Comparision of Conformable and Caputo fractional grey models

Abstract

In recent years, fractional order derivatives have been encountered in various fields of science, particularly in applied mathematics. Although there are many fractional derivative definitions in the literature, there are very few studies on which derivative definition works better in a mathematical model. In applications, it is seen that calculations are easier with the model using the Conformable derivative operator due to the simplicity of the derivative definition. However, the Caputo derivative operator, which is considered to be more effective in models related to time series due to its memory property, leads to more complex calculations. In this article, two fractional grey models were created in the same structure with Conformable and Caputo derivative operators and their applications were implemented on the same data sets to a performance comparison of the fractional operators. The working mechanisms of fractional grey models constructed with both Caputo and Conformable derivative operators were demonstrated in detail. Solution of the whitening differential equation in the Caputo fractional grey model was obtained using Laplace transforms. Here, Conformable and Caputo fractional grey models were applied to the forecast of three real time series and their forecast performances were compared. Data on China’s annual domestic energy consumption, annual wind energy consumption, and areas affected by drought disasters were utilized as real-time series. It has been observed that Conformable fractional grey models provide more accurate predictions with lower errors for certain datasets, while Caputo fractional grey models demonstrate better performance for others. This study is the first study in the literature that compared the Conformable and Caputo derivative operators on a grey model.