The graphical continuous Lyapunov model is a new approach to statistically model dependence structures that may include feedback loops in multivariate data. The covariance matrices of the distributions in such a model are parametrized as solutions of the continuous Lyapunov equation via suitable drift and volatility matrices that stem from a multivariate Ornstein-Uhlenbeck process. It was recently shown that two variables are independent in the Lyapunov model if the corresponding nodes in the graph are not connected by a trek. We conjecture that the opposite implication also holds, meaning that if two nodes are connected by a trek in the graph, then the corresponding variables cannot be conditionally independent in the Lyapunov model. In this thesis, we start the investigation of the conjecture by considering the Lyapunov model of a directed path of arbitrary length. We prove that no conditional independence statements that involve at most 100 conditioning variables between the two considered nodes hold in the path model. We further devise a way to extend any counterexample for the statement where the first and last node of the path are conditionally independent given all intermediate nodes to counterexamples for statements on any longer path. Additionally, we illustrate the challenges of working with singular covariance matrices that arise on the way to a full proof of the conjecture for the path model.     «

The graphical continuous Lyapunov model is a new approach to statistically model dependence structures that may include feedback loops in multivariate data. The covariance matrices of the distributions in such a model are parametrized as solutions of the continuous Lyapunov equation via suitable drift and volatility matrices that stem from a multivariate Ornstein-Uhlenbeck process. It was recently shown that two variables are independent in the Lyapunov model if the corresponding nodes in the gr...     »