## 2 Model Introduction
### AWOA-DBiLSTM Neural Network Modeling
#### 2.4.2 Adaptive Whale Optimization Algorithm (AWOA)  
The whale optimization algorithm (WOA) first sets up the parameters, such as the maximum number of iterations and the size of the whale population. Next, it sets up the position of the whale population, determines the fitness value of each whale, sorts the whales based on the size of the fitness value, and chooses the initial population. Finally, it determines the size of each individual's fitness value and determines the optimal position of the whale with the smallest fitness value. The algorithm changes the whales' positions in the fourth step. Lastly, it assesses if the need for termination is satisfied. The method outputs the ideal individual or optimal solution if the termination condition is met. If the termination condition is unsatisfied, the algorithm iteratively goes through step four again. In most situations, the whale algorithm is able to determine the ideal value. Nevertheless, local optima are frequently reached when the whale leader is given the best fitness value position in each cycle. This shortcoming is adequately addressed by the improved adaptive whale optimization algorithm (AWOA). Specifically, the individual fitness value is incorporated into the decision operator. The specific operation is to add the individual fitness value to the decision operator. The leader of the whale group records the fitness value and position of each individual before proceeding to the next iteration, compares the fitness through the decision probability so as to select a new leader, and finally selects the optimal position and fitness value under the action of the greedy mechanism so as to proceed to the next iteration. The three main AWOA behaviors are the following: searching, encircling, and hunting. 1. Searching behavior: \[D=|CX_{rand}-X(t)|\] \[X(t+1)=X_{rand}-AD\] (16) The algorithm is configured to randomly select a search agent when \(A\geq 1\) and update the positions of other whales based on the randomly selected whale position, forcing the whale to stray from the prey in order to find more suitable prey. This can strengthen the algorithm's exploration ability and, thus, allow it to conduct a global search. \(X_{rand}\) is the position vector of a randomly selected whale. 2. Encircling behavior: \[D=|CX(t)_{best}-X(t)|\] \[X(t+1)=X(t)_{best}-AD\] (17) The search range of the whale is the global solution space, and it needs to determine the position of the prey first in order to encircling the prey and update its own position by encircling the prey; Equation (17) shows the process of the whale encircling the prey. \(A\) and \(C\) represent the coefficients, and t is the current number of iterations. The whale's current position is represented by \(X(t)\), while its best position to date is shown by \(X(t)_{best}\). The following equations can be used to find \(A\) and \(C\): \[A=2 ar_{1}-a\] (18) \[C=2r_{2}\] (19)\[a=2-\frac{2t}{T_{max}} \tag{20}\]  
Here, \(t\) indicates the current iteration number and \(T_{max}\) indicates the maximum number of iterations. \(r_{1}\) and \(r_{2}\) represent random values within the range (0, 1), where the value of a drops linearly from 2 to 0 over the duration of iterations. 3. Hunting behavior: When a whale engages in hunting, it swims in a spiral motion towards its prey. This hunting behavior can be represented by the following model: \[X(t+1)=X(t)_{best}+D_{p}e^{bl}\cos{(2\pi l)}\] (21)  
The distance between the whale and its prey is shown above as \(D_{p}=|X(t)_{best}-X(t)|\), where \(X(t)_{best}\) is the position vector with the best current position. The spiral's form is determined by the maturity parameter, \(b\), which is a constant. \(b\) can take any positive real value, and this constant controls the shape and degree of expansion of the logarithmic spiral, with larger values leading to tighter spiral shapes and smaller values leading to looser spiral shapes. The random number \(l\) falls between \(-1\) and 1.