Background: This study addresses the challenge of predicting data sequences characterized by a mix of partial linearity and partial nonlinearity. Traditional forecasting models often struggle to accurately capture the complex patterns of change within the data. Methods: To this end, this study introduces a novel polynomial-driven discrete grey power model (PFDPGM(1,1)) that includes time perturbation parameters, enabling a flexible representation of complex variation patterns in the data. The model aims to determine the accumulation order, nonlinear power exponent, time perturbation parameter, and polynomial degree to minimize the fitting error under various criteria. The estimation of unknown parameters is carried out by leveraging a hybrid optimization algorithm, which integrates Particle Swarm Optimization (PSO) and the Grey Wolf Optimization (GWO) algorithm. Results: To validate the effectiveness of the proposed model, the annual total renewable energy consumption in the BRICS countries is used as a case study. The results demonstrate that the newly constructed polynomial-driven discrete grey power model can adaptively fit and accurately predict data series with diverse trend change characteristics. Conclusions: This study has achieved a significant breakthrough by successfully developing a new forecasting model. This model is capable of handling data sequences with mixed trends effectively. As a result, it provides a new tool for predicting complex data change patterns.

The purpose of this study is to construct a gray relational model based on information diffusion to avoid rank reversal when the available decision information is insufficient, or the decision objects vary. Considering that the sample dependence of the ideal sequence selection in gray relational decision-making is based on case sampling, which causes the phenomenon of rank reversal, this study designs an ideal point diffusion method based on the development trend and distribution skewness of the sample information. In this method, a gray relational model for sample classification is constructed using a virtual-ideal sequence. Subsequently, an optimization model is established to obtain the criteria weights and classification radius values that minimize the deviation between the comprehensive relational degree of the classification object and the critical value. The rank-reversal problem in gray relational models could drive decision-makers away from using this method. The results of this study demonstrate that the proposed gray relational model based on information diffusion and virtual-ideal sequencing can effectively avoid rank reversal. The method is applied to classify 31 brownfield redevelopment projects based on available interval gray information. The case analysis verifies the rationality and feasibility of the model. This study proposes a robust method for ideal point choice when the decision information is limited or dynamic. This method can reduce the influence of ideal sequence changes in gray relational models on decision-making results considerably better than other approaches.

How to upgrade the performance and suitability of the forecasting method and reduce the modelling error for nonlinear small sample data series has been one of the most key problems in grey system theoretical exploration. The main work of this study, a discrete grey power model is formulated that can not only simulate the development of exponential and power function systems but can also reflect the interaction between them. The new model not only ensures that the more recent information is given some priority but exhibits a lower simulation error and greater prediction accuracy. The findings indicated that the presented model minimizes the average relative error in theory, with better adaptability than previous models in simulation and prediction. The case study demonstrates that the new model is more appropriate for predicting small sample data than the previous grey power model. This research addresses limitations in existing grey models and expands the grey model system.