Mixed orthogonality graphs for continuous‐time state space models and orthogonal projections
本文为连续时间状态空间模型（包括多元连续时间ARMA过程）推导了局部正交性图，其中顶点表示过程分量，有向边表示因果影响，无向边表示同期相关，并通过控制器规范型参数和驱动过程的协方差矩阵刻画了边的特征。
In this article, we derive (local) orthogonality graphs for the popular continuous‐time state space models, including in particular multivariate continuous‐time ARMA (MCARMA) processes. In these (local) orthogonality graphs, vertices represent the components of the process, directed edges between the vertices indicate causal influences and undirected edges indicate contemporaneous correlations between the component processes. We present sufficient criteria for state space models to satisfy the assumptions of Fasen‐Hartmann and Schenk (2024a) so that the (local) orthogonality graphs are well‐defined and various Markov properties hold. Both directed and undirected edges in these graphs are characterised by orthogonal projections on well‐defined linear spaces. To compute these orthogonal projections, we use the unique controller canonical form of a state space model, which exists under mild assumptions, to recover the input process from the output process. We are then able to derive some alternative representations of the output process and its highest derivative. Finally, we apply these representations to calculate the necessary orthogonal projections, which culminate in the characterisations of the edges in the (local) orthogonality graph. These characterisations are given by the parameters of the controller canonical form and the covariance matrix of the driving Lévy process.