\section{Correlation structure illustration}
This subsection illustrates the results in Theorem \ref{th: pop_sparsity gnar} by analysing a $\GNAR$ model based on a static network. Let $\boldsymbol{X}_t$ be the GNAR process given by 
\begin{equation} \label{eq: example gnar corr structure}
    \boldsymbol{X}_t = \alpha_1 \boldsymbol{X}_{t - 1} + \sum_{r = 1}^{s_1} \beta_{kr} \boldsymbol{Z}_{t - 1}^{r} + \boldsymbol{u}_t,
\end{equation}
where $\alpha_1 = 0.25$, the $\beta_{kr}$ satisfy $\alpha_1 \geq \beta_{11} > \beta_{12} > \beta_{13}$, such that no active $\rstage$ neighbour at a larger depth has a larger autoregressive coefficient when multiplied by its respective $\beta_{kr}$ (i.e.,  $\beta_{1(r - 1)} w_{ij} > \beta_{11} w_{ij}$), and that $\boldsymbol{u}_t \sim N (0, \sigma^2 \bmat{I_d})$ is a white noise process. Also, assume that the underlying network structure is static and given by the network in Figure \ref{fig: underlying net}. 
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{images/underlying_network.pdf}
    \caption{Underlying network $\mathcal{G}$ for the GNAR process given by (\ref{eq: example gnar corr structure}).}
    \label{fig: underlying net}
\end{figure}
\subsection{Generalised partial correlation graph}
Under the assumptions above, the GNAR process given by (\ref{eq: example gnar corr structure}) satisfies the conditions in Theorem \ref{th: pop_sparsity gnar}. Therefore, the induced correlation structure is given by the node distances in the network shown in Figure \ref{fig: underlying net}. And, the generalised partial correlation graph is the one in Figure \ref{fig: generalised corr graph}.
\begin{figure}[H]
    \centering
    \includegraphics[scale=0.45]{images/generalised_corr_graph.pdf}
    \caption{Generalised partial correlation graph $G$ for the GNAR process given by (\ref{eq: example gnar corr structure}) with underlying network shown in Figure \ref{fig: underlying net}; see Definition \ref{def: conditional indep definition}.}
    \label{fig: generalised corr graph}
\end{figure}
%\newpage
By Theorem \ref{th: pop_sparsity gnar} the active $\rstage$ distance at which nodes become conditionally correlated is given by Table \ref{tab: corr structure} below.
\begin{table}[H]
    \centering
    \begin{tabular}{c|ccccccc}
         $\delta (i, j)$ & & &  & $\mathcal{G}$ & & & \\
         \hline
                 & $\ast$ & 1 & 2 & 3 & 4 & 5 & 6 \\ 
                 & 1 & $\ast$ & 1 & 2 & 3 & 4 & 5 \\ 
                 & 1 & 1 & $\ast$ & 1 & 2 & 3 & 4 \\ 
             $G$ & 2 & 1 & 1 & $\ast$ & 1 & 2 & 3 \\ 
                 & 2 & 2 & 1 & 1 & $\ast$ & 1 & 2 \\ 
                 & 3 & 2 & 2 & 1 & 1 & $\ast$ & 1 \\ 
                 & 3 & 3 & 2 & 2 & 1 & 1 & $\ast$ 
    \end{tabular}
    \caption{Distance between nodes, the distances below the diagonal correspond to the distances in the graph shown by Figure \ref{fig: generalised corr graph}, and the distances in the entries above the diagonal correspond to the graph shown by Figure \ref{fig: underlying net}.}
    \label{tab: corr structure}
\end{table}

\subsection{Correlation structure and the inverse spectral matrix}
Following \cite{Songsiri200989}, the inverse spectral matrix of the process given by (\ref{eq: example gnar corr structure}) can be computed as
\begin{equation*}
    \bmat{S} (\omega) = \bmat{A}_0 + \sum_{k = 1}^{p} ( e^{- \imath k \omega} \bmat{A}_k + e^{ \imath k \omega} \bmat{A}_k^T),
\end{equation*}
where $\bmat{A}_k = 2 \pi \sigma^{-2} \sum_{i = 1}^{p - k} \bmat{\Phi}_i^T \bmat{\Phi}_{i + k}$ and $\bmat{A}_0 =2 \pi \sigma^{-2}  \bmat{I_d}$. Thus, the real and imaginary parts are given by
\begin{align*}
    \mathrm{Re} \{ \bmat{S} (\omega) \} &= \bmat{A}_0 + \sum_{k = 1}^{p} \cos (k \omega) (\bmat{A}_k + \bmat{A}_k^T ), \\
    \mathrm{Im} \{ \bmat{S} (\omega) \} &= \sum_{k = 1}^{p} \sin (k \omega) ( - \bmat{A}_k + \bmat{A}_k^T ),
\end{align*}
and the partial spectral coherences are given by
\begin{equation*}
    \left| [\bmat{S} (\omega)]_{ij} \right | = \left [ \mathrm{Re} \{ [\bmat{S} (\omega)]_{ij} \}^2 + \mathrm{Im} \{ [\bmat{S} (\omega)]_{ij} \}^2 \right ]^{1 / 2}.    
\end{equation*}

The following plots illustrate the inner workings in Theorem \ref{th: pop_sparsity gnar} and this document, they compare the inverse spectral matrices selecting $r$ as the active $\rstage$ neighbourhood regression for $r \in \{1, 2, 3\}$ produce for the GNAR process given by (\ref{eq: example gnar corr structure}). 

\begin{figure}[H]
\centering
\begin{subfigure}{0.45\textwidth}
    \includegraphics[width=\textwidth]{images/partial_coherence_hierarchy_node_1.pdf}
    \caption{}
    \label{fig: psc node 1 all}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
    \includegraphics[width=\textwidth]{images/partial_coherence_hierarchy_node_1_zoom.pdf}
    \caption{}
    \label{fig: psc node collisions}
\end{subfigure}
\caption{
    \ref{fig: psc node 1 all} comparisons of $|[\bmat{S} (\omega)]_{ij}|$ when $i = 1$ and $r^* = 3$ across all nodes. \ref{fig: psc node collisions} comparisons of $|[\bmat{S} (\omega)]_{ij}|$ when $i = 1$ and $r^* = 3$ for $j = 4, 5, 6, 7$, where the dashed black line indicates $|[\bmat{S} (\omega)]_{ij}| \equiv 0$; see text below for description.
    }
\end{figure}
Figure \ref{fig: psc node 1 all} shows the magnitude of $|[\bmat{S} (\omega)]_{ij}|$ where $i = 1$ and $j$ varies across all other nodes, Figure \ref{fig: psc node collisions} highlights the magnitude of $|[\bmat{S} (\omega)]_{ij}|$, where $i = 1$ and $j = 4, 5, 6, 7$. 
By Theorem \ref{th: pop_sparsity gnar}, node $1$ decorrelates from nodes $j = 6, 7$ by decreasing $r^* = 3$ to $r^* = 2$, and from nodes $j = 4, 5, 6, 7$ by decreasing $r^* = 3$ to $r^* = 1$. 
\newpage

\begin{figure}[H]

\centering
\begin{subfigure}{0.45\textwidth}
    \includegraphics[width=\textwidth]{images/active_r=1_stop_light.pdf}
    \caption{Stoplight plot with active $r = 1$.}
    \label{fig:first}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
    \includegraphics[width=\textwidth]{images/active_r=2_stop_light.pdf}
    \caption{Stoplight plot with active $r = 2$.}
    \label{fig:second}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
    \includegraphics[width=\textwidth]{images/active_r=3_stop_light.pdf}
    \caption{Stoplight plot with active $r = 3$.}
    \label{fig:third}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
    \centering
    \includegraphics[scale=0.15]{images/stop_light_legend.pdf}
    \caption{Legend for Stoplight plots.}
    \label{fig: stoplight legend}
\end{subfigure}
\caption{
        Stoplight plots for the GNAR process given by (\ref{eq: example gnar corr structure}); see the text below for further description. }
\label{fig: stop light plots}
\end{figure}

Figure \ref{fig: stop light plots} shows the different correlation structures induced by changing the maximum active $\rstage$ neighbourhood regression depth. For each Stoplight plot - Figures \ref{fig:first}, \ref{fig:second}, \ref{fig:third} - the entries below the diagonal plot the values of $|[\bmat{S} (\omega)]_{ij}|$, and the entries above the diagonal plot the classification of the nodal pair according to the legend in Figure \ref{fig: stoplight legend}. 
\newpage

\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{images/partial_coherence_hierarchy_node_1.pdf}
    \caption{Partial coherence hierarchy for node 1 computed for the model given by \eqref{eq: example gnar corr structure}}
\end{figure}



\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{images/partial_coherence_hierarchy_node_1_zoom.pdf}
    \caption{Partial coherence hierarchy for node 1 computed for the model given by \eqref{eq: example gnar corr structure}. This plot highlights nodes 4, 5, 6 and 7, which activate for larger $\rstage$ values.}
\end{figure}



The Stoplight plots indicate whether or not the two nodal time series $X_{i, t}$ and $X_{j, t}$ are conditionally uncorrelated and their active/inactive $\rstage$ neighbourhood regression status. And, illustrate the results in Theorem \ref{th: pop_sparsity gnar} which agree with the ones in Table \ref{tab: corr structure}.

\begin{figure}[H]
    \centering
    \includegraphics[scale=0.30]{images/stop_light_legend.pdf}
    \caption{Legend for Stoplight plots. Each colour indicates the relationship between $X_{i, t}$ and $X_{j, t}$.}
    \label{fig: stoplight legend big}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{images/active_r=1_stop_light.pdf}
    \caption{Stoplight plot with active $r = 1$.}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{images/active_r=2_stop_light.pdf}
    \caption{Stoplight plot with active $r = 2$.}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[scale=0.40]{images/active_r=3_stop_light.pdf}
    \caption{Stoplight plot with active $r = 3$.}
\end{figure}