## 3 Model and Preliminaries  
### LSTAR Model Construction  
According to the STARIMA model defined in Reference [11], both space and time are considered. \[Z_{t}=\sum_{k=1}^{p}\sum_{l=0}^{\lambda_{k}}\phi_{kl}W_{l}z_{t-k}-\sum_{k=1}^{ q}\sum_{l=0}^{m_{k}}\theta_{kl}W_{l}\varepsilon_{t-k}+\varepsilon_{t}. \tag{1}\]  
\(Z_{t}\) is an \(N\)-dimensional column vector of road \(i\), while \(i=1,\,2,\,\ldots,\,N\), and \(W_{l}\) is an \(N\times N\) matrix with element \(w_{ij}^{l}\). \(\varepsilon_{t}\) is the residual vector. \(\phi_{kl}\), \(\theta_{kl}\) are the AR and MA parameters, respectively. Then, the observation of road \(i\), \(z_{i}(t)\) can be described as:  
\[z_{i}(t)=\sum_{k=1}^{p}\sum_{l=0}^{\lambda_{k}}\sum_{j=1}^{N}\phi_{kl}w_{ij}^{ (l)}z_{j}(t-k)-\sum_{k=1}^{q}\sum_{l=0}^{m_{k}}\sum_{j=1}^{N}\theta_{kl}w_{ij} ^{(l)}\varepsilon_{j}(t-k)+\varepsilon_{i}(t). \tag{2}\]  
In 2014, a new space-time model, the localized STARIMA (LSTARIMA) model, was proposed by [PERSON] et al. [13] to consider spatial heterogeneity and temporal non-stationarity. The model is described by the following form:  
\[Z_{i}(t)=\sum_{k=1}^{p_{i}}\sum_{h=0}^{\lambda_{k}(t-k,i)}\phi_{ij,kh}W^{(h,t -k,i)}Z_{i}(t-k)-\sum_{l=1}^{q_{i}}\sum_{h=0}^{n_{l}(t-l,i)}\theta_{l,ll}W^{( h,t-l,i)}\varepsilon_{i}(t-l)+\varepsilon_{i}(t). \tag{3}\]  
\(Z_{i}(t)\) is an \(N\)-dimensional column vector of the observation value on link 1, \(\ldots,\,N\) with tag \(i\) at time \(t\), which can be any prediction variable of roads such as speed, traffic flow, density, and so on. The term \(\varepsilon_{i}(t)\) is a residual on link 1, \(\ldots,\,N\) at time \(t\). The first term in Equation (3) is the AR component, whereas the second term is the MA. The parameters \(p_{i}\) and \(q_{i}\) are the AR and MA orders, respectively. \(h\) is the spatial order that represents the order of spatial separation between two locations. The parameters \(\lambda_{k}(t-k,i)\) and \(n_{l}(t-l,i)\) are the dynamic spatial orders associated with the \(k\)th and \(l\)th temporally lagged terms in the AR and MA components, respectively. They specify the size of the spatial neighborhood that could influence the link of interest \(i\) within temporal lags \(k\) and \(l\). The parameters \(W^{(h,t-k,i)}\) and \(W^{(h,t-l,j)}\) are the dynamic spatial weight matrices \(W^{(h,t,i)}\) pertaining to link \(i\) at temporal lags \(k\) and \(l\). \(\phi_{ij,kh}\) and \(\theta_{i,ll}\) are the AR and MA parameters for each link \(i\) (\(i=1,\,2,\,\ldots,\,N\)). Although spatial tag \(i\) was added to \(Z_{i}(t)\) in the LSTARIMA model construction, the spatial heterogeneity was not fully considered. As the \(Z_{i}(t)\) here is an \(N\)-dimension column vector which covers all of the roads (road 1, 2, \(\ldots,N\)) in the network, all of the roads will share the same \(\phi_{ij,kh}\) and \(\theta_{i,ll}\). In this LSTARIMA model, the same matrix W is used for both AR and MA components. As the weight matrix of AR and MA components is not always the same, using only one weight matrix \(W\) is not proper. In this paper, different weight matrices \(W\), \(U\) were used for AR and MA components and individual road traffic flow observation \(z_{i}(t)\), according to Equation (2), was defined to allow every road to have its own weight matrix \(W_{i}\), \(U_{i}\) according to spatial location, but not sharing the same weight matrix among all roads, as in Equation (3). Then, the LSTARIMA can be rewritten as follows:  
\[z_{i}(t)=\sum_{k=1}^{p_{i}}\sum_{h=0}^{\lambda_{i,k}}\sum_{j=1}^{N_{i}}\phi_{i,kh }w_{ij}^{(h)}(t-k)z_{j}(t-k)-\sum_{k=1}^{q_{i}}\sum_{h=0}^{m_{ijk}}\sum_{j=1}^{N _{i}}\theta_{i,kh}u_{ij}^{(h)}(t-k)\varepsilon_{j}(t-k)+\varepsilon_{i}(t). \tag{4}\]  
The parameters \(w_{ij}^{(h)}(t-k)\) and \(u_{ij}^{(h)}(t-l)\) are the elements of dynamic spatial weight matrices \(W_{i}^{(h)}(t-k)\) and \(U_{i}^{(h)}(t-l)\) pertaining to link \(i\) at temporal lags \(k\) and \(l\).