  Ordinary differential equations (ODE) have been widely used for modeling
dynamical complex systems. For high-dimensional ODE models where the number of
differential equations is large, it remains challenging to estimate the ODE
parameters and to identify the sparse structure of the ODE models. Most
existing methods exploit the least-square based approach and are only
applicable to Gaussian observations. However, as discrete data are ubiquitous
in applications, it is of practical importance to develop dynamic modeling for
non-Gaussian observations. New methods and algorithms are developed for both
parameter estimation and sparse structure identification in high-dimensional
linear ODE systems. First, the high-dimensional generalized profiling method is
proposed as a likelihood-based approach with ODE fidelity and sparsity-inducing
regularization, along with efficient computation based on parameter cascading.
Second, two versions of the two-step collocation methods are extended to the
non-Gaussian set-up by incorporating the iteratively reweighted least squares
technique. Simulations show that the profiling procedure has excellent
performance in latent process and derivative fitting and ODE parameter
estimation, while the two-step collocation approach excels in identifying the
sparse structure of the ODE system. The usefulness of the proposed methods is
also demonstrated by analyzing three real datasets from Google trends, stock
market sectors, and yeast cell cycle studies.
