Model Checking for Parametric Ordinary Differential Equations Systems

Ordinary differential equations have been used to model dynamical systems in a broad range. Model checking for parametric ordinary differential equations is a necessary step to check whether the assumed models are plausible. In this proposal, we will introduce a trajectory matching-based test for the whole system, which can also easily be applied to checking the parametric structure of partially observed systems. We also consider how to identify which component function would be wrongly modelled. To this end, we will first construct an integral matching-based test. To avoid cumulative error by estimation and integral in this test, we will propose another test, in terms of data splitting, that is bias corrected gradient matching-based. We investigate their asymptotic properties under the null, global and local alternative hypotheses. As there are no results of relevant parameter estimations with the alternative models in the literature, we also investigate the asymptotic properties of nonlinear least squares estimation and two-step collocation estimation under both the null and alternatives. The constructed tests by these two approaches can detect the alternatives distinct from the null at the typical nonparametric rate in hypothesis testing. Further, we will consider the case with diverging number of ordinary equations and diverging number of unknown parameters and construct a locally smoothing test and a globally smoothing test. W will also study the asymptotic properties of these two tests and expect that the locally smoothing test may still have normal distribution as its limiting null distribution, and unlike classical globally smoothing tests with fixed number of equations, the limiting null distribution may become to normal rather than that of weighted sum of chi-squared variables.