Learning Vector Autoregressive Models with Latent Processes
This addresses the challenge of accurate causal inference in time-series data with hidden variables, which is crucial for fields like economics and neuroscience, but the approach is incremental as it builds on existing VAR model frameworks.
The paper tackles the problem of learning the support of transition matrices in Vector Autoregressive models when some processes are latent, showing that under certain conditions, the influences among observed processes and latent path lengths can be identified, and the latent subgraph can be uniquely reconstructed if it is a directed tree, with experimental validation on synthetic and real-world datasets.
We study the problem of learning the support of transition matrix between random processes in a Vector Autoregressive (VAR) model from samples when a subset of the processes are latent. It is well known that ignoring the effect of the latent processes may lead to very different estimates of the influences among observed processes, and we are concerned with identifying the influences among the observed processes, those between the latent ones, and those from the latent to the observed ones. We show that the support of transition matrix among the observed processes and lengths of all latent paths between any two observed processes can be identified successfully under some conditions on the VAR model. From the lengths of latent paths, we reconstruct the latent subgraph (representing the influences among the latent processes) with a minimum number of variables uniquely if its topology is a directed tree. Furthermore, we propose an algorithm that finds all possible minimal latent graphs under some conditions on the lengths of latent paths. Our results apply to both non-Gaussian and Gaussian cases, and experimental results on various synthetic and real-world datasets validate our theoretical results.