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\title{Response to reviewers on `New tools for network time
series \ldots' ID JRSSA-Dec-2023-0316-R1}

\author{Nason, Salnikov and Cortina--Borja}

\date{18th December 2024}


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\noindent
{\color{blue} We would like to thank the AE and both reviewers very
much for reading our revised manuscript and for their helpful,
insightful and constructive comments. We address them below.
We have again used changebars in the very few places we have made changes
in Section~5.4, but did not use them for minor wording/typo changes.}





\noindent
\underline{\em ``Associate Editor}
{\em Comments to the Author:

\noindent
The authors have thoroughly addressed most of the points raised, and I personally have no further comments. I believe this paper will be a good candidate for a Discussion Meeting once the authors have addressed the final points raised by Reviewer \#2.''}

\noindent
{\color{blue} Thank you very much for your positive comments. We have
addressed Reviewer 2's comments below.}


\noindent
\underline{\em  ``Referee: 1}

\noindent
{\em Comments to the Author

\noindent
The authors have made a reasonable response to my review of the original submission. I have no objection to the paper going forward for reading. }

\noindent
{\color{blue} Thank you for your time and effort in rereading our manuscript.}

\noindent
\underline{\em Referee: 2}

\noindent
{\em Comments to the Author.\\
There is some nice work in this paper, especially on diagnostics and model selection. In comparison to alternatives, the predictive performance in the application of the extremely parsimonious GNAR model is impressive.

\noindent
I am content with the response to almost all my points.  I have just two issues remaining.}

\noindent
{\color{blue} Thank you for your positive comments, which we appreciate.}

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\noindent
{\em 
The first is relatively minor: neither panel of Figure 6 can be considered good presentation practice. I have seen the authors' comments and I have no solution to offer.  Still, neither is a good plot.}



\noindent
{\color{blue}
We agree. Our previous attempts to improve were not sufficient.
We have replaced Figure 6a (new Figure 6) by a map-based alternative
that shows the relevance colour-coded at each node. We think the plot
is better, but still some locations (particularly around London
and the North-West) are hard to read. In an attempt to mitigate
this, we have produced full-size A4 version
in Figure~14 in Appendix~D.

We have replaced figure 6b by a similar map-based alternative, but
relegated this to Appendix D, although we have referred to it
in the main paper, just prior to the start of Section~5.5.

We hope these figures are better and more intuitive.}

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\noindent
{\em 
The other is the discussion of influence. Again, I have seen the response and the interesting new plot.  I am not too convinced by the argument that (18) depends on r* and p and hence the data.  The issue might be that I am conditioned to use ``influence'' (and also ``local influence'') in a regression context to mean the effect on parameter estimates of removing or slightly altering an observation. That's not what is meant. Maybe stick to ``relevance''.}


\noindent
{\color{blue} Yes, we do not mean `influence' as in the regression context.
To prevent confusion, we have replaced `influence' by `relevance'
(and similar wording) as you suggested and think this clarifies matters.
Thank you for your comment. There is now only one instance of the word ``influence''
and it occurs in a context different to the one discussed here.}

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\noindent
{\em At r*=1, equation (18) surely is measuring connectivity in the application: how many nodes are within 120km of node i.  Do we really want to celebrate using a complicated model to find the trusts with largest values?  So far as I can tell, none of the new red dots in Fig 7(b) are within the convex hulls of the two red clusters in Fig 7(a), supporting the idea that the ``influential'' locations are artefactual consequences of the choice of a 120km threshold for direct connection.}


\noindent
{\color{blue} We want to celebrate  that statistical modelling has
guided us to select a good GNAR model, but, more than that, ruled out a vast array of other less suitable, more complicated, models. 

Our previous response noted that~(18) does indeed depend on the data via 
$r^\ast$and chosen $p$ both
being dependent on the data. Incidentally, the model selection process itself used to choose $(r^\ast, p)$ also depends on the
data and our judgement. 
However, it is not just about {\em ``how many nodes are within 120km of node $i$”}, because~(18) \underline{also} includes  weights via the $W$ matrix,
so it is not as simple as nodes being located next to each other on the network, or counting them, the distance is important too. 

We looked in more detail about the location of new red dots in
Figure~7(b). What you say about convex hulls is true about the southern cluster.
However, in the northern cluster,
some new points in Figure~7(b) do fall into the convex
hull of the previous points from Fig~7(a), which
indicates that it is not purely about connectivity.

Clearly, connectivity is important, but it is also down to
distances, dependence of $(r^\ast, p)$ on the data and
the model selection processes (e.g.\ would AIC/BIC have given
different answers?).

Overall, though, thank you very much indeed for raising these points.}









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