## 3 Methodology
### Prediction Model
\begin{array}{cccc}\frac{x_{1}^{(0)}(2)}{x_{2}^{(1)}(1)}&\frac{x_ {3}^{(0)}(2)}{x_{3}^{(1)}(1)}&\cdots&\frac{x_{n}^{(0)}(2)}{x_{n}^{(1)}(1)}\\ \frac{x_{2}^{(0)}(3)}{x_{2}^{(1)}(2)}&\frac{x_{3}^{(0)}(3)}{x_{3}^{(1)}(2)}& \cdots&\frac{x_{n}^{(0)}(3)}{x_{n}^{(1)}(2)}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{x_{2}^{(0)}(m)}{x_{1}^{(1)}(m-1)}&\frac{x_{3}^{(0)}(m)}{x_{3}^{(1)}(m-1) }&\cdots&\frac{x_{n}^{(0)}(m)}{x_{n}^{(1)}(m-1)}\end{array} \right] \tag{15}\]  
Because of the unknown variables in it, Equation (9) cannot be used to directly calculate the values of the dependent variable. As a result, a new time response function of FOBGM(1, N) should be inferred. Equation (9) can be rewritten as follows:  
\[\begin{array}{l}(1+a\xi)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\\ x_{1}^{(1)}(k)=\frac{1}{1+a\xi}\beta(1+\rho_{a}  
Since the expression is too complex, taking the iterative method to figure out \(\hat{x}_{1}^{(1)}(p)\). The final predicted value can be computed as follows.  
\[\hat{x}_{1}^{(0)}(t+1)=\hat{x}_{1}^{(1)}(t+1)-\hat{x}_{1}^{(1)}(t),t=1,2\cdots,m \tag{23}\]  
To avoid the influence of unreasonable background values on the prediction effect of the model, the Whale Optimization Algorithm (WOA) [50] will be used to optimize the dynamic background-value coefficient. The algorithm is inspired by the bubble-net hunting method and simulates humpback whale social behavior. The position of each whale in the whale algorithm symbolizes a plausible solution. The MAPE is used to measure the quality of the dynamic background value, as follows. \[\text{min}f(\xi)=\frac{100\%}{m-1}\underset{i=2}{\overset{m}{\sum}}\left| \frac{\hat{x}_{1}^{(0)}(k)-x_{1}^{(0)}(k)}{x_{1}^{(0)}(k)}\right| \tag{24}\]  
Throughout the procedure, each whale will exhibit three distinct behaviors. The first behavior is surrounding prey. \[ \begin{array}{c}\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C} \cdot\overset{\rightarrow}{X}^{*}(t)-\overset{\rightarrow}{X}(t)\right|\\ \overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X}^{*}(t)-\overset{ \rightarrow}{A}\cdot\overset{\rightarrow}{D}\end{array}  \tag{25}\]  
In which, the \(t\) is the current iteration, \(A\) and \(C\) are coefficient vectors, and \(X^{*}\) denotes the position vector of the best solution achieved thus far. The second behavior is the search for prey and the mathematical model is as follows. \[ \begin{array}{c}\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C} \cdot\overset{\rightarrow}{X}_{rand}-\overset{\rightarrow}{X}\right|\\ \overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X}_{rand}-\overset{ \rightarrow}{A}\cdot\overset{\rightarrow}{D}\end{array}  \tag{26}\]  
In this equation, \(\overset{\rightarrow}{X}_{rand}\) is a location vector picked at random from the current population. The last is called as spiral updating position, which updates the position by a spiral function, as Equation (27). \[ \begin{array}{c}\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{D} \prime\cdot e^{bl}\cdot\cos(2\pi l)+\overset{\rightarrow}{X}^{*}(t)\\ \overset{\rightarrow}{D}\prime=\left|\overset{\rightarrow}{X}^{*}(t)- \overset{\rightarrow}{X}(t)\right|\end{array}  \tag{27}\]  
The \(\overset{\rightarrow}{D}\prime\) is the distance between the best search agent and the prey, \(b\) controls the form of the logarithmic spiral and \(l\) is a random value ranging from \(-1\) to \(1\). The \(A\) and \(C\) appearing in Equations (25)-(27), respectively, are coefficient vectors that can be derived as follows. \[ \begin{array}{c}\overset{\rightarrow}{A}=2\overset{\rightarrow}{a} \cdot\overset{\rightarrow}{r}_{1}-\overset{\rightarrow}{a}\\ \overset{\rightarrow}{C}=2\cdot\overset{\rightarrow}{r}_{2}\end{array}  \tag{28}\]  
The \(\overset{\rightarrow}{a}\) is a variable that falls from \(2\) to \(0\) throughout the course of the whole iteration and \(\overset{\rightarrow}{r}\) is a random vector between \(0\) and \(1\).