The South West Local Group held a meeting on Wednesday, 3 April 2019 at the University of Plymouth with speaker Guy Nason, professor of statistics at the University of Bristol and vice president of the Royal Statistical Society.
Guy started with introducing mumps data as an example for network time series. In 2005, weekly cases of mumps disease in the UK were recorded for 47 counties. The individual time series of mumps cases for each county is linked within a network connecting the counties. The network time series models allow the weekly observation at each county to depend on observations at earlier times at the same county as well as at neighbour counties.
Guy then described a network autoregressive (NAR) process, where the observation on a node of the network depends linearly on earlier observations on the same node, on its neighbour nodes as well as on a random error term. Guy then introduced network autoregressive integrated moving average (NARIMA) processes, by adding the moving average term to the NAR process. Furthermore, the information about distance between neighbours can be incorporated into the NAR model via the parameters associated with earlier observations on the neighbour nodes. By using the inverse-distance weighting, the further the distance the less influential of the neighbours is on the observation.
Guy showed the implementation of the NAR model in R and that the parameters can be estimated by using least-squares, maximum likelihood or Bayesian methods. Guy illustrated the applications of NAR models with various orders for the autoregressive terms of the times series of the same node and for the neighbour nodes. These NAR models were then compared using scatter plots and autocorrelation plots of residuals. The NAR models were also compared with the vector autoregressive (VAR) model and the separate AR models using Bayesian information criterion and mean prediction error.
In some examples, neighbours can change over time, leading to, sometimes, unknown, dynamic networks. Guy discussed the network construction by selecting the best network among many different networks using the edges which help improve forecasting performance. The application of network construction is illustrated via modelling the Gross Domestic Product (GDP) time series across many countries. The following models are compared, the NAR models, separate AR models, and VAR, using the mean prediction error as well as the visualisation of the difference in the observed and predicted GDP.
Finally, Guy introduced trend removal by using lifting transform, which has the ability to detrend and decorrelate the time series. The resulting autocorrelation function plots illustrate the benefits of lifting transform and detrending for modelling. The correlation between time series is dramatically decreased, which then allows simpler stochastic models to be fitted.