\section{Conclusion and Further Work}
\label{sec:conc}
We have introduced new methods for modelling and inferring relationships present in network time series. Section~\ref{sec:gnar methodology} reviewed the GNAR model introduced by \cite{gnar_org, gnar_paper} and presents a new compact hierarchical matrix representation, which is easier to manipulate. 
Subsequently, we proposed the NACF and the PNACF as diagnostic statistics, which can be visualised with Corbit plots.
These choices for measuring correlation are not the only possible ones, however, their usefulness and ease of interpretation as graphical aids for model selection are illustrated in Section \ref{sec: application}. Moreover, the ideas in Section \ref{sec: graph model connection} show the connection between GNAR models and graphical models, which suggest a clear interpretation of the relationship between nodes depending on their distance on the graph, and how this affects conditional correlation; see Theorem \ref{th: pop_sparsity gnar}. Also, we exhibit a possible interpretation of GNAR models as a constrained VAR that exploits prior knowledge of the network for performing variable selection and shrinkage, which enables us to study the connection between model constraints, prior information and parameter updates. These ideas allow us to study the local interactions between nodes, and the properties of the network as a composite object. Further, it exhibits how we can extend results from VAR theory to the GNAR framework by properly adjusting the correspondence between VAR parametrizations and GNAR formulations.
\par
We remark that the GNAR framework is useful for specific problems where the data can be properly described by an underlying network, which induces a particular correlation structure. Section~\ref{sec: application} shows the advantages GNAR methods have  when studying problems similar to the demand for mechanical ventilation beds during the COVID--19 pandemic. Also, motivated by Section \ref{sub: subsect oracle selection}, investigating the connection between prior information and posterior distributions, such as assigning a prior to the association function between nodes might improve forecasting performance and complex models'  interpretability. Future work could focus on trend removal and tests of stationarity.
Also, a more thorough analysis of the NACF and PNACF might reveal interesting statistical properties for model selection, and aid in the study of AIC and BIC as criteria for parsimonious model selection; see \cite{aic, bic}. 
\par 
Another area for future exploration is that of the construction of the underlying network. We believe our current method
is
sensible, but it is not the only possibility. A more refined local geographically-derived network might be preferable such
as one that takes into account key transport connections. For example, some Trusts further apart than our 120~km might be
strongly connected by a commuter rail line or they are a regional centre, perhaps, for some kinds of healthcare from quite a 
large region. We also considered more general mathematical tessellations, such as Dirichlet, but these suffer from
the `long thin triangle' issue, which connects infeasibly far apart Trusts, such as on the Wirral and Devon in north-west and south-west England, respectively, and hence are not really viable for our problem. A potential issue is that we use the data several times for different purposes: to select
the $D=120$~km distance limit to connect NHS Trusts, to compute and assess models using NACF/PNACF
and predictive inference (properly using out-of-sample forecasts). Further work is necessary
to understand whether this `multiple-dipping' of the data is acceptable, whether it
affects reliability, and it might be preferable to investigate and implement methods
of post-selective inference such as splitting strategies, see, e.g.\ \cite{rasines}.
\par
Essentially, GNAR is a parsimonious model that enables us to estimate fewer parameters more efficiently and precisely, even in high-dimensional settings. Furthermore, it is more transparent than overparametrized VAR, which facilitates interpretation and replication. The code for our plots and NACF/PNACF functions are incorporated into the 
CRAN \texttt{GNAR} package. Many further examples and material relating to our work can be found in
%link can be found in the
an RMarkdown document (link provided in the supplementary material). 

The UK Covid--19 enquiry ({\tt https://covid19.public-inquiry.uk/}) highlights the importance of analysing all aspects of the pandemic. The enquiry’s remit specifically relates to “the state of the UK’s central structures and procedures for pandemic emergency preparedness, resilience and response”. Statistical modelling of complex health services data has an important role to play in this crucial initiative.
  
