions trading market, Chinese government can take effective measures to cut back Chinese CO 2 emissions. The proposed PSO-RBF method can be further extended to the future carbon price forecast for the nation's unified carbon emissions trading market in China.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was supported by the Foundation of Hebei Province: [grant number 2014BD0065].