componentsautoregression

Components autoregression is a time series modeling approach that represents a multivariate signal as a set of latent components, each evolving autoregressively. The observed series are formed by a mixing of these components, so temporal dependence can be captured within components rather than across observed series.

Formally, x_t = A s_t + e_t, with x_t in R^d, s_t in R^p, A a mixing matrix, and e_t observation noise. Each latent component s_t^i follows s_t^i = sum_{k=1}^{p_i} phi_{i,k} s_{t-k}^i + epsilon_t^i. The parameters include A, the AR coefficients phi, and the noise covariances.

Estimation typically involves a decomposition step (for example independent component analysis or probabilistic factor models) to obtain s_t and A, followed by fitting AR models to each component. Some approaches estimate all parameters jointly via likelihood or Bayesian methods.

Applications include signal processing, econometrics, and neuroscience, where separating independent sources before modeling improves interpretability and can reduce complexity. Limitations include identifiability of the latent components, sensitivity to noise, and nonstationarity.

Related topics include autoregressive models, independent component analysis, dynamic factor models, and vector autoregression.