# Inversion of Chlorophyll-a Concentration in Wuliangsu Lake Based on OGolden-DBO-XGBoost
## 3 Methods
### Improved Dung Beetle Optimization Algorithm
#### 3.2.2. Golden Sine Algorithm Update Location  
[PERSON] et al. [31] proposed the golden sine technique (Gold-SA) in 2017, a novel meta-heuristic algorithm that uses the mathematical sine function for computational iterative optimization search. The relationship between the sine function and the unit circle indicates that it is possible to search for all of the points on the unit circle by traversing all of the values on the sine function. At the same time, the golden section number is introduced to reduce the space of the solution while it is updating its position. This allows for the scanning of the region that may yield only positive results, which significantly speeds up the search and improves the balance between the ability to search globally and locally. The updating process of the solution of the Gold-SA is shown in Equations (5) and (6):  
\[X_{i}^{n}(t+1) =X_{i}^{n}(t)|\sin(\tau_{1})|+r_{2}\sin(\tau_{1})S \tag{5}\] \[S =|x_{1}B^{n}(t)-x_{2}X_{i}^{n}(t)| \tag{6}\]  
where \(r_{1}\) is a random number uniformly distributed in the range [0,2\(\pi\)], which determines the moving distance of the individual in the next iteration, and \(r_{2}\) is a random number uniformly distributed in the range [0,2\(\pi\)], which determines the direction of the individual's position update in the next iteration. \(B^{n}(t)\) is the globally optimal position in the tth iteration and \(x_{1}\) and \(x_{2}\) are the coefficients obtained by introducing the golden section number. These coefficients reduce the search space to lead the individual to gradually converge to the optimal value, which ensures the convergence of the algorithm, and the computational formula is shown in Equations (7)-(9):  
\[\tau=(\sqrt{5}-1)/2 \tag{7}\]  
\[x_{1}=-\pi\tau+\pi(1-\tau) \tag{8}\]  
\[x_{2}=-\pi(1-\tau)+\pi\tau \tag{9}\]  
The Gold-SA is introduced to improve the ability of global search and local exploitation of the DBO algorithm. In the rolling dung beetle behavioral update strategy, a random number \(r\) between 0 and 1, and a constant rate between 0 and 1, indicating the probability of encountering an obstacle, are generated. When \(r>rate\), the dung beetle encounters an obstacle and the dung beetle's position is updated using Gold-SA (Equations (5) and (6)), and when \(r<rate\), the update is performed according to the original strategy (Equations (1) and (2)). The improved rolling behavior update strategy is shown in Figure 4.  
Figure 4: Flowchart of the improved DBO algorithm with Gold-SA.  
The improved dung beetle rolling behavior position update strategy is shown in Equation (10):  
\[X_{i}^{n}(t+1)=\left\{\begin{array}{l}X_{i}^{n}(t)+\alpha\times k\times X_{i}^{ n}(t-1)+b\times\Delta x,r\leq rate\\ X_{i}^{n}(t)\left|\sin(r_{1})\right|+r_{2}\sin(r_{1})\big{|}x_{1}B^{n}(t)-x_{2 }X_{i}^{n}(t)\big{|},r>rate\end{array}\right. \tag{10}\]  
where \(r\) is a random uniformly distributed random number in the range (0,1) and rate is a fixed value in the range (0,1). When \(r<rate\), it means that the dung beetle rolls with a goal, and when \(r>rate\), it means that the dung beetle encounters an obstacle, which needs to be iteratively updated in position by the Gold-SA.  
By introducing the golden sine algorithm, the DBO algorithm improves the position update strategy of the dancing behavior, so that dung beetle individuals will communicate with the current optimal individual \(B^{n}(t)\) for information exchange, which promotes information exchange between individuals.