Kernel Estimation for Nonlinear Dynamics

TL;DR

This paper develops a kernel-based estimation method for nonlinear dynamic models with dependent data, providing probabilistic bounds and conditions for optimal convergence in a complex dependent setting.

Contribution

It introduces a novel concentration bound for quadratic forms of stochastic matrices with dependent data and characterizes kernel conditions for optimal convergence rates.

Findings

Derived nonasymptotic probabilistic bounds for kernel estimators.

Established a concentration bound for quadratic forms in dependent data.

Identified kernel conditions ensuring optimal convergence rates.

Abstract

Many scientific problems involve data exhibiting both temporal and cross-sectional dependencies. While linear dependencies have been extensively studied, the theoretical analysis of regression 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 vector autoregressive models. We derive nonasymptotic probabilistic bounds on the deviation between a regularized kernel estimator and the nonlinear regression function. A key technical contribution is a concentration bound for quadratic forms of stochastic matrices in the presence of dependent data, which is of independent interest. Additionally, we characterize conditions on multivariate kernels that guarantee optimal convergence rates.