Learning the Dependence Graph of Time Series with Latent Factors

TL;DR

This paper introduces a convex optimization method to learn the dependency graph of observed variables in linear stochastic differential systems with latent factors, ensuring accurate structure recovery in high-dimensional settings.

Contribution

It proposes a novel convex optimization approach for identifying observed variable dependencies while accounting for latent variables, with theoretical guarantees for sparse structures.

Findings

Successful structure recovery in high-dimensional regimes

Theoretical guarantees for sparse dependency graphs

Validated results on synthetic and stock market data

Abstract

This paper considers the problem of learning, from samples, the dependency structure of a system of linear stochastic differential equations, when some of the variables are latent. In particular, we observe the time evolution of some variables, and never observe other variables; from this, we would like to find the dependency structure between the observed variables - separating out the spurious interactions caused by the (marginalizing out of the) latent variables' time series. We develop a new method, based on convex optimization, to do so in the case when the number of latent variables is smaller than the number of observed ones. For the case when the dependency structure between the observed variables is sparse, we theoretically establish a high-dimensional scaling result for structure recovery. We verify our theoretical result with both synthetic and real data (from the stock market).