A novel fractional grey system model and its application

• Extending from GM(1,1) to FGM(q, 1) via fractional accumulated generating matrix.
• Decomposing the parameter estimation matrix and discussing the relationship among GM(1,1), FAGM(1,1) and FGM(q, 1).
• Solving the grey fractional differential equation via finite difference method.
• Overcoming the limit of class ratio in origin grey model.

Of the grey models proposed for making predictions based on small sample data, the GM(1,1) model is the most important because of its low demands of data distribution, simple operation, and calculation requirements. However, the classical GM(1,1) model has two disadvantages: it cannot reflect the new information priority principle, and, if it is necessary to obtain the ideal effect of modeling, the original data must meet the class ratio test. This paper presents a new fractional grey model, FGM(q, 1), which is an extension of the GM(1,1) model in that first-order differential equations are transformed into fractional differential equations. Decomposition of the data matrix parameters during the process of solution shows that the new model follows the new information priority principle. For modeling, the mean absolute percentage error (MAPE) is established as the objective function of the optimization model, and a particle swarm algorithm is used to calculate the accumulation number and the order of the differential equation that can minimize the MAPE. Finally, the results from three groups of data modeling show that, compared with other classical grey models, FGM(q, 1) has higher modeling precision, can overcome the GM(1,1) model class ratio test restrictions and has a wider adaptability.