The GNAR model of order \((p,S)\) is defined as
$$X_{i,t} = \sum_{j=1}^p \left( \alpha_{i,j} X_{i,t-j} +\sum_{c=1}^C \sum_{r=1}^{S_j} \beta_{j,r, c}\sum_{q \in {N}^{(r)}_t(i)} \omega_{i,q,c}^{(t)} X_{q,t-j} \right) + u_{i,t}$$
where \(p\) is the maximum time lag, \(S=(S_1,...,S_p)\) and \(S_j \) is the maximum stage of neighbour dependence for time lag \(j\), \({N}^{(r)}_t(i)\) is the \(r\)th stage neighbour set of node \(i\) at time \(t\), \(\omega_{i,q,c}^{(t)}\) is the connection weight between node \(i\) and node \(q\) at time \(t\) if the path corresponds
to covariate \(c\). Here, we consider a sum from one to zero to be zero and \(\{u_{i,t}\}\), are assumed to be independent and identically distributed at each node \(i\), with mean zero and variance \(\sigma_i^2\).
Currently, only a single network GNAR model can be fitted.
The connection weight, \(\omega_{i,q,c}^{(t)}\), is the inverse of the distance between nodes i and q, normalised so that they sum to 1 for each i,t.
See is.GNARnet for GNARnet object information and example construction.
A GNARX process of order \(p\), neighbourhood order vector \(s\) of length \(p\) and \(H\) node-specific time series exogenous variables with order vector \(p'\), denoted
GNARX \((p, s, p')\), is given by
$$
Y_{i,t}=\sum_{j=1}^{p}\left(\alpha_{i,j}Y_{i,t-j} + \sum_{r=1}^{s_j} \beta_{j,r} \sum_{q\in\mathcal{N}^{(r)}(i)}\omega_{i,q}Y_{q,t-j}\right)+ {\color{blue} \sum_{h=1}^H
\sum_{j'=0}^{p'_h}
\lambda_{h,j'} X_{h,i,t-j'}+u_{i,t}},
$$
where \( u_{i, t}\) are assumed to be set of mutually uncorrelated random variables with
mean zero and variance of \(\sigma^2\).