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To effectively solve the environmental pollution caused by greenhouse gases, countries around the world have successively set up carbon emission trading markets in recent years. At present, China's carbon trading market is only in its infancy, making various mechanisms and policies insufficient. As the core issue of carbon trading market, the fluctuations of carbon trading price are related to the investment decisions of market participants, the formulation of enterprise production and operation plans, and the realization of global emission reduction targets. Therefore, it is of great practical significance to study the influencing factors of carbon price fluctuations and to predict future carbon prices. In this paper, we take the carbon trading week data of the Beijing carbon trading market in China from May to September 2020 as the research object. Firstly, the gray correlation technique is employed to measure the rationality of the selected factors affecting carbon price fluctuations. Secondly, through the principal component analysis method to analyze the various influencing factors, it is found that energy prices and macroeconomic development are the main factors affecting carbon prices, and weather conditions will also cause carbon price fluctuations. Then, four different models are proposed to predict carbon trading prices, and the forecast results are evaluated through the performance evaluation index system. The results show that the GA-ELM model has the best prognosis effect. Finally, according to the analysis results, it provides a useful theoretical reference for carbon market decision makers and participants. [ABSTRACT FROM AUTHOR]