  Since most of the research about grey forecasting models is focused on
developing novel models and improving accuracy, relatively limited attention
has been paid to the modelling mechanism and relationships among diverse kinds
of models. This paper aims to unify and reconstruct continuous-time grey
models, highlighting the differences and similarities among different models.
First, the unified form of grey models is proposed and simplified into a
reduced-order ordinary differential equation. Then, the integral matching that
consists of integral transformation and least squares, is proposed to estimate
the structural parameter and initial value simultaneously. The cumulative sum
operator, an essential element in grey modelling, proves to be the discrete
approximation of the integral transformation formula. Next, grey models are
reconstructed by the integral matching-based ordinary differential equations.
Finally, the existing grey models are compared with the reconstructed models
through extensive simulation studies, and a real-world example shows how to
apply and further verify the reconstructed model.
