  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.
