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.

翻译：本文针对多变量连续时间随机过程建立了偏相关图，仅要求底层过程是平稳且均方连续的，具有零期望和谱密度函数。在偏相关图中，顶点表示过程的各个分量，无向边表示顶点间的偏相关关系。为定义此图，我们首先引入连续时间过程的偏相关关系，并给出若干等价刻画。特别地，我们证明偏相关关系构成一个graphoid结构。偏相关图还满足通常的马可夫性质，且其边可通过谱密度函数的逆矩阵轻松确定。本文始终将偏相关图与Fasen-Hartmann和Schenk（2023a）提出的混合（局部）因果图进行比较与关联。最后，以流行的多变量连续时间AR（MCAR）过程为例，我们明确刻画并阐释了偏相关图中边的具体含义。