Modelling, Detrending and Decorrelation of Network Time Series

A network time series is a multivariate time series augmented by a graph that describes how variables (or nodes) are connected. We introduce the network autoregressive (integrated) moving average (NARIMA) processes: a set of flexible models for network time series. For fixed networks the NARIMA models are essentially equivalent to vector autoregressive moving average-type models. However, NARIMA models are especially useful when the structure of the graph, associated with the multivariate time series, changes over time. Such network topology changes are invisible to standard VARMA-like models. For integrated NARIMA models we introduce network differencing, based on the network lifting (wavelet) transform, which removes trend. We exhibit our techniques on a network time series describing the evolution of mumps throughout counties of England and Wales weekly during 2005. We further demonstrate the action of network lifting on a simple bivariate VAR(1) model with associated two-node graph. We show theoretically that decorrelation occurs only in certain circumstances and maybe less than expected. This suggests that the time-decorrelation properties of spatial network lifting are due more to the trend removal properties of lifting rather than any kind of stochastic decorrelation.