Developing a robust, accurate forecasting model and improving the prediction abilities of the limited historical data that lacks statistical rules has become a top priority. To address this problem, an improved conformable fractional non-homogeneous gray model, namely CFONGM(1,1,k,c), is proposed. Combining the dynamic background-value and particle swarm optimization algorithm to further improve forecasting ability of the existing gray model. In which matrix perturbation theory is employed to prove that the novel model conforms the principle of...