Directed Acyclic Graphical (DAG) models efficiently formulate causal relationships in complex systems. Traditional DAGs assume nodes to be scalar variables, characterizing complex systems under a facile and oversimplified form. This paper considers that nodes can be multivariate functional data and thus proposes a multivariate functional DAG (MultiFun-DAG). It constructs a hidden bilinear multivariate function-to-function regression to describe the causal relationships between different nodes. Then an Expectation-Maximum algorithm is used to learn the graph structure as a score-based algorithm with acyclic constraints. Theoretical properties are diligently derived. Prudent numerical studies and a case study from urban traffic congestion analysis are conducted to show MultiFun-DAG's effectiveness.

翻译：有向无环图模型能有效刻画复杂系统中的因果关系。传统有向无环图假设节点为标量变量，以简化形式表征复杂系统。本文考虑节点可为多变量函数型数据，进而提出多变量函数有向无环图（MultiFun-DAG）。该模型构建了隐藏的双线性多变量函数到函数回归，用以描述不同节点间的因果关系，并采用期望最大化算法作为带无环约束的评分学习方法进行图结构推断。本文严格推导了理论性质，通过审慎的数值实验及城市交通拥堵分析案例研究，验证了MultiFun-DAG的有效性。