Independent Innovation Analysis for Nonlinear Vector Autoregressive Process

TL;DR

This paper introduces a novel independent innovation analysis framework for nonlinear vector autoregressive models, enabling the identification of innovations with arbitrary nonlinearities under broad conditions.

Contribution

It provides the first rigorous identifiability results for general NVAR processes with nonlinear innovations, extending beyond additive assumptions.

Findings

Guarantees identifiability of innovations up to permutation and invertible nonlinearities.

Proposes three estimation frameworks based on auxiliary variables.

Enables analysis of nonlinear interactions in multivariate time series.

Abstract

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.