Continuously Indexed Graphical Models

TL;DR

This paper introduces a novel framework for modeling and estimating the dependence structure of continuous-index Gaussian processes using a kernel-based characterization, enabling consistent structure recovery from finite samples.

Contribution

It provides a new characterization of infinite-dimensional Gaussian processes that does not rely on inverse covariance zeros, and proposes a consistent structure estimation method with finite sample guarantees.

Findings

Method achieves consistent graph recovery with increasing sample size.

Convergence rates and finite sample guarantees are established.

Illustrated through simulations and real data analyses.

Abstract

Let $X = \{X_{u}\}_{u \in U}$ be a real-valued Gaussian process indexed by a set $U$. It can be thought of as an undirected graphical model with every random variable $X_{u}$ serving as a vertex. We characterize this graph in terms of the covariance of $X$ through its reproducing kernel property. Unlike other characterizations in the literature, our characterization does not restrict the index set $U$ to be finite or countable, and hence can be used to model the intrinsic dependence structure of stochastic processes in continuous time/space. Consequently, this characterization is not in terms of the zero entries of an inverse covariance. This poses novel challenges for the problem of recovery of the dependence structure from a sample of independent realizations of $X$, also known as structure estimation. We propose a methodology that circumvents these issues, by targeting the recovery of the underlying graph up to a finite resolution, which can be arbitrarily fine and is limited only by the available sample size. The recovery is shown to be consistent so long as the graph is sufficiently regular in an appropriate sense. We derive corresponding convergence rates and finite sample guarantees. Our methodology is illustrated by means of a simulation study and two data analyses.