  We propose a method for adaptive nonlinear sequential modeling of vector-time
series data. Data is modeled as a nonlinear function of past values corrupted
by noise, and the underlying non-linear function is assumed to be approximately
expandable in a spline basis. We cast the modeling of data as finding a good
fit representation in the linear span of multi-dimensional spline basis, and
use a variant of l1-penalty regularization in order to reduce the
dimensionality of representation. Using adaptive filtering techniques, we
design our online algorithm to automatically tune the underlying parameters
based on the minimization of the regularized sequential prediction error. We
demonstrate the generality and flexibility of the proposed approach on both
synthetic and real-world datasets. Moreover, we analytically investigate the
performance of our algorithm by obtaining both bounds of the prediction errors,
and consistency results for variable selection.
