Cubic spline estimation for non parametric uncertain differential equation

In the history of researching to estimate the unknown parameters that were in the uncertain differential equation (UDE), the problem of parameter estimation with known functional forms is often studied. However, in practical situations, its functional form is often unknown. In order to deal with this problem, this article proposes the cubic spline method to approximate the autonomous UDE, and perform non parametric estimation on it. The cross validation is introduced to determine the number of term (J), which is in the approximate cubic spline. In addition, the uncertain hypothesis testing is given to verify the rationality of this method. Finally, some numerical examples are given. Then this method is applied to a case study of the Beijing Air Quality Index, and a comparative analysis is given to verify the practicability and superiority of the cubic spline method.