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.

Ordinary differential equations (ODEs) are widely used to characterize the dynamics of complex systems in real applications. In this article, we propose a novel joint estimation approach for generalized sparse additive ODEs where observations are allowed to be non-Gaussian. The new method is unified with existing collocation methods by considering the likelihood, ODE fidelity and sparse regularization simultaneously. We design a block coordinate descent algorithm for optimizing the non-convex and non-differentiable objective function. The global convergence of the algorithm is established. The simulation study and two applications demonstrate the superior performance of the proposed method in estimation and improved performance of identifying the sparse structure.