# Efficient Feature Selection Using Weighted Superposition Attraction Optimization Algorithm
## 2 Methodology
### Metaheuristics
#### 2.3.6 Marine Predators' Algorithm  
The MPA is a population-based metaheuristic formulated by drawing inspiration from marine predators. For the first trial, the initial solution is randomly spread over a search area, as depicted by the following equation [49]:  
\[X_{0}=X_{min}+rand\ (X_{max}-X_{min}) \tag{12}\]  
where \(X_{min}\) and \(X_{max}\), respectively, represent the minimum and maximum boundary for the variables, and rand is a random value vector within the range of 0 and 1. Based on the theory of survival of the fittest, the fittest solution is selected as the top predator for developing a matrix that is known as the elite [49]. The elite's matrix arrays oversee the search and finding of prey based on the information on the position of the prey. \[\text{Elite}= \begin{bmatrix}X_{1,1}^{I}&X_{1,2}^{I}&\ldots&X_{1,d}^{I}\\ X_{2,1}^{I}&X_{2,2}^{I}&\ldots&X_{2,d}^{I}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ X_{n,1}^{I}&X_{n,2}^{I}&\ldots&X_{n,d}^{I}\end{bmatrix}  \tag{13}\]  
where \(\overrightarrow{X}^{I}\) denotes the top predator that is duplicated n number of times to make the elite matrix, \(n\) stands for the population of agents, and d represents the dimension. After each computation, the elite is updated in the event a better predator substitutes the top predator. The predators move into new positions based on the prey matrix [30; 50]. The prey matrix is indicated as shown below:  
\[\text{Prey}= \begin{bmatrix}X_{1,1}&X_{1,2}&\ldots&X_{1,d}\\ X_{2,1}&X_{2,2}&\ldots&X_{2,d}\\ \vdots&\vdots&\vdots&\vdots\\ X_{n,1}&X_{n,2}&\ldots&X_{n,d}\end{bmatrix}  \tag{14}\]  
where \(X_{i,j}\) represents \(j\)th dimension of \(i\)th prey. The entire optimization process is majorly and directly associated with the elite and prey matrixes. It is divided into three key phases based on the ratio of velocity and emulates the entire life of the predator and prey [51]. Phase 1: Prey is moving faster compared to the predator (i.e., in high-velocity ratio) [51]. The mathematical model is:  
\[ \begin{array}{l}\text{If }Iter<\frac{1}{3}MaxIter\\ \overrightarrow{stepsize_{i}}=\overrightarrow{R_{B}}\otimes&\bigg{(} \overrightarrow{Elite_{i}}-\overrightarrow{R_{B}}\otimes \overrightarrow{Prey_{i}}\bigg{)}(i=1,2,\ldots,n)\\ &\overrightarrow{Prey_{i}}=\overrightarrow{Prey_{i}}+P\cdot \overrightarrow{R}\otimes\overrightarrow{\tau^{*}stepsize_{i}}\end{array} \]  
where \(\overrightarrow{R}_{B}\) is a vector consisting of random numbers, \(\otimes\) denotes entry-wise multiplication, \(\overrightarrow{R}\) is a random number vector uniformly distributed in the range [0, 1], and \(P\) is a constant with a value of 0.5. Phase 2: Both predator and prey are moving at the same speed (i.e., in unit velocity ratio). This can be modeled as shown below [51]:  
If \(\frac{1}{3}MaxIter<Iter<\frac{2}{3}MaxIter\)  
For the first half of the population:  
\[\overrightarrow{stepsize_{i}}=\overrightarrow{R_{L}}\otimes\bigg{(} \overrightarrow{Elite_{i}}-\overrightarrow{R_{L}}\otimes\overrightarrow{Prey _{i}}\bigg{)}(i=1,2,\ldots,n/2)\]  
\[\overrightarrow{Prey_{i}}=\overrightarrow{Prey_{i}}+P\cdot \overrightarrow{R}\otimes\overrightarrow{\tau^{*}stepsize_{i}}\]  
For the second half of the population:  
\[\overrightarrow{Prey_{i}}=\overrightarrow{Elite_{i}}+P\cdot CF\otimes \overrightarrow{\tau^{*}stepsize_{i}}\]where \(\overset{\rightarrow}{R}_{L}\) denotes a vector of random numbers representing the Levy movement, and \(CF\) is an adaptive parameter to control the predator's step size. \[CF=\left(1-\frac{Iter}{MaxIter}\right)^{(\frac{2\times Iter}{MaxIter})}\]  
Phase 3: Predators move faster than prey (i.e., in low-velocity ratio). It is the last stage of the process of optimization, and it is associated with high exploitation ability [40]. \[