Partial correlation graphs for continuous-parameter time series

TL;DR

This paper introduces the concept of partial correlation graphs for multivariate continuous-time stochastic processes, providing theoretical foundations, properties, and practical examples, including AR processes, to understand dependencies in such systems.

Contribution

It defines the partial correlation graph for continuous-time processes, proves it forms a graphoid with Markov properties, and relates it to existing causality graphs.

Findings

Partial correlation graph is characterized via spectral density inverse.

The graph satisfies Markov properties and is easy to determine.

Edges in MCAR processes are explicitly characterized.

Abstract

In this paper, we establish the partial correlation graph for multivariate continuous-time stochastic processes, assuming only that the underlying process is stationary and mean-square continuous with expectation zero and spectral density function. In the partial correlation graph, the vertices are the components of the process and the undirected edges represent partial correlations between the vertices. To define this graph, we therefore first introduce the partial correlation relation for continuous-time processes and provide several equivalent characterisations. In particular, we establish that the partial correlation relation defines a graphoid. The partial correlation graph additionally satisfies the usual Markov properties and the edges can be determined very easily via the inverse of the spectral density function. Throughout the paper, we compare and relate the partial correlation graph to the mixed (local) causality graph of Fasen-Hartmann and Schenk (2023a). Finally, as an example, we explicitly characterise and interpret the edges in the partial correlation graph for the popular multivariate continuous-time AR (MCAR) processes.