## 2 Material and Methods
### Landslide Prediction Models
#### 2.4.3 Grey Wolf Optimizer (GWO)  
The GWO algorithm is another meta-heuristic algorithm [104] and is derived from the hunting behavior of the grey wolves and the social hierarchy in nature [116]. The Grey wolves live in pack by a strict social dominant hierarchy and imitate the leadership hierarchy [117]. Figure 6: Flowchart of the Whale Optimization Algorithm (WOA) algorithm. The grey wolves have various groups for activities including hunting the prey. The social dominant hierarchy of the grey wolves shown in Figure 7 has four forms: alpha (\(\alpha\)), beta (\(\beta\)), delta (\(\delta\)), and omega (\(\omega\)). The \(\alpha\)-wolves are the top of the social hierarchy as leaders for making decisions whereas other wolves follow them [118]. The \(\beta\)-wolves help the leaders and devise them. The \(\delta\)-wolves situate the next level and obey the \(\alpha\)- and \(\beta\)-wolves. Finally, the \(\omega\)-wolves have to submit to all of them [117]. The GWO technique has an optimization process similar to other meta-heuristic algorithms through the collection of random candidate solutions [119], thus it is designed as a mathematically model for grey wolves. Specifically, \(\alpha\), \(\beta\), \(\delta\) and \(\omega\) are the fittest, second optimal, third optimal and the rest of solutions, respectively [120]. The process of hunting behavior of GWO is shown in Figure 7 and this algorithm includes four phases described as follows:  
(1) Encircling of prey  
In the first phase, the grey wolves harass and encircle the prey during hunting. The parameter \(D\) measures the distance between the grey wolf and the prey and is given by:  
\[D=|\vec{C}.\vec{X}_{p}(t)-\vec{X}(t)| \tag{13}\]  
where \(t\) represents the current iteration, and \(\vec{X}_{p}\) and \(\vec{X}\) denote the position vectors of the prey and the grey wolves, respectively, and the coefficient vector is defined by \(\vec{C}\) as follows:  
\[\vec{C}=2.\vec{r}_{1} \tag{14}\]  
where \(r_{1}\) is a random vector which is in the interval [0, 1]. The prey's location can be represented as follows:  
\[\vec{X}(t+1)=\vec{X}_{P}(t)-\vec{A}.\vec{D} \tag{15}\]  
The value of the coefficient \(\vec{A}\) is computed as follows:  
\[\vec{A}=2\pi.r_{2}-a \tag{16}\]  
where \(a\) is a linearly decreasing variable and \(r_{2}\) is a random vector between [0, 1] as \(r_{1}\). (2) Hunting  
After the phase of encircling the prey, the hunting behavior is guided by \(\alpha\), \(\beta\) and \(\delta\), respectively, since they have compressive information about the prey's position. This behavior is shown as follows:  
\[\vec{D}_{\alpha}=|\vec{C}_{1}.\vec{X}_{\alpha}-\vec{X}_{|},\vec{D}_{\beta}=| \vec{C}_{2}.\vec{X}_{\beta}-\vec{X}_{|},\vec{D}_{\delta}=|\vec{C}_{3}.\vec{X} _{\delta}-\vec{X}| \tag{17}\]  
\[\vec{X}_{1}=\vec{X}_{\alpha}-\vec{A}_{1}.\vec{D}.\vec{X}_{2}=\vec{X}_{\beta} -\vec{A}_{2}.\vec{D},\vec{X}_{3}=\vec{X}_{\delta}-\vec{A}_{3}.\vec{D} \tag{18}\]  
Figure 7: The social hierarchy of grey wolves [104]. where \(\overset{\rightarrow}{X_{1}}\), \(\overset{\rightarrow}{X_{2}}\) and \(\overset{\rightarrow}{X_{3}}\) indicate the position vectors of \(\alpha\), \(\beta\) and \(\delta\), respectively. \(\overset{\rightarrow}{A_{1}}\), \(\overset{\rightarrow}{A_{2}}\), \(\overset{\rightarrow}{A_{3}}\) and \(\overset{\rightarrow}{C_{1}}\), \(\overset{\rightarrow}{C_{2}}\), \(\overset{\rightarrow}{C_{3}}\) are the coefficients that can be computed using (14) and (16). The position of a grey wolf in search space can be updated as follows: \[\overset{\rightarrow}{X}(t+1)=\frac{\overset{\rightarrow}{X_{1}}+\overset{ \rightarrow}{X_{2}}+\overset{\rightarrow}{X_{3}}}{3}\] (19) The positions of other wolves update randomly according to the position of the prey. 3. Attacking prey (exploitation) The process of hunting finishes when the prey stops moving. Mathematically, \(\overset{\rightarrow}{A}\) is linearly decreased from 2 to 0. The exploration trend happens when \(|\overset{\rightarrow}{A}|<1\) and \(|\overset{\rightarrow}{C}|<1\).