As the complexity of the system increases, real numbers often fail to capture the inherent uncertainty in the data. However, time delay and nonlinear interdependence pose further challenges to accurate modeling. To address these issues, this paper thoroughly investigates the time-delayed and nonlinear effects of related factor sequences on system behaviour sequences within the interval sequence framework, and proposes an interval grey BP neural network combination model with time-delay causal term, named BP-TDIGM(1,1). Firstly, considering the uncertainty and time-delay characteristics of data, this study integrates model and introduces a time-delay parameter, extending the traditional time-delay prediction model to an interval time-delay prediction model. The Kramer method is then employed to solve the proposed model. Secondly, combined with the BP neural network, a serial grey BP neural network hybrid model is constructed to characterize the nonlinear impacts of related factor sequences on system behaviour sequences. Subsequently, the least squares method and particle swarm optimization algorithm are utilized to determine the structural parameters and time-delay parameters of the model, respectively. Finally, the effectiveness of BP-TDIGM(1,1) is verified through fitting and prediction experiments on two numerical examples and a practical case concerning R&D expenditure in the Yangtze River Delta. The results are compared with six other grey prediction models, demonstrating that the proposed model outperforms other grey models in terms of prediction accuracy for multivariable interval time-delay data series.

Grey prediction, Interval grey number, Time delay effect, BP neural network, R&, D expenditure prediction