An IDE-based nonlinear grey Bernoulli model and applications to daily traffic flow pattern identification

Abstract

The nonlinear grey Bernoulli model is a powerful tool for modelling and forecasting time series exhibiting a rough inverted U-shape behaviour. Traditionally, it is considered an indirect dynamical model, informed by empirical knowledge that the accumulation of the inverted U-shape series leads to a growth behaviour, which is subsequently characterized by the Bernoulli ordinary differential equation (ODE). However, its mechanism remains incomplete, particularly concerning the modelling paradigm and parameter estimation. To this end, we propose an equivalent and simplified variant in the form of an integro-differential equation (IDE), enabling direct modelling of the inverted U-shaped series and thereby enhancing the interpretability. By taking the IDE as a state equation in the state space framework, separable nonlinear least squares (SLS) and nonlinear least squares (NLS) are respectively formulated to simultaneously estimate structural parameters and the initial condition. Then, by combining the state equation (ODE/IDE) and parameter estimation approach (SLS/NLS), the generated ODE-based indirect paradigms are comprehensively compared with the IDE-based direct ones. Numerical simulations show that (i) IDE-based models outperform ODE-based ones and (ii) NLS outperforms SLS. Real-world applications indicate that the nonlinear grey Bernoulli model combining IDE and NLS can accurately discover the underlying dynamics of daily traffic flow, outperforming the competitive models.