## 3 Methodology
### Prediction Model
In the control feedback process of the traditional multivariate discrete grey prediction model, the system development parameters are often set in constant coefficients, and may deviate from the actual situation. The actual system developmental dynamics parameters are often multivariate influenced and time-varying. First, the developmental dynamics are not only related to the development of the system itself but are also affected by the development dynamics of other influencing factors of the system. Second, the developmental dynamics of the influencing factors make the system's developmental dynamics constantly change over time. Considering the above two features, a grey model with the trend information of development factors and a dynamic background-value coefficient has been proposed, or, N-variable Feedback Optimizing Background Grey Model, abbreviated as FOBGM (1, N) for convince. As shown in Figure 6, AGO is the accumulating generation operator which can reduce the effects of stochastic variation in system behavior factors and their influences by cumulative generation. The \(\rho\), defined in Equation (8), is called an extractor which is used to extract information on the development trend of influencing factors. The \(\gamma_{i}(i=1,2,\ldots,\)\(m-1)\) are named as parameters of developmental states. \[\rho=\frac{x^{(0)}(t+1)}{x^{(1)}(t)} \tag{8}\]  
Figure 6: Details of multi-factor information collection. Assume that X1(0) and Xj(0) are the same as they were previously defined. Equation (9) is the OGM(1, N) with development factor trend information and a dynamic background value, which is the FOBGM(1, N)  
\[(1+a\xi)\dot{x}_{1}^{(1)}(k)=\beta(1+\rho_{a}(k))x_{1}^{(1)}(k-1)+(1-a+a\xi)x_{1 }^{(1)}(k-1)+kc+d \tag{9}\]  
where the \(\xi\), is the background value ranging from 0 to 1. The \(\rho\)a is the integrated system development situation, defined in Equation (10). \[\rho_{a}(k)=\gamma_{1}\frac{x_{2}^{(0)}(k)}{x_{2}^{(1)}(k-1)}+\gamma_{2}\frac{ x_{3}^{(0)}(k)}{x_{3}^{(1)}(k-1)}+\cdots+\gamma_{n-1}\frac{x_{n}^{(0)}(k)}{x_{n}^ {(1)}(k-1)} \tag{10}\]  
The \(p_{1}\) and \(p_{2}\) are the parameter sequences of the new model, as stated in Equation (11). The least-square method may be used to estimate both sequences. Once the \(p_{1}\) and \(p_{2}\) have already known, the \(\xi\) will take the minimum case of the Mean Absolute Percentage Error (MAPE) between the calculated values and actual series \(X_{1}^{(0)}\) as its value, establishing the OGM(1, N) model with trend information of development components and a dynamic background-value coefficient. \[ \begin{array}{l}p_{1}=[\beta,a,c,d]\\ p_{2}=[\gamma_{1},\gamma_{2},\cdots,\gamma_{n-1}]\end{array}  \tag{11}\]  
The parameters of FOBGM(1, N) are going to be estimated by ordinary least squares(OLS). Let the \(X_{i}^{(0)}(i=1,2,\,\ldots,\,\text{N})\) and \(X_{i}^{(1)}(j=1,2,\,\ldots,\,\text{N})\) are defined in Equation (5), and the \(p_{1}\) can satisfy Equation (12) by least squares estimation. \[p_{1}=(E^{T}E)^{-1}E^{T}S \tag{12}\]  
The \(E\) and \(S\) are defined as follows. \[ \begin{array}{l}E=\left[\begin{array}{cccc}(1+\rho_{a}(2))x_{1}^{(1)}(1)&( \xi-1)x_{1}^{(1)}(1)-\xi x_{1}^{(1)}(2)&2&1\\ (1+\rho_{a}(3))x_{1}^{(1)}(2)&(\xi-1)x_{1}^{(1)}(2)-\xi x_{1}^{(1)}(3)&3&1\\ \vdots&\vdots&\vdots&\vdots\\ (1+\rho_{a}(m))x_{1}^{(1)}(m-1)&(\xi-1)x_{1}^{(1)}(m-1)-\xi x_{1}^{(1)}(m)&m&1 \end{array}\right]\\ \text{S}=\left[\begin{array}{c}x_{1}^{(1)}(2)\\ x_{1}^{(1)}(3)\\ \vdots\\ x_{1}^{(1)}(m)\end{array}\right]\end{array}  \tag{13}\]  
The \(p_{2}\) in Equation (11) can use the same way to figure out  
\[p_{2}=\left(B_{\rho}^{T}B_{\rho}\right)^{-1}B_{\rho}^{T}Y_{\rho} \tag{14}\]where  
\[B_{\rho}=\left[