  The nonlinear vector autoregressive (NVAR) model provides an appealing
framework to analyze multivariate time series obtained from a nonlinear
dynamical system. However, the innovation (or error), which plays a key role by
driving the dynamics, is almost always assumed to be additive. Additivity
greatly limits the generality of the model, hindering analysis of general NVAR
processes which have nonlinear interactions between the innovations. Here, we
propose a new general framework called independent innovation analysis (IIA),
which estimates the innovations from completely general NVAR. We assume mutual
independence of the innovations as well as their modulation by an auxiliary
variable (which is often taken as the time index and simply interpreted as
nonstationarity). We show that IIA guarantees the identifiability of the
innovations with arbitrary nonlinearities, up to a permutation and
component-wise invertible nonlinearities. We also propose three estimation
frameworks depending on the type of the auxiliary variable. We thus provide the
first rigorous identifiability result for general NVAR, as well as very general
tools for learning such models.
