Generalized fractional grey system models: The memory effects perspective.

Area of Science:

• Mathematical Modeling
• Time Series Analysis
• Grey System Theory

Background:

• Fractional-order grey models offer advantages over integer-order models for uncertain, limited data.
• Existing fractional grey models have limitations, necessitating further improvements.

Purpose of the Study:

• To develop a more flexible and general fractional grey model structure.
• To introduce the generalized fractional grey model (GFGM(1,1)) incorporating generalized fractional-order derivatives (GFOD).
• To analyze the GFGM(1,1) model's mechanism, parameter estimation, and time response.

Main Methods:

• Incorporation of generalized fractional-order derivatives (GFOD) into the grey model framework.
• Analysis of model parameter estimation and time response functions.
• Application of four metaheuristic algorithms to optimize fractional orders.
• Simulation studies and a real-world case study for validation.

Main Results:

• The proposed GFGM(1,1) model demonstrates enhanced performance compared to existing benchmarks.
• The GFOD effectively captures memory effects, improving model flexibility.
• Metaheuristic algorithms successfully optimized fractional orders for better accuracy.
• The model proved applicable and advantageous in both simulation and real-world scenarios.

Conclusions:

• The generalized fractional grey model (GFGM(1,1)) represents a significant advancement in grey system modeling.
• The GFGM(1,1) offers superior performance for time series forecasting with limited and uncertain data.
• The integration of GFOD and metaheuristic optimization enhances the model's applicability and predictive power.