# Neural Multivariate Grey Model and Its Applications
## 2 Literature Review
### GM(1, N) Model  
The GM(1, N) model operates as a prediction model with N variables [6]. It is designed to handle N-1 influencing factors alongside one main system behavior variable. The GM(1, N) model differs from traditional univariate models like GM(1, 1) by incorporating multiple relevant factors, thus enhancing prediction accuracy in complex systems. Then, a time series of the \(m\) dimension or an original sequence  
\[X_{i}^{(0)}=(x_{i}^{(0)}(1),x_{i}^{(0)}(2),\cdots,x_{i}^{(0)}(m)), \tag{1}\]  
where the system characteristic sequence is \(i=1\), and the relevant factor sequences are i = 2, 3, \(\cdots\), N, \(X_{j}^{(1)}\) is the 1-AGO (accumulating generation operator) sequences of \(X_{j}^{(0)}\), j = 1, 2, \(\cdots\), N, which are defined as  
\[X_{j}^{(1)}=\big{(}x_{j}^{(1)}(1),x_{j}^{(1)}(2),\cdots,x_{j}^{(1)}(m)\big{)}, \tag{2}\]  
where \(x_{j}^{(1)}(i)=\sum_{k=1}^{i}x_{j}^{(0)}(k),i=1,2,\cdots,m\). An advantageous aspect of AGO is its ability to uncover potential hidden patterns within data sequences, facilitating the identification of underlying regularities [8]. The mean sequence that is created by consecutive neighbors of \(X_{1}^{(1)}\) is denoted by the \(Z_{1}^{(1)}\):  
\[Z_{1}^{(1)}=\big{(}z_{1}^{(1)}(2),z_{1}^{(1)}(3),\cdots,z_{1}^{(1)}(m)\big{)}, \tag{3}\]where \(z_{1}^{(1)}(k)=0.5*(x_{1}^{(1)}(k)+x_{1}^{(1)}(k-1)),k=2,3,\cdots,m\), then the GM(1, N) model [6] can be defined as:  
\[x_{1}^{(0)}(k)+az_{1}^{(1)}(k)=\sum_{j=2}^{N}b_{j}x_{j}^{(1)}(k),k=2,3,\ldots,m, \tag{4}\]  
where \(-a\) represents the development coefficient, \(b_{i}\) stands for the driving coefficient, and \(b_{i}x_{i}(k)\) denotes the driving item. The parameter sequence is denoted by \(\hat{a}=[a,b_{2},b_{3},\cdots,b_{N}]^{T}\). Then, the GM(1, N) model's whitening equation [6] is denoted by  
\[\frac{dx_{1}^{(1)}}{dt}+ax_{1}^{1}=\sum_{j=2}^{N}b_{j}x_{j}^{(1)}. \tag{5}\]  
Ordinary least squares (OLS) are used by the GM (1,N) to calculate \(a\),\(b_{2}\),\(b_{3}\),\(\cdots\), \(b_{N}\)[6]:  
\[[a,b_{2},\ldots,b_{N}]=(\textbf{B}^{T}\textbf{B})^{-1}\textbf{B}^{T}\textbf{Y}, \tag{6}\]  
\[\text{where }\textbf{B}= \begin{bmatrix}-z_{1}^{(1)}(2)&x_{2}^{(1)}(2)&\cdots&x_{N }^{(1)}(2)\\ -z_{1}^{(1)}(3)&x_{2}^{(1)}(3)&\cdots&x_{N}^{(1)}(3)\\ \vdots&\vdots&\ddots&\vdots\\ -z_{1}^{(1)}(m)&x_{2}^{(1)}(m)&\cdots&x_{N}^{(1)}(m)\end{bmatrix} ,\textbf{Y}= \begin{bmatrix}x_{1}^{(0)}(2)\\ x_{1}^{(0)}(3)\\ \vdots\\ x_{1}^{(0)}(m)\end{bmatrix} .\]  
With the initial condition, \(x_{1}^{(1)}(0)\) is taken to \(x_{1}^{(0)}(1)\). The whitening equation (5), has a solution [6] that can be obtained by:  
\[x_{1}^{(1)}(t)=e^{-ak}\Big{[}x_{1}^{(0)}-t\sum_{j=2}^{N}b_{j}x_{j}^{(1)}(0)+ \sum_{j=2}^{N}\int b_{j}x_{j}^{(1)}(t)e^{at}dt\Big{]}. \tag{7}\]  
When there are tiny changes in \(X_{i}^{(1)}(i=1,2,\cdots,N)\), \(\sum_{j=2}^{N}b_{j}x_{j}^{(1)}(0)\) represents a grey constant. We can obtain the approximate time-response sequence [6]\(\hat{x}_{k}^{(1)}\) of the GM(1, N) model:  
\[\hat{x}_{1}^{(1)}(1)=\Big{[}x_{1}^{(0)}-\frac{1}{a}\sum_{j=2}^{N}b_{j}x_{j}^{ (1)}(k+1)\Big{]}e^{-ak}+\frac{1}{a}\sum_{j=2}^{N}b_{j}x_{j}^{(1)}(k+1). \tag{8}\]  
Ultimately, by employing the inverse accumulated generation operation (IAGO), the forecasting value for the raw data \(\hat{x}_{1}^{(0)}(k)\) of \(x_{1}^{(0)}(k)\) can be reconstructed using the following equation:  
\[\hat{x}_{1}^{(0)}(k)=\hat{x}_{1}^{(1)}(k)-\hat{x}_{1}^{(1)}(k-1). \tag{9}\]  
By extensively investigating the definition and modeling procedure of the GM(1, N) model, it has been demonstrated that the model's structure, parameter application, and modeling mechanism have several shortcomings [20]. The conventional GM(1, N) model faces limitations in its application and stability primarily because of these critical challenges.