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.
翻译:非线性矢量自动递减模式为分析从非线性动态系统获得的多变量时间序列提供了一个有吸引力的框架。然而,创新(或错误)在驱动动态方面起着关键作用,但几乎总是被假定为添加剂。增殖性极大地限制了模型的普遍性,阻碍了对非线性创新相互作用的通用NVAR进程的分析。在这里,我们提议了一个称为独立创新分析的新的一般框架,从全局性NVAR中估算创新。我们承担创新的相互独立性,并以辅助变量(通常作为时间指数,简单地解释为非静止性)来调节这些创新。我们表明,IIA保证了创新的可识别性,其任意的非线性最高为可移动性,且不具有可视性。我们还根据辅助变量的类型提出了三个估算框架。我们因此为通用的NVAR提供了第一个严格的可识别性结果,以及学习此类模型的非常一般的工具。