Kernel Estimation for Nonlinear Dynamics

Music City Center 

Many problems involve data exhibiting both temporal and cross-sectional dependencies. While linear dependencies have been extensively studied, the theoretical analysis of estimators under nonlinear dependencies remains scarce. This work studies a kernel-based estimation procedure for nonlinear dynamics within the reproducing kernel Hilbert space framework, focusing on nonlinear stochastic regression and nonlinear vector autoregressive models. We derive nonasymptotic probabilistic bounds on the deviation between a kernel estimator and the true nonlinear regression function. A key technical contribution is a concentration bound for quadratic forms of stochastic matrices in the presence of dependent data, which may be of independent interest. Additionally, we characterize conditions on multivariate kernels required to achieve optimal convergence rates.

nonlinear dynamics

vector autoregressive model

reproducing kernel Hilbert space

concentration inequality

time series

machine learning 

Section on Nonparametric Statistics