Theoretical foundations of autoregressive models for time series on acyclic directed graphs

Three classes of models for time series on acyclic directed graphs are considered. At first a review of tree-structured models constructed from a nested partitioning of the observation interval is given. This nested partitioning leads to several resolution scales. The concept of mass balance allowing to interpret the average over an interval as the sum of averages over the sub-intervals implies linear restrictions in the tree-structured model. Under a white noise assumption for transition and observation noise there is an change-of-resolution Kalman filter for linear least squares prediction of interval averages (Chou 1991). This class of models is generalized by modeling transition noise on the same scale in linear state space form. The third class deals with models on a more general class of directed acyclic graphs where nodes are allowed to have two parents. We show that these models have a linear state space representation with white system and coloured observation noise.