Analysis and forecasting of the carbon price using multi—resolution singular value decomposition and extreme learning machine optimized by adaptive whale optimization algorithm

Highlights: • The adaptive weight strengthens search ability of whale optimization algorithm. • The carbon price is decomposed via multi-resolution singular value decomposition. • An improved extreme learning machine is built to predict the carbon price. • The proposed approach can be applied in China and EU ETS simultaneously. Precise prediction of carbon prices by means of single forecasting models may be difficult due to the inherent non-stationary and nonlinearity characteristics of the carbon price. This paper i proposes an innovative hybrid model for predicting the carbon price. The prediction was made through the extreme learning machine optimized by the adaptive whale optimization algorithm based on the multi-resolution singular value decomposition. The multi-resolution singular value decomposition was used to eliminate the high frequency components of the previous carbon price data. Afterwards, the carbon price was successfully decomposed into two time series—the approximation series and the detailed series. The partial auto-correlation function was employed in the approximation series for determining the input variables of the extreme learning machine. The adaptive whale optimization algorithm was utilized to optimize both the input weight matrix and the bias matrix to improve the robustness and accuracy of extreme learning machines. The empirical simulation based on four diverse types of carbon prices under carbon trading pilot programs in China found that the proposed model outperformed the other benchmark methods. Four different matured carbon future prices under the European Union national emissions trading scheme (EU ETS) were also selected for forecasting. The results showed that the proposed model performed fairly well in forecasting the EU carbon price.