Carbon price forecasting with complex network and extreme learning machine

Journal Pre-proof Carbon price forecasting with complex network and extreme learning machine Hua Xu, Minggang Wang, Shumin Jiang, Weiguo Yang

PII: DOI: Reference:

S0378-4371(19)31609-7 https://doi.org/10.1016/j.physa.2019.122830 PHYSA 122830

To appear in:

Physica A

Received date : 30 June 2019 Revised date : 14 August 2019 Please cite this article as: H. Xu, M. Wang, S. Jiang et al., Carbon price forecasting with complex network and extreme learning machine, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.122830. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

*Highlights (for review)

Journal Pre-proof Highlights

Jo

urn

al

Pr e-

p ro

of

The highlights of this paper list as follows: 1.A new prediction paradigm (CPN-ELM) is proposed based on complex network and extreme learning machine. 2.The carbon price network is constructed by using the carbon price data. 3.The information of carbon price fluctuation is extracted by using network topology. 4.Empirical results demonstrate the effectiveness and robustness of CPN-ELM.

*Manuscript Click here to view linked References

Journal Pre-proof Carbon price forecasting with complex network and extreme learning machine Hua Xua,c, Minggang Wangb,c*, Shumin Jianga,Weiguo Yanga Faculty of Science, Jiangsu University, Zhenjiang, 212013, Jiangsu China

b

School of Mathematical Science, Nanjing Normal University, Nanjing, 210042, Jiangsu, China

c

Taizhou College, Nanjing Normal University, Taizhou 225300, Jiangsu, China

of

a

*

Corresponding author: [email protected] (M. Wang)

al

Pr e-

p ro

Abstract: Carbon emission price mechanism is the core issue in carbon emission trading. The carbon emission price fluctuation trend is related to the play of the effectiveness of carbon emission trading market, and directly affects the green and low-carbon behavior of enterprises and residents. Therefore, the prediction of carbon price is of great practical significance. This study presents a new carbon price prediction model by using time series complex network analysis technology and extreme learning machine algorithm (ELM). In our model, we first map the carbon price data into a carbon price network (CPN), and then extract the effective information of carbon price fluctuations by using the network topology，and use the extracted effective information to reconstruct the carbon price sample data. With the reconstructed data and the extreme learning machine algorithm, the carbon price network extreme learning machine model (CPN-ELM) is built. To test the validity of the model, we selected the carbon emission price data of the second, third and transition stages of the European Union Emissions Trading System (EU ETS) for empirical analysis, the results show that CPN-ELM can improve the predictive accuracy of ELM in both level accuracy and directional accuracy. Meanwhile, CPN-ELM prediction model has better robustness when facing the random samples, sample data with different frequencies or sample data with structural changes.

urn

Keywords: carbon price prediction, complex network, extreme learning machine, green and low-carbon behavior

1. Introduction

Jo

Currently, climate change and carbon dioxide emission reduction have caused great concern globally. How to reduce greenhouse gas emissions and curb the global warming trend effectively has become a common challenge facing all countries in the world. Carbon trading market is an effective policy tool to control and reduce greenhouse gas emissions, which core is to form an effective carbon trading price through the supply and demand mechanism to guide enterprises to make emission reduction decisions. Since 16 February 2005, the international carbon market has shown a rapid growth trend and plays an increasingly important role in curbing global carbon emissions effectively. Carbon market price forecast is the basis of making carbon financial market policy and improving risk management ability. The reasonable prediction of carbon market price is not only conducive to deeply understanding and grasping the price fluctuation rules of carbon

Journal Pre-proof market, the establishment of an effective carbon market price stability mechanism, but also conducive to the investors to avoid carbon market risks, to achieve the preservation and appreciation of carbon assets. The trend of carbon price can provide insight into the changes of green and low-carbon behaviors of enterprises and residents. Therefore, carbon market price prediction has become a hot and difficult point in the research of international energy and climate economy.

Pr e-

p ro

of

At present, a lot of scholars have carried out researches on carbon price prediction, such as Zhu et al. (2013) conducted an algorithm to predict price by mixing Autoregressive Integrated Moving Average model (ARIMA) and Least Squares Support Vector Machines (LSSVM). Sun et al. (2016) built a hybrid carbon price prediction algorithm based on variational mode decomposition (VMD) and spiking neural networks (SNNs). Based on Empirical Mode Decomposition (EMD), Zhu et al. (2017) established an evolutionary least squares support vector regression multi-scale integrated prediction model. By using phase reconstruction, Fan et al. (2015) proposed a multilayer perceptron neural network prediction model to characterize the strong non-linearity of carbon price. Koop et al. (2012) established the dynamic model averaging to predict the carbon market, and found that this method could predict the carbon market more accurately than the traditional method. Zhu et al. (2018) proposed a multi-scale nonlinear integrated learning model to predict carbon price. Atsalakis(2016) built a new neural fuzzy controller for predicting carbon emission price. Zhao et al. (2018) proposed a real-time prediction program to predict the weekly carbon emission price and found that the combined-MIDAS model was better than the traditional model. Segnon et al. (2017) summarized existing methods for predicting price fluctuations of carbon emission. Based on GARCH, CIM, CEEMD and GNN optimized by ant colony algorithm (ACA), Zhang et al.(2018) established a hybrid model. Chevallier(2011) conducted a non-parametric modeling on the daily frequency of carbon price.

Jo

urn

al

Based on the above research on carbon price prediction, we can see that the existing prediction models about carbon price can be mainly divided into three types. (1) Prediction model based on econometric method（Koop,2013; Segnon et al.,2017）. This type of model can also be divided into two categories according to different ranges of data used. One is the structure model of carbon price, such as ECM, VECM and VAR model, etc, which use the method of linear regression to process data. In these models, historical carbon price data can be included as well as corresponding explanatory variables. The other is the prediction model based on time series, including ARIMA and GARCH model, etc., which only rely on historical carbon price data for prediction. The advantage of the prediction model based on the econometric method is that it can capture the time-varying volatility of carbon price, but such model cannot accurately describe the nonlinear characteristics of data. (2) Prediction model based on artificial intelligence algorithms （Atsalakis, 2016; Chevallier, 2011; Fan et al.,2015; Zhu et al.,2018）. Compared with the econometric prediction models, the prediction models based on the artificial intelligence algorithms have stronger non-linear ability to process data, but this kind of model often contains many parameters, which may lead to overfitting or poor convergence. (3) Combined prediction model (Sun et al.,2016; Zhang et al.,2018; Zhu et al.,2013; Zhu et al.,2017 ). This kind of model can make good use of the advantages of only using one prediction model, and the accuracy of prediction is higher than that of single prediction model. However, the structure of this kind of

Journal Pre-proof model is relatively complex, and more parameters need to be choosed in the actual use process. Therefore, there are still many questions worth discussing about the prediction of carbon price.

urn

al

Pr e-

p ro

of

As a matter of fact, to predict the carbon emission price accurately, the fluctuation rule of carbon emission price time series must be described firstly and extract the effective fluctuation information of carbon price. Over the past 20 years, the use of complex network theory to mine the hidden information of nonlinear time series has caused great concern（Lacasa et al.,2008; Xu et al.,2008; Zhang et al.,2006; Wang et al.,2016; Wang et al.,2017）, The main idea is as follows. Firstly, the nonlinear time series is transformed into a complex network by some algorithms, and then describe the fluctuation law of the nonlinear time series with the help of the topology structure of the complex network. Many studies have shown that the use of complex network technology can obtain the properties of time series effectively, and many new algorithms have been generated, such as visibility methods including NVG (Lacasa et al.,2008), HVG (Luque et al.,2009), PNVG (Bezsudnov et al., 2014), LPHVG (Wang et al., 2018) and PLVG (Li et al., 2018), the mapping algorithm for pseudo-periodic time series (2008; Xu et al), phase space based method (Zhang et al.,2006; Donner et al., 2010; Gao et al.,2017), phase space roughening algorithm (Wang et al.,2016), etc. Recently, domestic and foreign scholars have also introduced this technology into the field of energy economy (Chen et al.,2010; An et al., 2014; Wang et al., 2016; Chen et al.,2017; Chen et al.,2018) to study the characteristics of energy prices. For example, Chen et al.(2010) constructed the oil price complex network and studied the dynamic characteristics of oil price by using the oil price network. Wang et al., 2016 built the directed weighted crude oil and gasoline network and discussed the nonlinear fluctuation characteristics in different periods. Chen et al.,2017 set up the heating oil spot and futures price network and analyzed the dynamic evolutionary behavior of them. The above researches used the theory of network to discuss the characteristics of price fluctuations and obtain abundant significant results. Recently, by using complex network theory, Wang et al. (2018) established a new method to predict the fluctuation of oil price, its basic idea is as follows. Firstly, the energy prices are transformed into a directed weighted price network by some algorithms. Then, the energy price network is used to extract the characteristics of energy price fluctuations, and the extracted characteristics of energy price fluctuations are used to predict the future energy price fluctuations. Following this method, Zhang et al. (2019) developed a hybrid approach that combines the DFN and artificial intelligence (AI) techniques to forecast the Baltic Dry Index (BDI). Wang et al. (2019) developed a hybrid predictive technique combining complex network and traditional artificial neural network (ANN) techniques for copper price forecasting.

Jo

Based on time series complex network analysis technology and ELM, we present a new carbon price forecasting model. In this model, we first map the carbon price into a complex network (CPN), then we use the network topology to extract the effective information of carbon price fluctuations, and use the extracted effective information to reconstruct the original carbon price data. Finally, CPN-ELM model is constructed by using the reconstructed data and the extreme learning machine. To test the model's efficiency, we select the price data of carbon of the second, third and transition stages of the EU carbon trading market for empirical analysis, which indicate that the CPN-ELM model is superior to pure ELM in both horizontal accuracy and directional accuracy. It can improve the predictive accuracy of ELM model effectively and

Journal Pre-proof meanwhile, CPN-ELM prediction model has better robustness when facing the random samples、 sample data with different frequencies or sample data with structural changes. This paper mainly studies from the following aspects. Part 2 provides the methods applied in this paper. Part 3 builds a carbon price complex network and analyzes the structure of the carbon price complex network. Part 4 studies the carbon price prediction in different stages. Finally, the conclusions of this paper are given.

of

2. Methodology

p ro

2.1 The CPN-ELM prediction model

The carbon price prediction model based on complex network and ELM constructed in this paper contains three main steps, which are introduced as follows. 2.1.1 Build carbon price network（CPN）

Pr e-

Based on the Coarse graining method, Wang et al.(2016) proposed the crude oil price fluctuation network construction method. This method converts the crude oil price volatility sequences into the characters composed by five symbols {R, r, e, d, D}, which can better reflect complexity of energy price fluctuations. In this paper, we use this method to build a carbon price fluctuation network. Steps are as follows: Step 1: Calculate the volatility of carbon price





Let the carbon price sequence be X cp = xcp  t  , t  1, 2,..., N ,then we calculate the





al

volatility sequence of carbon price Fcp  f cp  t  ,

xcp (t )  xcp (t  1) xcp (t  1)

urn

f cp (t ) 

where xcp  0   xcp 1 ，

,

(1)

fcp 1  0 .

Step 2: Calculate the carbon price symbol sequence





Formula (2) is used to symbolize the volatility sequence Fcp  f cp  t 



and obtain the



Jo

symbol sequence corresponding to the carbon price Scp = scp  t  ， scp  t  R, e, D ，

 R, f cp (t )  0,  scp (t )  e, f cp (t )  0,   D, f cp (t )  0,

In formula (2) ， R indicates that the carbon price is rising, relatively stable, D indicates that the carbon price is falling.

(2)

e indicates that the carbon price is

Journal Pre-proof Step 3: Construct the carbon price fluctuation network In the process of carbon price volatility convert into a symbol sequences, different time intervals are selected, and the resolution of time series is also different. For the same time series, its corresponding symbol sequence length, the number of the characters Num  R  ， Num  e  ，

Num  D  and the correlation between these characters will be different. We take the weekly

carbon price change (a group of five days) as the modal research object. The above three symbols

of

R, e, D are combined into 5 strings to represent a node, representing the change mode of the week. The weekly fluctuation mode is taken as the node, and the conversion from this week to the next week is taken as the edge in time order to build a relatively independent carbon price

Pr e-

p ro

fluctuation network, the network connection diagram is shown as following.

Fig.1 Schematic diagram of carbon price network construction

2.1.2 Reconstruction of carbon price data

According to the carbon price network construction method, we take a group of five days as a fluctuation mode, and take a day as the sliding step size, the following data matrix can be got from





the carbon price sequence X cp = xcp  t  , t  1, 2,..., N ,

xcp  2 

xcp  M    1 2    X cp X cp xcp  M  4  

al

X cp

 xcp 1    xcp  5  

xcp  6 

X cpM  ，

(3)

urn

Where M is the number of fluctuating modes. From the local structure of the carbon emission price network, the carbon emission price data corresponding to all nodes connected to the target node are extracted, and the carbon price data extracted based on the network topology structure is denoted as EX cp ,

Jo

 excp 1 excp  2   EX cp   excp  5  excp  6  



excp  q    i1 i2    X cp X cp excp  q  4  

X cpiq  ，

(4)



in Eq. (4), q  M , ik  1, M ， k  1, 2,..., q . Denote the partial data of the original carbon price data as SX cp ,

SX cp   xcp  t  ， t  N     N   1,..., N ,    0,1 ,

(5)

Journal Pre-proof where  is an integral function,

 is the scale of the original data. Using the carbon price data

EX cp extracted from the network topology and part of the raw carbon price data SX cp , we can get the new data set of carbon price, denoted as RX cp ,

excp  q 

xcp  N    N   1

excp  q  4  xcp  N    N   L 

xcp  M     xcp  M  4   

（6）

of

 excp 1  RX cp   ex  5   cp

Let

Y , K 

T

i

i

i 1

Pr e-

p ro

2.1.3 Build carbon price prediction model ELM is a machine learning algorithm developed on the basis of feedforward neuron network. Compared with other Machine Learning algorithms, its core feature is that parameters of hidden layer nodes do not need to be adjusted, and in the application, we only need to calculate its output weight. ELM has obvious advantages: high learning efficiency and strong generalization ability, and it is widely used in classification, regression, clustering, feature learning and other problems (Huang et al., 2006; Huang et al., 2012; Huang et al., 2015). be training samples, in which

classes.

T represents samples, K represents

  i gi Yj     i g Wi  Yj  bi   Z j , j  1, 2,..., T , h

h

i 1

(7)

i 1

 where g  x  is activation function, Y j   y j1 , y j 2 ,..., y jn  , Wi   wi1 , wi 2 ,..., win  and bi are

al

the bias of the corresponding the output hidden node i , respectively, and  i   i1 ,  i 2 ,...,  im  are

urn

the weights which connecting the hidden neuron i and output neurons, and Z j are the output corresponding to the input Y j . ELM can be expressed as follows,

min H   K

with

F

,

(8)

Jo

H W1 ,W2 ,...,Wh , b1 , b2 ,..., bh   g W1  Y1  b1     g W1  YT  b1 

   k   g Wh  Y1  bh    1  1 .   ,    , K        g Wh  YT  bh    h   kT  

(9)

In Eq. (9), H represents the hidden layer matrix,  represents the output data. In fact, Eq. (9) is a solution to ˆ  H 1 K . The advantages of ELM algorithm are obvious, i.e., only the output weight is needed, but the disadvantage is that it cannot process the noise time series well.

Journal Pre-proof

p ro

of

By combining the above three steps, we can construct the CPN-ELM prediction model. CPN-ELM can overcome the defects of ELM very well, whose basic framework is shown in Fig 2.

Fig.2. CPN-ELM prediction algorithm

Pr e-

It can be seen from Fig.2, the CPN-ELM prediction framework mainly consists of three parts : (1) mapping carbon price data into carbon price network based on coarse granulation method; (2) Extracting the effective information of the carbon price fluctuation by using the topology structure of carbon price network, and reconstructing the original carbon price data; (3) Adopting ELM model to process the reconstructed carbon price data. 2.2 Evaluation index of prediction model

al

To test the prediction effect of the model, we use MAPE and RMSE as the loss functions to measure the level accuracy,

urn

MAPE 

Jo

where xˆt is the estimated value and

1 N xt  xˆt  N t 1 xt N

  xˆ  x 

RMSE 

t 1

t

(10)

2

t

(11)

N

xt the real value.

The directional accuracy of the model can be measured by the following indicator,

Dstat 

1 N

N

a t 1

t

1,  xt 1  xt  xˆt 1  xt   0 at   0, otherwise

,

(12)

the closer Dstat gets to 1, the more accurate the model is in predicting volatility trends, whereas the closer Dstat gets to 0, the less accurate the model is in predicting volatility trends.

Journal Pre-proof To verify the statistical advantages of the prediction model, we select DM test for verification. DM is a kind of statistics used to measure the difference in prediction accuracy of prediction models (Diebold et al., 2015;Wang et al., 2018). MSPE under test shall not be lower than the null hypothesis of the benchmark model. The following is the definition of DM statistic.

DMS 

(13)

 1 E 2 2 e t ˆ ˆ , , e t  x t  x t  x t  x t V    2 q ,        test      bench     0 D E t 1 q 1

of

where D 

D , VD / E

p ro

q  cov  et , et q  .

3. Data selection and topology structure of carbon price network

Jo

urn

al

Pr e-

The carbon futures price data of the second phase from 13 December 2010 to 31 December 2012 and the third phase from 1 January 2013 to 27 December 2018 of EU ETS were selected as sample data, we show the data in Fig 3 (a). The descriptive statistical characteristics of the selected carbon emission price data are shown in Figs 3 (b), (c) and (d). The average carbon emission price is 15.2047, the variance is 26.1299, present the right distribution (skewness value is 0.4876), the probability density with thin tail characteristics (kurtosis value is 1.7935), while the average carbon price of the third phase is 7.9546, the variance is 17.0440, present the right distribution (skewness value is 2.1219), the probability density with thick tail characteristics (kurtosis value is 6.9114). Compared with the two stages, the carbon price in the third stage is smaller and fluctuates more smoothly.

Fig 3 (a) Carbon price data of EU ETS, (b) the box plot of carbon price in the second stage and the third stage, (c) the probability density of the second stage of EU carbon price, (d) the probability density of the third stage of EU carbon price

Journal Pre-proof

p ro

of

To further compare the carbon emission price in the EU ETS in two different stages, we build the carbon price fluctuation network in the second and third stages respectively based on the method proposed in 2.1. We show the network structure in Fig 4 .

Fig 4 Carbon price network of EU ETS, (a) the second stage, (b) the third stage

Jo

urn

al

Pr e-

According to Fig 4(a), in the second stage of carbon price fluctuation, the probability of the occurrence of character e is 0.0134, the probability of the occurrence of character R is 0.4733, and the probability of the occurrence of character D is 0.5134. We can see that in the second stage the process of carbon emission price fluctuation is mainly the transition between the rising state and the falling state. The probability of stable state is very low (only 0.0134), and the probability of falling state is greater than that of rising state, indicating that the carbon price in this stage presents an irregular fluctuation state, but still has a downward trend on the whole. According to figure 4(b), the probability of occurrence of character e , character R and character D during the carbon price fluctuation of the third stage is 0.0345, 0.5039 and 0.4616 respectively. We can find that the fluctuation of carbon price at this stage is also complicated, and the probability of rising state in the process of carbon price fluctuation at this stage is greater than that of falling state, indicating that the carbon price at this stage, like the second stage, shows an irregular fluctuation state, but still an upward trend on the whole. From the perspective of the overall network structure, the carbon price fluctuation network in the second and third stages presents the characteristics of the number of large nodes is small and the number of small nodes is large, but the carbon price fluctuation network in different stages has different network structure. The number of different modes in the first place, according to the network building method and in theory should be 243 different wave mode, in the second stage in the carbon price network actually appeared 54 different modal, the third stage, in fact, there were 126 different modal, visible in the carbon price fluctuation in the process of a few typical modal can reflect the complex characteristics of the carbon price fluctuations.

4. Empirical analysis of carbon emission price prediction In this section，we choose MATLAB 2017a software for numerical simulation. To illustrate the prediction effect, the carbon emission price data is first divided into three parts. The first part is the second stage of EU ETS from 13 December 2010 to 31 December 2012. The second part is the price data of the third stage of the EU ETS from 1 January 2013 to 27 December 2018. The

Journal Pre-proof third part is the carbon price data of the transition stage between the second stage and the third from 30 July 2012 to 10 May 2013 and 4 April 2012 to 15 January 2013. We use ELM and CPN-ELM built in this manuscript respectively to predict carbon price data in different periods, and made a comparative analysis of the prediction accuracy.

4.1 Carbon price forecast for the second stage of EU ETS

of

The second stage of EU ETS is the emission reduction period, that is, the development period of EU ETS. Within this stage, the price mechanism of carbon emission trading has taken shape, and the price can reflect the market information to some extent. In this stage, we first selected the

p ro

carbon price data from 13 January 2010 to 26 August 2011 as the training sample, and selected the carbon price data from 29 August 2011 to 23 September 2011 as the test data for prediction. We use MATLAB simulation for ten times, and take the average value to get the prediction results, as shown in Fig 5 (a). To verify the applicability of the prediction model to the randomly selected carbon price data, we take the time window of 200 and the sliding step length of 1, and let time window traverse the carbon price data of the whole EU carbon trading market in the second stage,

Pr e-

then a total of 325 time windows can be obtained. In each time window, the first 180 carbon emission price are selected as the training set, and the last 20 carbon emission price are selected as the test set. ELM and CPN-ELM are used to predict and calculate the accuracy indicators MAPE, RMSE and Dstat in every time window respectively. We show the results in Figs 5 (b), ( c) and (d),

Jo

urn

al

and the mean values are shown in Tab 1.

Fig 5 (a) The carbon emission price forecast during 29 August 2011 to 23 September 2011, (b) MAPE within each time window, (c) RMSE within each time window, (d) Dstat within each time window

As can be seen from Fig 5 (a) and Tab 1, both ELM and CPN-ELM prediction models have good prediction ability for the carbon price data during the period from 29 August 2011 to 23

Journal Pre-proof September 2011. The comparison between the two shows that the CPN-ELM prediction model is superior to the ELM prediction model in both level accuracy and directional accuracy, and CPN-ELM is superior to the ELM especially in the direction of carbon price fluctuation. The core reason is that we use the carbon price fluctuation network to extract the information of carbon price fluctuation effectively. The following conclusions can be drawn from Figs 5 (b), (c), (d) and Tab 1. From the aspect of MAPE accuracy index, the prediction accuracy of CPN-ELM model in

of

the 63.38% time window is better than that of ELM. At the same time, the CPN-ELM model's mean value of MAPE index in all time windows is 0.0206, which was less than the mean value of ELM prediction model of 0.0229. In terms of RMSE accuracy index, CPN-ELM model is better

p ro

than ELM model in the time window of 53.85%. And meanwhile, the mean value of RMSE index in all time windows of CPN-ELM model is 0.3102, which is less than the mean value of ELM model of 0.3417. For the aspect of directivity accuracy index, the accuracy of CPN-ELM model in the time window of 69.23% is better than that of ELM prediction model, and the mean value of MAPE index of CPN-ELM model in all time windows is 0.5292, which is greater than the mean value of ELM model of 0.4982. It can be seen that within the second stage of EU ETS, for

Pr e-

randomly selected carbon price data, CPN-ELM is better than ELM in both level and directional accuracy. However, from the aspect of prediction time consuming, time required for CPN-ELM model is about 4 to 5 times larger than ELM, the core reason is that the CPN-ELM needs to spend time to extract effective information from the original data of carbon price and reconstruct the data, but the CPN-ELM prediction model’s problem with long time consuming can be solved by improving the computer performance.

al

4.2 Carbon price forecast for the third stage of the EU ETS The EU ETS in this stage is the mature development stage of emission reduction, the salient

urn

feature of this stage is to summarize the development problems of the first two stages. In terms of allocation method, innovative baseline allocation method is adopted for each transaction subject, and the original free allocation method is gradually transferred to auction form. Meanwhile, EU ETS monitoring, reporting and verification，MRAV are regulated. At this stage, the carbon price can effectively reflect the market information. In this stage, we first select the carbon price data from 2 January 2013 to 12 September 2013 as the training sample, and the carbon price data from

Jo

13 September 2013 to 10 October 2013 as the test data for prediction, and the predicted results shown in Fig 6 (a). To test the applicability of CPN-ELM to the carbon price data randomly selected in the third stage, we take the time window of 200 and the sliding step length of 1, and let time window traverse the carbon price data within the third stage of the carbon market, a total of 1336 time Windows can be obtained. In each time window, the first 180 carbon emission price are selected as the training set, and the last 20 carbon price are selected as the test set. ELM and CPN-ELM are used to predict and calculate the precision indicators MAPE, RMSE and Dstat in every window. We show the results in Figs 6 (b), (c) and (d), and the mean values shown in

Journal Pre-proof

p ro

of

Table 1.

Pr e-

Fig 6 (a) the carbon price forecast from 13 September 2013 to 10 October 2013, (b) MAPE within each time window, (c) RMSE within each time window, (d) Dstat within each time window

Jo

urn

al

From Fig 6 (a) and Table 1, for the carbon emission price data from 13 September 2013 to 10 October 2013, the CPN-ELM prediction model is superior to the ELM prediction model in both level and directional accuracy. The following conclusions can be drawn from Figs 6 (b), (c), (d) and Table 1. From the aspect of MAPE precision index, the prediction accuracy of CPN-ELM model in the time window of 68.11% is better than that of ELM model. Meanwhile, the average value of MAPE index in all time windows of CPN-ELM model is 0.0199, which is less than the average value of 0.0236 of ELM model. In terms of RMSE accuracy index, CPN-ELM model is better than ELM model in 59.96% of time windows according to prediction accuracy. Meanwhile, the mean value of RMSE index in all time windows of CPN-ELM model is 0.2089, which is less than the mean value of ELM model, which is 0.2479. For the aspect of directivity accuracy index, the accuracy of CPN-ELM model in the time window of 73.20% was better than that of ELM model, and the mean value of MAPE index of CPN-ELM model in all time windows is 0.5333, which is greater than the mean value of ELM model of 0.4916. It can be seen that in the third stage of EU carbon emission market, for randomly selected carbon price data, CPN-ELM is better than ELM in both horizontal and directional accuracy. Comparing the carbon price forecast results of the third stage and the second stage, we can see that the prediction accuracy of carbon price of CPN-ELM prediction model for the third stage is higher than that for the second stage. The main reason is that the carbon market in the third stage is more mature, and the carbon emission price can reflect the carbon market information more effectively. Therefore, we can use the carbon emission price network to extract more effective data information, thus improving the prediction ability of CPN-ELM. 4.3 Carbon price prediction for the transitional phase of EU ETS

Journal Pre-proof

p ro

of

To test the predictive ability of the model for data with structural mutations, we select carbon price data from 30 July 2012 to 12 April 2013 as the training data. At this time, the training sample set contains the carbon price data of the second stage and the third stage. The carbon price data from 15 April 2013 to 10 May 2013 are selected as the prediction test set, and the prediction results as shown in Fig 7 (a) and Table 1. Data from 4 April 2012 to 13 December 2012 are selected as training data, and data from 14 December 2012 to 15 January 2013 are selected as test data. At this time, the carbon price data of the second stage and the third stage are included in the prediction data. We show the prediction results in Fig 7 (b) and Table 1.

Fig 7 (a) Carbon price forecast from 15 April 2013 to 10 May 2013, (b) Carbon price forecast from 14 December

Pr e-

2012 to 15 January 2013

urn

al

From Fig 7 (a) and Table 1, for the CPN-ELM model, the level prediction accuracy indexes MAPE and RMSE of carbon price data from 15 April 2013 to 10 May 2010 are 0.0673 and 0.4510 respectively, and the directional accuracy index Dstat is 0.65, both higher than the prediction effect of ELM model. It shows that for the training sample set, which contains carbon price data of the second stage and the third stage, the CPN-ELM is stronger than the ELM. From Fig 7 (b) and Table 1, for the CPN-ELM model, the level prediction accuracy indexes MAPE and RMSE of carbon price data from 14 December 2012 to 15 January 2013 are 0.0198 and 0.2070 respectively, and the directional accuracy index Dstat is 0.65, both higher than the prediction effect of ELM model. It shows that for the testing sample set, which contains both the carbon price data of the second stage and the third stage, and the CPN-ELM is also stronger than the ELM. Therefore, it can be seen that for data with structural mutations (whether in training set or test set), the accuracy of CPN-ELM model is higher than that of ELM. However, for the prediction results of carbon emission price data in the second and third stages, we can see that the prediction accuracy of ELM and CPN-ELM models is lower than that of the carbon emission price in second and third stages, indicating that the structural mutation of carbon market will affect the prediction effect of the model. Table 1．Prediction accuracy of ELM and CPN-ELM in different periods

Testing

Jo

Training

MAPE

RMSE

Dstat

Time(s)

2010/12/13 -2011/8/26

2011/8/29 -2011/9/23

ELM

0.0138

0.2966

0.5

1.563

CPN-ELM

0.0129

0.2889

0.65

6.253

2013/01/02 -2013/09/12

2013/09/13 -2013/10/10

ELM

0.0193

0.1626

0.6

1.079

CPN-ELM

0.0147

0.1329

0.8

4.641

2012/07/30 -2013/04/12

2013/04/15 -2013/05/10

ELM

0.1433

0.6921

0.55

1.422

CPN-ELM

0.0673

0.4510

0.65

5.376

2012/04/04 -2012/12/13

2012/12/14 -2013/01/15

ELM

0.0273

0.2682

0.5

1.203

CPN-ELM

0.0198

0.2070

0.65

5.329

Journal Pre-proof

4.4 Diebold-Mariano (DM) test

p ro

of

To illustrate the superiority of the model we built from a statistical sense, we use DM test to test the prediction effect of the model in this part. By using Formula (13) to calculate the value of DMS, and we calculate the corresponding p value, and then we use these two indicators to test the model. We show the calculated results in Table 2, where the p value is bold in parentheses. We can see from the Table 2, in three different periods, p values are less than 5%, which shows that in three different periods, the prediction effect of CPN-ELM prediction model on carbon price is significantly better than ELM. At the same time, it also shows that the carbon emission price fluctuation information extracted from the carbon emission price network is used to reconstruct the carbon emission price data, which can effectively improve the prediction effect of the model at the 95% confidence level. The above DM test results show that CPN-ELM is statistically superior to ELM prediction model. Table.2 DM test results for CPN-ELM and ELM

Reference model

CPN-ELM

5. Discussion and conclusion

Second stage ELM

Third stage ELM

Transition stage ELM

-2.16810 (0.03640)

-2.73311 (0.01160)

-2.65390 (0.01284)

Pr e-

Tested model

urn

al

Based on complex network and ELM, this paper proposes a new carbon price forecasting model, its core idea is that the carbon price is first mapped to the carbon price complex networks, then use the carbon price network topology extraction carbon price fluctuations of effective information. With the reconstructed the carbon price data and the extreme learning machine CPN-ELM model is constructed. This model is a new kind of combined prediction model which combines time series complex network analysis technology with artificial intelligence algorithm.

Jo

We selected the carbon emission price data of the second, third and transition stages of the EU ETS for empirical analysis. In different stages of the EU ETS, carbon emission price during the second stage from 29 August 2011 to 23 September 2011, carbon price during the third stage from 13 September 2013 to 10 October 2013, and carbon price during the transition stage from 15 April 2013 to 10 May 2013 and 14 December 2012 to 15 January 2013 were predicted and analyzed. It was found that the level prediction accuracy and direction prediction accuracy of CPN-ELM model were higher than those of ELM model. However, CPN-ELM prediction model takes longer time than ELM model. We point out that in the era of rapid development of computer computing technology, Time-consuming problem of CPN-ELM model is not the core problem of carbon price prediction, and this problem can be solved by improving computer performance. To further verify the prediction ability of CPN-ELM model for randomly selected carbon price data, we constructed 325 and 1336 time windows in the second and third stages of the

Journal Pre-proof

p ro

of

development of the EU ETS. In each time window, we predict the carbon price. The study find that in the second stage of the EU ETS, the accuracy indexes MAPE, RMSE and Dstat of the CPN-ELM model for carbon price are 63.38%, 53.85% and 69.23% respectively, better than the ELM model, and the average accuracy of CPN-ELM model are 10.04%, 9.22% and 6.22% higher than that of ELM prediction model. In the third stage of the EU ETS, the accuracy indexes MAPE, RMSE and Dstat of the CPN-ELM model for carbon price prediction are better than the ELM model within the time Windows of 68.11%, 59.96% and 73.20% respectively, and the average accuracy of the CPN-ELM is higher than the ELM by 15.68%, 15.73% and 8.48% respectively. In the transition stage of EU carbon market, the average accuracy of CPN-ELM model is higher than 48.94%, 31.48% and 23.81% respectively. Therefore, the CPN-ELM model is superior to pure ELM model in both horizontal accuracy and directional accuracy. It can improve the predictive accuracy of ELM model effectively, and meanwhile, CPN-ELM model has better robustness when facing the random samples、sample data with different frequencies or sample data with structural changes.

Acknowledgements

Pr e-

The idea of building CPN-ELM model can be applied to the prediction of time series data in other fields. From the characteristics of the studied data, the corresponding data fluctuation network can be built, and the network topology can be used to extract the effective information implied by the studied data. Recently, link prediction has been widely used in network [Liben et al.,2007; Lü et al.,2011; Zhang et al.,2017]. How to combine link prediction methods with artificial intelligence algorithms to build a more accurate prediction model is worth further research.

References

urn

al

The Research was supported by the following foundations: The Philosophy and Social Science Research Project in Colleges and Universities of Jiangsu Province (2018SJA2123), The National Natural Science Foundation of China (71503132, 71811520710, 11571142), Qing Lan Project of Jiangsu Province (2017), Six talent peaks project in Jiangsu Province (JY-055).

Jo

An Haizhong, Xiangyun Gao, Wei Fang, Xuan Huang,Yinghui Ding. The role of fluctuating modes of autocorrelation in crude oil prices. Physica A,2014; 393:382-390 Atsalakis G S. Using computational intelligence to forecast carbon prices. Applied Soft Computing, 2016, 43: 107-116. Bezsudnov I V, Snarskii A A. From the time series to the complex networks: The parametric natural visibility graph. Physica A: Statistical Mechanics and its Applications, 2014, 414: 53-60. Chen H, Tian L, Wang M, et al. Analysis of the dynamic evolutionary behavior of American heating oil spot and futures price fluctuation networks. Sustainability, 2017, 9(4): 574. Chen L, Qiao Z, Wang M, et al. Which artificial intelligence algorithm better predicts the Chinese stock market? IEEE Access.2018,6:48625-48633.

Journal Pre-proof

Jo

urn

al

Pr e-

p ro

of

Chen W D,Xu H,Guo Q. Dynamic analysis on the topological properties of the complex network of international oil prices. ACTA PHYSICA SINICA, 2010 (7): 4514-4523. Chevallier J. Nonparametric modeling of carbon prices. Energy Economics, 2011, 33(6): 1267-1282. Diebold F X. Comparing predictive accuracy, twenty years later: A personal perspective on the use and abuse of Diebold–Mariano tests. Journal of Business & Economic Statistics, 2015, 33(1): 1-1. Donner R V, Zou Y, Donges J F, et al. Recurrence networks—a novel paradigm for nonlinear time series analysis. New Journal of Physics, 2010, 12(3): 033025. Fan X, Li S, Tian L. Chaotic characteristic identification for carbon price and an multi-layer perceptron network prediction model. Expert Systems with Applications, 2015, 42(8): 3945-3952. Gao Z K, Small M, Kurths J. Complex network analysis of time series. EPL (Europhysics Letters), 2017, 116(5): 50001. Koop G, Tole L. Forecasting the European carbon market. Journal of the Royal Statistical Society: Series A (Statistics in Society), 2013, 176(3): 723-741. Lacasa L, Luque B, Ballesteros F, et al. From time series to complex networks: The visibility graph. Proceedings of the National Academy of Sciences, 2008, 105(13): 4972-4975. Li X, Sun M, Gao C, et al. The parametric modified limited penetrable visibility graph for constructing complex networks from time series. Physica A: Statistical Mechanics and its Applications, 2018, 492: 1097-1106. Liben‐Nowell D, Kleinberg J. The link‐prediction problem for social networks. Journal of the American society for information science and technology, 2007, 58(7): 1019-1031. Luque B, Lacasa L, Ballesteros F, et al. Horizontal visibility graphs: Exact results for random time series. Physical Review E, 2009, 80(4): 046103. Lü L, Zhou T. Link prediction in complex networks: A survey. Physica A: statistical mechanics and its applications, 2011, 390(6): 1150-1170. Segnon M, Lux T, Gupta R. Modeling and forecasting the volatility of carbon dioxide emission allowance prices: A review and comparison of modern volatility models. Renewable and Sustainable Energy Reviews, 2017, 69: 692-704. Sun G, Chen T, Wei Z, et al. A carbon price forecasting model based on variational mode decomposition and spiking neural networks. Energies, 2016, 9(1): 54. Xu X, Zhang J, Small M. Superfamily phenomena and motifs of networks induced from time series. Proceedings of the National Academy of Sciences, 2008, 105(50): 19601-19605. Zhang J, Li D, Hao Y, et al. A hybrid model using signal processing technology, econometric models and neural network for carbon spot price forecasting. Journal of Cleaner Production, 2018, 204: 958-964. Zhang J, Small M. Complex network from pseudoperiodic time series: Topology versus dynamics. Physical review letters, 2006, 96(23): 238701. Zhang R, Ashuri B, Deng Y. A novel method for forecasting time series based on fuzzy logic and visibility graph. Advances in Data Analysis and Classification, 2017, 11(4): 759-783. Zhang X, Chen M.Y., Wang M.G., et al. A novel hybrid approach to Baltic Dry Index forecasting based on a combined dynamic fluctuation network and artificial intelligence method. Applied Mathematics and Computation, 2019,361:499-516.

Journal Pre-proof

Jo

urn

al

Pr e-

p ro

of

Zhao X, Han M, Ding L, et al. Usefulness of economic and energy data at different frequencies for carbon price forecasting in the EU ETS. Applied Energy, 2018, 216: 132-141. Zhu B, Han D, Wang P, et al. Forecasting carbon price using empirical mode decomposition and evolutionary least squares support vector regression. Applied energy, 2017, 191: 521-530. Zhu B, Wei Y. Carbon price forecasting with a novel hybrid ARIMA and least squares support vector machines methodology. Omega, 2013, 41(3): 517-524. Zhu B, Ye S, Wang P, et al. A novel multiscale nonlinear ensemble leaning paradigm for carbon price forecasting. Energy Economics, 2018, 70: 143-157. Wang C, Zhang X.Y., Wang M.G., et al. Predictive analytics of the copper spot price by utilizing complex network and artificial neural network techniques. Resources Policy, 2019,63: 101414. Wang M, Vilela A L M, Du R, et al. Exact results of the limited penetrable horizontal visibility graph associated to random time series and its application. Scientific reports, 2018, 8(1): 5130. Wang M, Tian L, Xu H, et al. Systemic risk and spatiotemporal dynamics of the consumer market of China. Physica A: Statistical Mechanics and its Applications, 2017. Wang M, Tian L. From time series to complex networks: The phase space coarse graining. Physica A: Statistical Mechanics and its Applications, 2016, 461: 456-468. Wang M, Tian L, Du R. Research on the interaction patterns among the global crude oil import dependency countries: A complex network approach. Applied Energy, 2016, 180: 779-791. Wang M, Chen Y, Tian L, et al. Fluctuation behavior analysis of international crude oil and gasoline price based on complex network perspective. Applied Energy, 2016, 175: 109-127. Wang M, Zhao L, Du R, et al. A novel hybrid method of forecasting crude oil prices using complex network science and artificial intelligence algorithms. Applied energy, 2018, 220: 480-495. Wang M, Tian L, Zhou P. A novel approach for oil price forecasting based on data fluctuation network. Energy Economics, 2018, 71: 201-212. Huang G B, Zhu Q Y, Siew C K. Extreme learning machine: theory and applications. Neurocomputing, 2006, 70(1-3): 489-501. Huang G B, Zhou H, Ding X, et al. Extreme learning machine for regression and multiclass classification. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 2012, 42(2): 513-529. Huang G, Huang G B, Song S, et al. Trends in extreme learning machines: A review. Neural Networks, 2015, 61: 32-48.