Xiaowu Dai, UCLA

Title: Kernel ordinary differential equations

Abstract: The ordinary differential equation (ODE) is widely used in modelling biological and physical processes in science. A new reproducing kernelbased approach is proposed for the estimation and inference of ODE given noisy observations. The functional forms in ODE are not assumed to be known or restricted to be linear or additive, and pairwise interactions are allowed. Sparse estimation is performed to select individual functionals and construct confidence intervals for the estimated signal trajectories. The estimation optimality and selection consistency of kernel ODE are established under both the low-dimensional and high-dimensional settings, where the number of unknown functionals can be smaller or larger than the sample size. The proposal tackles several important problems that are not yet fully addressed in smoothing spline analysis of variance (SS-ANOVA) framework, and extends the existing methods of dynamic causal modeling.