The Northern Ireland local group of the RSS held an online meeting using MS Teams on Wednesday, 10 February 2021, at 2pm. The speaker was Professor Guy Nason from the Department of Mathematics and Statistics at Imperial College, London.
Most of the audience new something about time series and others, probably a lesser number, knew something about networks, but the idea of Network Time Series was intriguing. Professor Nason opened his talk by giving several concrete examples. The first was weekly cases of mumps disease in England at county level for 2005. Thus, we have: t = 1,...,T(=52) weeks and p = 1,...,P (=47) counties – a multivariate time series. The idea of an imposed network is to capture some relevant structure of spread amongst the multivariate components over time. Networking possibilities were: transport corridors, weather or geography. The latter was chosen and a minimum spanning tree was adopted, augmented by close town links.
To model these data we focus on simple models which model dependence between the value of node i at time t to node i at earlier times and neighbours of node i at earlier times. We would also like to cope with neighbours that drop in/out, and /or change their neighbourhood in dynamic networks and/or cope with links between neighbours of different types. To accomplish this, Guy proposed a Generalised Network Autoregressive (GNAR) class of models given by:
This is an auto-regression model based on the network nodes and their neighbours. There several components: (a) the αi,j measure dependence between node i at time t and node i at the earlier time t – j, (b) the common βj,r,c measure the relationship strength between nodes that are separated by lag j in time, are r-stage (nested) neighbours away and of (categorical) covariate type c, (c) the ω(t)(i,q,c) (possibly deterministic) parameters measure the strength of the type c connection between node i and neighbourhood node q at time t. Hence, these ωs can change over time and reflect local connectivity. Finally, the εi,t are i.i.d. mean-zero random variables with variance, σ2.
This is a very general model and Guy discussed various simplifications of it, including models with temporal stationarity and spatial homogeneity. He pointed out that the model could be fitted by Least Squares using the GNARfit package in R and illustrated this by fitting the much simpler NAR(1,[1]) (AR1 with 1st stage neighbours, no weights) model to the mumps data. He wrapped up with another example and concluded by discussing the connection with VAR times series and possible extensions of the GNAR class.
This was an excellent talk on a non-trivial subject and it was well received by the online audience of 45. There were several queries. Guy was asked about integer time series the examples were treated as continuous outcomes (for example rates). He was also asked about covariate structures which were categorical in the models as formulated and finally there was a technical question about the inclusion of weights in the model. At this time in model development these were largely open issues and no doubt would be addressed as the subject expanded.The audience showed their appreciation by thanking the speaker for a very illuminating exposition.
Author
Gilbert MacKenzie, written on 10 April, 2021.