Nonparametric estimation for uncertain differential equations

In recent years, the researches on parameter estimation of uncertain differential equations have developed significantly. However, when we deal with some nonparametric uncertain differential equations, the parameter estimation may not be used directly. To deal with these uncertain differential equations, it is important to consider the nonparametric estimation with the help of the observations. As an important branch of uncertain differential equation, autonomous uncertain differential equation may be properly applied to model some uncertain autonomous dynamic systems. In this paper, we propose a Legendre polynomial based method for the nonparametric estimation of autonomous uncertain differential equations. After that, some numerical examples are given and the residuals as well as uncertain hypothesis tests are used to prove the acceptability of these estimations. In application, we consider an atmospheric carbon dioxide model by the proposed method of nonparametric estimation.