Nonlinear grey Bernoulli model with physics-preserving Cusum operator

The Cusum (cumulative sum) operator is a fundamental prerequisite for the nonlinear grey Bernoulli model. Traditionally, it is believed to visually identify the underlying dynamic pattern of the original time series. This paper presents the misconceptions concerning the Cusum operation and the over-optimization of the initial condition in the classical nonlinear grey Bernoulli model, both of which inspire the proposal of a physics -preserving Cusum operator. Under a state space framework, separable nonlinear least squares and nonlinear least squares are formulated to generate simultaneous estimates of structural parameters and initial condition. By combining the Cusum operators and parameter estimation methods, four modeling paradigms are generated and comprehensively compared. The simulation results show that (i) the physics-preserving Cusum outperforms the traditional Cusum and (ii) nonlinear least squares outperforms separable nonlinear least squares, especially in irregular sampling settings with large time intervals and high noise levels. Finally, the proposed approach is used to identify the underlying dynamics from short-term traffic flow data, and the results validate its effectiveness.