Independent Innovation Analysis for Nonlinear Vector Autoregressive Process
This provides a foundational method for analyzing multivariate time series from nonlinear dynamical systems, addressing a key bottleneck in general NVAR processes.
The paper tackles the limitation of assuming additive innovations in nonlinear vector autoregressive (NVAR) models by proposing independent innovation analysis (IIA), a framework that estimates innovations with arbitrary nonlinearities and guarantees identifiability up to permutation and invertible nonlinearities.
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.