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\begin{document}
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\title{\textbf{Supplementary material for: \\
    New tools for network time series with an application
    to COVID-19  hospitalisations}
    }
%\author{Daniel Salnikov}
\author{G. P. Nason\footnotemark[1] \footnotemark[2],
    D. Salnikov\thanks{Dept.\ Mathematics, Huxley Building, Imperial College, 180 Queen's Gate, South Kensington, London, SW7 2AZ, UK.} \footnotemark[2]\\
    Imperial College London\\
    and \\
    M. Cortina-Borja\thanks{Great Ormond Street Institute of Child Health, 30 Guilford Street, London WC1N 1EH} \footnotemark[1] \\
    Great Ormond Street Institute of Child Health,\\
    University College London}
\date{}

\maketitle
\section*{Introduction}
This supplementary document contains technical proofs and further simulation results mentioned in {\bf New tools for network time series with an application
to COVID-19  hospitalisations}; see \cite{corbit_paper}. 
\par
\subsection*{Notation}
A network time series $\mathcal{X} = (\boldsymbol{X}_t, \mathcal{G})$, where $\boldsymbol{X}_t \in \mathbb{R}^d$ is a multivariate time series, and the network $\mathcal{G}$ has a weights matrix $\mathbf{W}_{t} \in \mathbb{R}^{d \times d}$. 
\par
Also, each \mbox{$r$-stage} neighbourhood has its own adjacency matrix $\mathbf{S}_r \in \mathbb{R}^{d \times d}$; see \cite{corbit_paper} for details. 
\par
The GNAR model selects \mbox{$r$-stage} neighbourhoods for building regression paths. These regression paths are linear combinations at \mbox{$k$-lag} and \mbox{$r$-stage} depth pairs. The network autocorrelation function and partial network autocorrelation function introduced in \cite{corbit_paper} are given below.

\begin{definition}[network autocorrelation function (NACF)] \label{def: nacf definition}
\begin{equation} \label{eq:nacf definition}
    \nacf_t (h, r) := \frac{\sum_{t=1}^{N - h} ( \boldsymbol{X}_{t + h} - \boldsymbol{\overline{X}})^{T} \big ( \mathbf{W}_t \odot \bmat{S_r} + \bmat{I_d} \big )
    ( \boldsymbol{X}_{t} - \boldsymbol{\overline{X}})}
    {\sum_{t=1}^{N}  ( \boldsymbol{X}_{t} - \boldsymbol{\overline{X}})^{T} \big \{ \big (1 + \lambda_t \big) \bmat{I_d} \big \} ( \boldsymbol{X}_{t} - \boldsymbol{\overline{X}})},
\end{equation}
where
$$ \lambda_t := \underset{j = 1, \dots, d}{\max} \bigg \{ \sum_{i = 1}^d [\bmat{W}_{t} \odot \bmat{W}_t]_{i j} \bigg \}^{1/2} . $$
\end{definition}
Note that the NACF in Definition \ref{def: nacf definition} satisfies the following proposition.
\begin{proposition} \label{prop: nacf one}
Let $\{ \boldsymbol{X}_{t} \}_{t = 1}^{T}$ be a realisation of length $T$ of a \mbox{$d$-dimensional} network time series with associated graph $\mathcal{G}$, network weights matrix $\bmat{W}_{t}$, stage adjacency matrices $\bmat{S}_r$ and autocovariance bound $\lambda_t$. Then, for all \mbox{$h$-lag} and \mbox{$r$-stage} pairs we have that the network autocorrelation function defined by \text{(\ref{eq:nacf definition})} satisfies
\begin{equation*}
    -1 \leq \nacf_t (h, r) \leq 1 .
\end{equation*}
\end{proposition}
See Section \ref{sec: derivation} for a derivation of \eqref{eq:nacf definition}, and Section \ref{sec: bounds} for a proof of Proposition \ref{prop: nacf one}. Our PNACF is defined below.
\begin{definition} \label{def: pnacf}
For a stationary $\GNAR$ process $\boldsymbol{X}_t$ compute the residuals coming from a $\GNAR(h - 1, [(r - 1)])$ fit, the corresponding residual mean $\overline{\boldsymbol{u}}$, and the residuals with $\hlag$s between them
    $$\boldsymbol{\hat{u}}_{t + h} = \boldsymbol{X}_{t + h} - \hat{\boldsymbol{X}}_{t + h} (h - 1, r - 1), $$
    $$\boldsymbol{\hat{u}}_{t} = \boldsymbol{X}_{t} - \hat{\boldsymbol{X}}_{t} (h - 1, r - 1).$$
Then, the sample partial network autocorrelation function is given by
\begin{equation} \label{eq: pnacf}
    \pnacf(h, r) := \frac{\sum_{t=1}^{T - h} ( \boldsymbol{\hat{u}}_{t + h} - \boldsymbol{\overline{u}})^{T} \big ( \bmat{W} \odot \bmat{S}_r + \bmat{I_d} \big )
    ( \boldsymbol{\hat{u}}_{t} - \boldsymbol{\overline{u}})}
    {\sum_{t=1}^{T}  ( \boldsymbol{\hat{u}}_{t} - \boldsymbol{\overline{u}})^{T} \big \{ \big (1 + \lambda \big) \bmat{I_d} \big \} ( \boldsymbol{\hat{u}}_{t} - \boldsymbol{\overline{u}})}, 
\end{equation}
where $\lambda$ is the same as in Definition \ref{def: nacf definition},
$$\hat{\boldsymbol{X}}_{t} (h - 1, r - 1) = \sum_{k = 1}^{h - 1} \left ( \hat{\alpha}_k \boldsymbol{X}_{t - k} + \sum_{s = 1}^{r - 1} \hat{\beta}_{ks} \boldsymbol{Z}_{t - k}^s) \right )$$
is the best linear predictor constrained to $(h - 1)$ lags and $(r - 1)$ $\rstage$ depth for all lags, and $\boldsymbol{Z}_{t - k}^r = (\bmat{W} \odot \bmat{S}_r) \boldsymbol{X}_{t- k}$ is the vector of $\rstage$ neighbourhood regressions. 
\end{definition}


\appendix
\setcounter{equation}{0}
\counterwithin{equation}{section}

\input{nacf_derviation}

\input{nacf_bounds}

\input{r_stage_adjacency}

\input{correlation_structure}

\input{simulations}

\input{corr_structure_illustration}

\bibliographystyle{agsm}
\bibliography{references}


\end{document}
