Nonlinear grey system models, serving to time series forecasting, are extensively used in diverse areas of science and engineering. However, most research concerns improving classical models and developing novel models, relatively limited attention has been paid to the relationship among diverse models and the modelling mechanism. The current paper proposes a unified framework and reconstructs the unified model from an integro-differential equation perspective. First, we propose a methodological framework that subsumes various nonlinear grey system models as special cases, providing a cumulative sum series-orientated modelling paradigm. Then, by introducing an integral operator, the unified model is reduced to an equivalent integro-differential equation; on this basis, the structural parameters and initial value are estimated simultaneously via the integral matching approach. The modelling procedure comparison further indicates that the integral matching-based integro-differential equation provides a direct modelling paradigm. Next, large-scale Monte Carlo simulations are conducted to compare the finite sample performance, and the results show that the reduced model has higher accuracy and robustness to noise. Applications of forecasting the municipal sewage discharge and water consumption in the Yangtze River Delta of China further illustrate the effectiveness of the reconstructed nonlinear grey models.

翻译：用于时间序列预测的非线性灰色系统模型广泛用于科学和工程的不同领域,然而,大多数研究都涉及改进古典模型和开发新型模型,对不同模型和建模机制之间的关系的注意相对有限。本文件提议了一个统一框架,并从内地差异方程的角度对统一模型进行重新构建。首先,我们提出一个方法框架,将各种非线性灰色系统模型作为特例进行分解,提供一个累积和一系列的建模模式。然后,通过引入一个综合操作者,统一模型被减少到一个等同的内地差异方程式;在此基础上,结构参数和初始值通过综合匹配方法同时估算。建模程序比较进一步表明,基于内地差异方程的一体化匹配式模型提供了直接建模模式。接下来,将大规模蒙特卡洛模拟用于比较有限的样本性能,结果显示,降低的模型对噪音的准确性和坚固度更高。中国长江三角市污水排放和水消费量的预测应用进一步说明非线性灰色模型的实效。