Prediction error identification with rank-reduced output noise

In data-driven modelling in dynamic networks, it is commonly assumed that all measured node variables in the network are noise-disturbed and that the network (vector) noise process is full rank. However when the scale of the network increases, this full rank assumption may not be considered as realistic, as noises on different node signals can be strongly correlated. In this paper it is analyzed how a prediction error method can deal with a noise disturbance whose dimension is strictly larger than the number of white noise signals than is required to generate it (rank-reduced noise). Based on maximum likelihood considerations, an appropriate prediction error identification criterion will be derived and consistency will be shown, while variance results will be demonstrated in a simulation example.