## 4 Main Results
\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],R_{i}= \left[ \begin{array}{c}r_{i}(0)\\ r_{i}(1)\\ r_{i}(2)\\ \vdots\\ r_{i}(n-1)\end{array} \right],\Sigma=\left[ \begin{array}{c}\sigma_{i}^{2} \\ 0\\ 0\\ \vdots\\ 0\end{array} \right].\]  
Then, the augmented matrix of Equation (12) is:  
\[\overline{A}=\left[ \begin{array}{cccc}S(0,1)&S(1,1)&\dots&S(\lambda_{i,k}, p_{i})&r_{i}(0)-\sigma_{i}^{2}\\ S(0,0)&S(1,0)&\dots&S(\lambda_{i,k},p_{i}-1)&r_{i}(1)\\ S(0,1)&S(1,1)&\dots&S(\lambda_{i,k},p_{i}-2)&r_{i}(2)\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ S(0,n-1)&S(1,n-1)&\dots&S(\lambda_{i,k},n-1-p_{i})&r_{i}(n-1)\end{array} \right]= \left[ \begin{array}{cc}A&R_{i}&-\Sigma\end{array} \right].\]  
If \(Rank(A)=n\), then \(Rank(\overline{A})=n\) and we will have a unique solution of parameters \(\Phi_{i}\):  
\[\Phi_{i}=A^{-1}(R_{i}-\Sigma) \tag{13}\]  
Then, we can uniquely define the LSTAR model and predict traffic flow with it. If \(R(A)\
eq R(\overline{A})\), there is no solution for Equation (12). If \(Rank(A)=Rank(\overline{A})<n\), there are infinite solutions for \(\Phi_{i}\).\(\square\)  
**Remark 1**. : _In case the spatial weight matrix \(W_{i}^{(h)}(t)\) is time invariant, we can determine the LSTAR prediction model by the correlation of roads. We can uniquely define the LSTAR model when \(Rank(A)=Rank(\overline{A})=n\). When \(Rank(A)=Rank(\overline{A})<n\), there will be many solutions for parameter \(\Phi\). This means that we have defined more parameters than are actually needed. We can reduce \(\lambda_{i,k}\) and/or \(p_{i}\) to obtain a unique LSTAR model._  
**Definitions 2**. : _Let \(r_{i}^{\prime}(m)\) be the autocorrelation coefficient of \(ith\) element of \(U(t)=W(t)Z(t)\), where the matrix \(W(t)=\sum_{h=0}^{\lambda_{i,k}}W_{i}^{(h)}(t)\), which combines all spatial effect defined in \(W_{i}^{(h)}(t)\) to one matrix, and \(Z(t)=\left[z_{1}(t),z_{2}(t),\dots,\,z_{N_{i}}(t)\right]^{T}\) is the vector form of \(z_{i}(t)\). We define \(\Phi_{i}=\left[\phi_{i,1},\phi_{i,2},\dots,\phi_{i,p_{i}}\right]^{T}\), \(\overline{R}_{i}^{\prime}=\left[\overline{r}_{i}^{\prime}(0),\overline{r}_{i}^ {\prime}(1),\dots,\overline{r}_{i}^{\prime}(p_{i}-1)\right]^{T}\), \(\Sigma=\left[\sigma_{i}^{2},0,\dots,\,0\right]^{T}\),_  
\[A^{\prime}=\left[ \begin{array}{cccc}r_{i}^{\prime}(1)&r_{i}^{\prime}(2)& \dots&r_{i}^{\prime}(p_{i})\\ r_{i}^{\prime}(0)&r_{i}^{\prime}(1)&\dots&r_{i}^{\prime}(p_{i}-1)\\ \vdots&\vdots&\vdots&\vdots\\ r_{i}^{\prime}(p_{i}-2)&r_{i}^{\prime}(p_{i}-3)&\dots&r_{i}^{\prime}(1)\end{array}  \right],\overline{A^{\prime}}=\left[ \begin{array}{cc}A^{\prime}&\overline{R}_{ i}^{\prime}-\Sigma\end{array} \right].\]  
We then present Theorem 2. **Theorem 2**. : _If \(W_{i}^{(h)}(t)\) is time variant, the combined weight matrix \(W(t)\) is full ranked, and \(Rank(A^{\prime})=Rank(\overline{A^{\prime}})=p_{i}\), the LSTAR model can be uniquely determined._  
**Proof. ** According to the weight matrix \(W_{i}^{(h)}(t)\) construction in the LSTAR model, the element \(w_{ij}^{(h)}(t)\) will always be zero when \(i\), \(j\) is not at spatial order \(h\). We combine all of the \(\lambda_{i,k}\) weight matrix into one weight matrix, \(W(t)=\sum_{h=0}^{\lambda_{i,k}}W_{i}^{(h)}(t)\).