Selection of uncertain differential equations using cross validation

Abstract

Uncertain differential equations have been widely applied to modelling dynamic phenomena with noises. Facing with the same dynamic phenomenon, different scholars may adopt different uncertain differential equations. Naturally, there is a crucial question about which of these equations is most suitable to describe the given dynamic phenomenon. A desired uncertain differential equation should have a good prediction ability, that is, it performs well in a new dataset. As a model selection technique, cross validation assesses the model’s ability to predict new data in order to avoid overfitting. This paper applies k fold cross validation to the selection of uncertain differential equations. Observations in training sets and testing sets are used to calculate generalized moment estimations of unknown parameters and testing errors for uncertain differential equations, respectively. Then the uncertain differential equation with the smallest total testing error is selected. A numerical example and a real data example using closing prices of Industrial and Commercial Bank of China (ICBC) stock illustrate our methods in detail.