  Consider a multi-variate time series $(X_t)_{t=0}^{T}$ where $X_t \in
\mathbb{R}^d$ which may represent spike train responses for multiple neurons in
a brain, crime event data across multiple regions, and many others. An
important challenge associated with these time series models is to estimate an
influence network between the $d$ variables, especially when the number of
variables $d$ is large meaning we are in the high-dimensional setting. Prior
work has focused on parametric vector auto-regressive models. However,
parametric approaches are somewhat restrictive in practice. In this paper, we
use the non-parametric sparse additive model (SpAM) framework to address this
challenge. Using a combination of $\beta$ and $\phi$-mixing properties of
Markov chains and empirical process techniques for reproducing kernel Hilbert
spaces (RKHSs), we provide upper bounds on mean-squared error in terms of the
sparsity $s$, logarithm of the dimension $\log d$, number of time points $T$,
and the smoothness of the RKHSs. Our rates are sharp up to logarithm factors in
many cases. We also provide numerical experiments that support our theoretical
results and display potential advantages of using our non-parametric SpAM
framework for a Chicago crime dataset.
