Partial correlation graphs for continuous-parameter time series

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) orthogonality graph of Fasen-Hartmann and Schenk (Stoch Process Appl 179:104501, 2024. https://doi.org/10.1016/j.spa.2024.104501 ). 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.