The quantification of the uncertainty in crude oil price is of significance to improve the related financial decision-making. However, studies in this field have remained limited because the nonlinearity inherent in the crude oil price makes it challenging to model its uncertainty. In this paper, a novel learning system of ensemble probabilistic prediction combining five popular machine learning methods and an improved optimizer is presented to effectively model the uncertainty in crude oil price and establish the corresponding prediction interval with satisfactory reliability and resolution. An improved grey wolf optimizer based on the adaptive Cuckoo search algorithm (AGWOCS) is proposed in the learning system to integrate the prediction intervals produced by the above machine learning methods. In addition, the superiority of the proposed AGWOCS is validated based on an algorithm test, compared to three benchmark optimizers. To validate the effectiveness of the proposed learning system, the uncertainties in daily and weekly Europe Brent spot prices are modeled as a case study. The evaluation results based on the reliability, resolution, and sharpness demonstrate that the proposed learning system can yield the prediction interval with a higher quality, which has distinct advantages over eight benchmarks as a whole. The convergence and scalability of the learning system are also investigated, which reveals its feasibility.

量化原油价格的不确定性对于改善相关的财务决策具有重要意义。但是，该领域的研究仍然有限，因为原油价格固有的非线性特性使其建模不确定性具有挑战性。本文提出了一种新颖的集成概率预测学习系统，该系统结合了五种流行的机器学习方法和改进的优化器，可以有效地对原油价格的不确定性进行建模，并以令人满意的可靠性和分辨率建立相应的预测区间。在学习系统中，提出了一种基于自适应杜鹃搜索算法（AGWOCS）的改进的灰狼优化器，以整合上述机器学习方法产生的预测间隔。此外，与三个基准优化程序相比，基于算法测试验证了所提出的AGWOCS的优越性。为了验证所提议的学习系统的有效性，以案例为模型对每日和每周欧洲布伦特原油现货价格的不确定性进行建模。基于可靠性，分辨率和清晰度的评估结果表明，所提出的学习系统可以产生更高质量的预测区间，相对于整个八个基准具有明显的优势。研究了学习系统的收敛性和可扩展性，揭示了其可行性。欧洲布伦特原油现货价格的每日和每周不确定性均以案例研究为模型。基于可靠性，分辨率和清晰度的评估结果表明，所提出的学习系统可以产生更高质量的预测区间，相对于整个八个基准具有明显的优势。研究了学习系统的收敛性和可扩展性，揭示了其可行性。欧洲布伦特原油现货价格的每日和每周不确定性均以案例研究为模型。基于可靠性，分辨率和清晰度的评估结果表明，所提出的学习系统可以产生更高质量的预测区间，相对于整个八个基准具有明显的优势。研究了学习系统的收敛性和可扩展性，揭示了其可行性。