Forecasting carbon futures price: a hybrid method incorporating fuzzy entropy and extreme learning machine

Abstract

In this paper, we propose a novel hybrid model that extends prior work involving ensemble empirical mode decomposition (EEMD) by using fuzzy entropy and extreme learning machine (ELM) methods. We demonstrate this 3-stage model by applying it to forecast carbon futures prices which are characterized by chaos and complexity. First, we employ the EEMD method to decompose carbon futures prices into a couple of intrinsic mode functions (IMFs) and one residue. Second, the fuzzy entropy and K-means clustering methods are used to reconstruct the IMFs and the residue to obtain three reconstructed components, specifically a high frequency series, a low frequency series, and a trend series. Third, the ARMA model is implemented for the stationary high and low frequency series, while the extreme learning machine (ELM) model is utilized for the non-stationary trend series. Finally, all the component forecasts are aggregated to form final forecasts of the carbon price for each model. The empirical results show that the proposed reconstruction algorithm can bring more than 40% improvement in prediction accuracy compared to the traditional fine-to-coarse reconstruction algorithm under the same forecasting framework. The hybrid forecasting model proposed in this paper also well captures the direction of the price changes, with strong and robust forecasting ability, which is significantly better than the single forecasting models and the other hybrid forecasting models.

Keywords: ARMA; Carbon futures price; EEMD; Extreme learning machine; Fuzzy entropy; K-means clustering method.

Fig. 1

The network of extreme learning machine (ELM)

Fig. 2

The overall framework of the proposed hybrid model

Fig. 3

The EEMD decomposition of carbon futures prices

Fig. 4

The fuzzy entropy of the IMFs and the residue

Fig. 5

The reconstructed high frequency, low frequency and trend series

Fig. 6

Out-of-sample forecasting result of high frequency series

Fig. 7

Out-of-sample forecasting result of low frequency series

Fig. 8

The RMSE of the cross-validation and grid search methods

Fig. 9

Out-of-sample forecasting result of trend frequency series

Fig. 10

Actual price and forecasting prices for Dec16 and Dec17

Fig. 11

The direction prediction of high frequency series under different reconstruction algorithms Notes: “Proposed” refers to the proposed reconstruction algorithm, using the Fuzzy Entropy and K-means clustering methods in the paper; "FTC" represents the traditional fine-to-coarse reconstruction algorithm

Fig. 12

The forecasting results of different models across different testing sample sizes

Fig. 13

Actual price and forecasting prices for Dec20

Fig. 14

Time-varying cross-validation method

Fig. 15

The closing price of the Dec16 and Dec17