## 4 Main Results  
For the LSTAR model, in this section we discuss how to determine the parameter \(\phi_{i,bh}\) with a given weight matrix \(W_{i}^{(h)}(t)\). With the determined weight matrix \(W_{i}^{(h)}(t)\) and \(\phi_{i,kh}\), the future traffic observation \(\hat{z}(t)\) of road \(i\) can be predicted. Normally, \(w_{ij}^{(h)}(t)\) is time variant. When the traffic flow distribution is stable in the road network, \(w_{ij}^{(h)}(t)\) will be time invariant since \(\frac{Q^{(h)}{}_{ij}(t)}{\sum Q^{(h)}(t)}\) is constant. Considering the weight matrix differences of being time invariant or not, two theorems are discussed, and the LSTAR parameters estimation can be conducted accordingly. **Definitions 1. **\(r_{ij}(m)\) _is the correlation coefficient of road \(i\) and \(j\), \(r_{i}(m)=r_{ii}(m)\) is the autocorrelation coefficient of road \(i\). We define \(\Phi_{i}=\left[\phi_{i,10},\phi_{i,11},\ldots\phi_{i,1\lambda_{i0}},\phi_{i,20 },\phi_{i,21},\ldots,\phi_{i,p_{i}\lambda_{i}}\right]^{T}\)  
Figure 1: Traffic information collection. _let \(n=\sum_{k=1}^{p_{i}}\lambda_{i,k}\) to be the dimension of \(\Phi_{i}\). \(R_{i}=\left[r_{i}(0),r_{i}(1),\ldots,r_{i}(n-1)\right]^{T}\), \(\Sigma=\left[\sigma_{i}^{2},0,\ldots,0\right]^{T}\), \(S(h,m)=\sum_{j=1}^{N_{i}}w_{ij}^{(h)}r_{ij}(m)\),_  
\[A=\left[ \begin{array}{cccc}S(0,1)&S(1,1)&\cdots&S(\lambda_{i,k},p_{i})\\ S(0,0)&S(1,0)&\cdots&S(\lambda_{i,k},p_{i}-1)\\ S(0,1)&S(1,1)&\cdots&S(\lambda_{i,k},p_{i}-2)\\ \vdots&\vdots&\vdots&\vdots\\ S(0,n-1)&S(1,n-1)&\cdots&S(\lambda_{i,k},n-1-p_{i})\end{array} \right],\overline {A}=\left[ \begin{array}{cc}A&R_{i}-\Sigma\end{array} \right]\). Theorem 1 follows:  
**Theorem 1. **_If \(W_{i}^{(h)}(t)\) is time invariant and \(Rank(A)=Rank(\overline{A})=n\), the LSTAR model can be uniquely determined._  
**Proof. ** As \(W_{i}^{(h)}(t)\) is time invariant, we can observe that \(W_{i}^{(h)}(t)=W_{i}^{(h)}(t-k)\) and it can be rewritten to \(W_{i}^{(h)}\). The model of Equation (5) will be a time invariant system. Since \(z_{i}(t)\) is a stationary random signal, Equation (5) will be:  
\[z_{i}(t)=\sum_{k=1}^{p_{i}}\sum_{h=0}^{\lambda_{i,k}}\sum_{j=1}^{N_{i}}\phi_{i,kh}w_{ij}^{(h)}z_{j}(t-k)+\varepsilon_{i}(t). \tag{7}\]  
Pre-multiplying both sides of the model (Equation (7)) by \(z_{i}(t-m)\):  
\[z_{i}(t)z_{i}(t-m)=\sum_{k=1}^{p_{i}}\sum_{h=0}^{\lambda_{i,k}}\sum_{j=1}^{N_{ i}}\phi_{i,kh}w_{ij}^{(h)}z_{j}(t-k)z_{i}(t-m)+\varepsilon_{i}(t)z_{i}(t-m). \tag{8}\]  
Taking the expected values in both sides, we obtain an equation similar to the Yule-Walker equation:  
\[r_{i}(m)=\sum_{k=1}^{p_{i}}\sum_{h=0}^{\lambda_{i,k}}\sum_{j=1}^{N_{i}}\phi_{ i,kh}w_{ij}^{(h)}r_{ij}(m-k)+\sigma_{i}^{2}\delta(m) \tag{9}\]  
where the expected value is:  
\[E(\varepsilon_{i}(t)z_{i}(t-m))=\sigma_{i}^{2}\delta(m). \tag{10}\]  
As \(\phi_{i,hk}\) does not have tag \(j\), we obtain:  
\[r_{i}(m)=\sum_{k=1}^{p_{i}}\sum_{h=0}^{\lambda_{i,k}}\phi_{i,kh}\sum_{j=1}^{N_ {i}}w_{ij}^{(h)}r_{ij}(m-k)+\sigma_{i}^{2}\delta(m). \tag{11}\]  
For Equation (11), rewrite \(\phi_{i,kh}\) to a column vector as \(\Phi_{i}=\left[\phi_{i,10},\phi_{i,11},\ldots,\phi_{i,1\lambda_{i,0}},\phi_{ i,20},\phi_{i,21}\ldots\phi_{i,k\lambda_{i,k}}\right]^{T}\). In addition, \(n=\sum_{k=1}^{p_{i}}\lambda_{i,k}\) is the number of parameters \(\phi_{i,kh}\) to be estimated. As \(S(h,m)=\sum_{j=1}^{N_{i}}w_{ij}^{(h)}r_{i,j}(m)\), we obtain:  
\[\left[ \begin{array}{c}r_{i}(0)\\ r_{i}(1)\\ r_{i}(2)\\ \vdots\\ r_{i}(n-1)\end{array} \right]=\left[ \begin{array}{cccc}S(0,1)&S(1,1)&\dots&S( \lambda_{i,k},p_{i})\\ S(0,0)&S(1,0)&\dots&S(\lambda_{i,k},p_{i}-1)\\ S(0,1)&S(1,1)&\dots&S(\lambda_{i,k},p_{i}-2)\\ \vdots&\vdots&\vdots&\vdots\\ S(0,n-1)&S(1,n-1)&\dots&S(\lambda_{i,k},n-1-p_{i})\end{array} \right]\left[ \begin{array}{c}\phi_{i,10}\\ \phi_{i,11}\\ \phi_{i,12}\\ \vdots\\ \phi_{i,k\lambda_{i}}\end{array} \right]+\left[ \begin{array}{c}\sigma_{i}^{2 }\\ 0\\ 0\\ \vdots\\ 0\end{array} \right]. \tag{12}\]  
\[\text{Let }A=\left[