Novel grey wolf optimizer based parameters selection for GARCH and ARIMA models for stock price prediction

GWO method for parameter selection in ARIMA and GARCH model

In the GWO method, the population of candidate solutions is represented by a pack of “wolves”, which communicate with one another and adjust their locations based on the actions of the pack’s alpha, beta, and delta wolves. In this article, the fitness levels of the wolves are used to classify them as alpha, beta, or delta. The wolf with the highest fitness is chosen as the alpha wolf, the second-best wolf is the beta wolf, and the third-best wolf is the delta wolf. These wolves are essential because they help direct the search and shape the updates the other wolves receive. MSE and RMSE metric is considered to calculate the fitness value. The proposed work GWO-based parameter selection for ARIMA and GARCH model steps are described in Algorithm 1 . The selection of parameters for ARIMA and GARCH using GWO is as follows. The first is to initialize the population of wolves with random positions using random numbers. Each wolf represents a candidate solution, and the aim is to identify the optimal solution. In the second step, define the initial value for a = 2 and encircle the prey. The third is to evaluate each wolf’s fitness by applying the RMSE and MSE metrics to its corresponding position. The fitness represents the quality of the solution. The fourth is to identify the wolves with the first-highest, second-highest, and third-highest wolves using the fitness values of wolves. The fifth is to update the positions of the remaining wolves. The sixth is to select the best solution by taking the average of three wolves. The positions are adjusted to explore the solution space and converge toward better solutions.

 
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Algorithm 1 GWO-based parameter selection for ARIMA and GARCH model 
algorithm___________________________________________________________________________________________ 
  1:  The population of grey wolves Gi (i = 1,2,...,n) is created in a random 
     manner. 
  2:  Assign the initial value of the variable ”a” as 2 using − → A = − → 2a− → r1 −− → a1 and 
     − → C = 2−→ r 
2 
  3:  Compute the fitness level of every individual within the population using 
     RMSE, MSE, and MAE metrics for below wolves. − → 
  G1 = − → Gα −− → A1.− →Dα 
    − → G2 = − → Gβ −− → A2.− →Dβ 
    − → G3 = − → Gδ −− → A3.− →Dδ 
 4:  Identify the wolves with the highest, second-highest, and third-highest fit- 
     ness values, respectively, as the alpha, beta, and delta wolves. 
  5:  For the range of values of t from 1 to the maximum number of iterations, 
     update the parameters: 
     − → D 
α = ∣ ∣ 
∣− − → C1.− − − → G 
α−− → G∣ ∣ 
        ∣ 
  − → D 
β = ∣ ∣ 
∣− − → C2.− − → G 
β−− → G∣ ∣ 
        ∣ 
  − → D 
δ = ∣ ∣ 
∣− − → C3.− − → G 
δ−− → G∣ ∣ 
       ∣ 
 6:  Select the best solution by taking the average of 3 wolves.  − → G(t + 1) = 
     − → G1 
+− →G2 
+− →G3 
       3_______________________________________________________________________________________________________________    

In this expression, t represents the current iteration, $\vec{A}$ and $\vec{C}$ are coefficient vectors, $\overrightarrow{G_p}$ is the prey’s position vector, $\vec{G}$ shows the grey wolf’s position vector. The variable “D” represents the distance between the grey wolf and its prey, and it is defined in Eqs. (1) and (2). Iteratively reducing the number of components in $\vec{a}$ from 2 to 0 while generating random $\overrightarrow{r1}$ and $\overrightarrow{r2}$ in the interval [0, 1] and it is defined in Eqs. (3) and (4). (1)${\displaystyle \vec{D}=|\overrightarrow{C.}\overrightarrow{G_{p\left(t\right)}}-\overrightarrow{G_{\left(t\right)}}|}$ (2)${\displaystyle \vec{G}\left(t+1\right)={\vec{G}}_{p\left(t\right)}-\overrightarrow{A.}\vec{D}}$ (3)${\displaystyle \vec{A}=\overrightarrow{2a}\overrightarrow{r1}-\overrightarrow{a1}}$ (4)${\displaystyle \vec{C}=2{\vec{r}}_2}$

Where r1 and r2 are random variables between zero and one, and a convergence factor falls from two to zero as n iterations increase. The alpha, beta, and delta wolves direct the omega wolves in their pursuit of the prey, and the omega wolves recalculate the prey’s location based on the best estimates of the alpha, beta, and delta wolves. The pack of grey wolves is positioned as shown by Eqs. (5) to (11). (5)${\displaystyle \overrightarrow{D_{\alpha }}=\left|\overrightarrow{C1.}\overrightarrow{G_{\alpha }-}\vec{G}\right|}$ (6)${\displaystyle \overrightarrow{D_{\beta }}=\left|\overrightarrow{C2.}\overrightarrow{G_{\beta }-}\vec{G}\right|}$ (7)${\displaystyle \overrightarrow{D_{\delta }}=\left|\overrightarrow{C3.}\overrightarrow{G_{\delta }-}\vec{G}\right|}$ (8)${\displaystyle \overrightarrow{G1}=\overrightarrow{G\alpha }-\overrightarrow{A1}.\overrightarrow{D\alpha }}$ (9)${\displaystyle \overrightarrow{G2}=\overrightarrow{G\beta }-\overrightarrow{A2}.\overrightarrow{D\beta }}$ (10)${\displaystyle \overrightarrow{G3}=\overrightarrow{G\delta }-\overrightarrow{A3}.\overrightarrow{D\delta }}$ (11)${\displaystyle \vec{G}\left(t+1\right)=\frac{\overrightarrow{G1}+\overrightarrow{G2}+\overrightarrow{G3}}{3}.}$

During each iteration, omega wolves adjust their positions based on the positions of alpha, beta, and delta wolves, as these wolves possess superior knowledge regarding the potential location of prey.