This ML model includes heuristic algorithms based on statistics (Abedi et al., 2012; Geranian et al., 2016; Gonbadi et al., 2015; Maepa et al., 2021). A dataset of vectors (training data) with predefined (priori knowledge) class labels is commonly used to design the SVM hyperplane for delineating classes (Geranian et al., 2016). This approach employs different kernel functions for transferring the training data to a higher-dimension space (Abedi et al., 2012). A support vector in the SVM method represents a point that has the least distance from the hyperplane and regulates the orientation of the hyperplane (Figure 2). The decision boundary of the hyperplane is commonly determined using a number of input features. The two-class SVM entails training data with l feature vectors, where x _i ∈ R and i = (1, …, n) is the number of the feature vector in the training data. The class label y _i is assigned to each sample and is equal to 1 or −1 (Zuo and Carranza, 2011). The separating hyperplane is formulated as the following decision function: f x = s g n w x + b , (1) where parameters w and b are optimized using the following equation: L w , b , α = 1 2 w 2 − ∑ i = 1 l α i y i x i w + b − 1 . (2) Here, α i is the Lagrange multiplier. The Lagrange function is minimized with respect to w and b and maximized with respect to α i > 0. The Lagrange multipliers are determined as follows: M a x i m i z e   ∑ i = 1 l α i − 1 2 ∑ i , j = 1 l α i α j y i y j x i x j . (3)

Next, the classification based on the optimal hyperplane is obtained using the following decision function: f x = s g n ∑ i = 1 l y i α i x x i + b . (4)

Positive slack variables ξ i along with a penalty value C for misclassification errors are considered to improve the classification. Accordingly, the following objective function should be minimized: S = 1 2 w 2 + C ∑ i = 1 l ξ i k . (5)

When the training data are not linearly separable, an RBF kernel K x i ,   x j is employed to transfer the training data to the Hilbert space. Then, Equations 3, 4) can be reformulated as follows: M a x i m i z e   ∑ i = 1 l α i − 1 2 ∑ i , j = 1 l α i α j y i y j K x i x j . (6) f x = s g n ∑ i = 1 l y i α i K x x i + b . (7)

In addition to the RBF kernel, the PF kernel was applied to the SVM model in this study and is expressed as follows: K x i ,   x j = x i x j + 1 d . (8)

Then, the relevant parameters like the kernel size (σ) of the RBF, kernel order (d) of the PF, and objective function parameter (penalty value C) for both kernels must be optimized for credible results.

Grasshoppers are generally observed individually but belong to one of the largest swarm of creatures. Exploitation and exploration of the search space are the two logical attitudes of most nature-inspired optimization algorithms. Here, the search agents are supported for sudden movements in the exploration tendency and local movements in the exploitation tendency. Grasshoppers naturally demonstrate these two tendencies to seek their targets. Therefore, mathematical modeling of their behaviors has been used to construct a new nature-inspired algorithm (Saremi et al., 2017). The simulated behaviors of grasshoppers can be expressed as follows: Y i = P i + S i + W i , (9) where W _i , P _i , S _i , and Y _i refer to the wind advection, gravity intensity, social interaction, and position of the ith grasshopper, respectively. Moreover, their random behavior can be expressed as Y _i = r ₁ P _i + r ₂ S _i + r ₃ W _i , where r ₃ , r ₂ , and r ₁ are random values in [0, 1]. The social interaction (S _i ) can be calculated as follows: S i = ∑ j = 1 N s d i j ď i j j ≠ i , (10) where d _ij is the distance between the ith and jth grasshoppers computed as d _ij =|x _j − x _i |, N is the number of grasshoppers, and s is the strength of social forces depicted as a function and determined using Equation 11. Moreover, ď i j = x j − x i d i j is a unit vector between the jth and ith grasshoppers. s r = f e − r l − e − r , (11) where l and f are the attractive length scale and intensity of attraction, respectively. The repulsion and attraction of grasshoppers as social interactions are exhibited in Figure 3A, where the distances are considered in the range of [0, 15] and repulsion commonly occurs in [0, 2.079]. When two grasshoppers are separated by a distance of 2.079 from each other, there is neither repulsion nor attraction (Saremi et al., 2017); in this case, both are at a comfortable distance or in the comfort zone. Furthermore, attraction increases in the range of [2.079, 4] and then decreases slowly. In Equation 11, variations in the factors f and l can result in different social behaviors of the grasshoppers. The s function is independently replotted based on changes in f and l, as shown in Figure 4. Thus, the factors f and l can meaningfully change the repulsion region, attraction region, and comfort zone between the grasshoppers. It is worth noting that factors l and f are empirically considered as equal to 1.5 and 0.5, respectively, for better results (Saremi et al., 2017). Figure 5 conceptually depicts the comfort zone based on the function s and interactions between the grasshoppers; the spaces between the grasshoppers for the repulsion region, comfort zone, and attraction region can be discriminated from the function s, which assigns distances higher than 10 to values closer to 0 (Figures 3, 4). Hence, the function s is not appropriate when the forces between the grasshoppers are strong and the individuals are separated by large distances from each other. Therefore, this problem is solved by plotting the distances between the grasshoppers in the range of [1, 4]. Thus, the function s for the range [1, 4] is as shown in Figure 3B. The gravity intensity (P _i ) in Equation 9 is calculated as follows: P i = − g h ^ p . (12) where ĥ p is a unit vector into the center of the earth and g is the gravity coefficient. The wind advection factor is computed as follows: W i = z h ^ w , (13) where ĥ w and z are the unit vectors in the wind direction and constant value of the drift, respectively. Expanding Equation 9 using Equations 10–13, we obtain Y i = ∑ j = 1 N s x j − x i x j − x i d i j − g h ^ p + z ĥ w . (14)

This model cannot be optimized because it is meant for swarms in free space, and its application avoids the exploration and exploitation procedures around a solution in the desired space (Saremi et al., 2017). Therefore, a modified version of Equation 14 is presented as follows: Y i d = c ∑ j = 1 N c u b d − l b d 2 s   x j d − x i d x j − x i d i j + C ^ d , (15) where c is the optimizer coefficient related to the modified attraction zone (promising exploitation region), repulsion zone (exploration of the search space), and comfort zone (balance between exploration and exploitation). Here, lb _d and ub _d are the lower and upper bounds of s(r), respectively, and Ĉ d is the value of the target. Figure 6 shows an example of the swarm behavior based on Equation 15. Here, s is calculated similarly based on the factor S in Equation 10. Moreover, the components W _i and P _i are always considered toward a target ( Ĉ d ) in this work. The grasshoppers should achieve balance between the exploration and exploitation procedures to reach the optimal comfort zone such that the GOA does not fall into a local minimum and provides an accurate estimation of the global optimum point. Factor c is the most significant director parameter of the GOA and is updated as follows: c = c max − l c max − c min , L (16) where c _min = 0.00004, c _max = 1, and L and l are the total number of iterations and current iteration, respectively.

A total of 800 stream sediment samples were collected to check the changes rates of the concentrations of 38 elements using the induced coupled plasma mass spectrometry (ICP-MS) technique with precision less than 10% (Sabbaghi, 2018). The isometric log-ratio (ilr) transformation was employed to remove the data closure problem as per Equation 22 (Sadeghi et al., 2024; Wang et al., 2014). Then, the geochemical data of the stream sediments were transformed into the range of [0, 1] to train the applied models. ilr x = D − i D − i + 1 log x i g x   i = 1 , 2 , … , D − 1 , (22) where x, x _D , and g(x) indicate the vector of a composition with D dimensions, Euclidean distances between the distinct variables, and geometric mean of the composition x, respectively (Aitchison et al., 2000). Accordingly, a geochemical data table comprising the transformed values of elements Pb, Zn, Ag, and Cd was constructed to delineate the geochemical populations in the study area. Thus, the geochemical data table had four columns, including the transformed values of the pathfinder elements, with one column including the labels and 800 rows including the individual collected samples to train the model. To assign the predefined labels of the training data, we applied several ranges of transformed values (Table 1). Accordingly, the label 2 was assigned to geochemical samples with the criterion Pb + Zn + Ag + Cd ≥ 2.762 and label 1 was allocated to samples with the criterion 2.726 > Pb + Zn + Ag + Cd > 0. For example, a geochemical sample with the transformed values of Pb = 0.612, Zn = 0.751, Ag = 0.648, and Cd = 0.806 is a member of the geochemical anomaly population and has the label 2.

The ability of the GOA to seek search spaces for discovering the optimal solutions of challenging problems is shown in Figure 6. Here, the GOA is fittingly attracted to optimal solutions in the best locations of the search spaces, and the number of random parameters is restricted in the GOA. Therefore, the initial population of grasshoppers is significant in this algorithm. The relevant hyperparameters, including kernel size (σ) of the RBF, kernel order (d) of the PF, and objective function parameter (penalty value C), are important such that differences in their values can produce various classification conclusions of the SVM model. Furthermore, trial-and-error procedures are time-consuming and onerous that may not always provide reliable results. Therefore, the hyperparameters of the RBF-SVM and PF-SVM models should be optimized using a metaheuristic optimization algorithm that is quick, easy, and also provides reliable results. The present study entails reliable optimization of the aforementioned hyperparameters using a known swarm-intelligence optimization algorithm that is not time-consuming. A schematic of the GSVM framework is presented in Figure 7; here, MATLAB R2022a was applied to the GSVM model. After preprocessing, the 800 collected samples were divided into the testing (30%) and training (70%) data in terms of the percentage of anomaly and background population. For model training, the required parameters are number of grasshoppers, maximum number of iterations, K-fold cross validation, lower bound, and upper bound, which were empirically set to 20, 100, 10, 0, and 10, respectively. The optimized values of the hyperparameters are presented in Table 2, and the results of the training data classification are shown in Figure 8. Accordingly, the precision and accuracy of training the GSVM with the RBF kernel are 90.48% and 98.21% (Figure 8A), while the results with the PF kernel are 92.86% and 98.75% (Figure 8B), respectively. Furthermore, the specificity and sensitivity of the trained GSVM are suitable. In fact, the specificity (99.22% and 99.42%) as well as sensitivity (86.36% and 90.7%) of the GSVM with the corresponding RBF (Figure 8A) and PF (Figure 8B) kernels prove that the GSVM model has been trained efficiently.

In the present work, we also decreased the root mean-squared error (RMSE) associated with the cost function values, as shown in Equation 23, during training. R M S E = 1 n ∑ i = 1 n C R − C p , (23) where n, C _R , and C _p are the number of geochemical samples, real class allocated to the sample, and predicted class of the sample, respectively. Figure 9 shows that the GSVM models with both kernels can achieve the most optimized states of classification. SVM optimization with the RBF kernel was implemented to 42 iterations, at which point the value converged to 0.088 (Figure 9A). Figure 9B depicts that SVM optimization with the PF kernel was implemented to 45 iterations and that the converged value is 0.052. Both cost functions (Figure 9) are noted to have been in their steady states for more than 40 iterations, thereby ensuring good model optimization. The effects of these procedures on the mapping of the geochemical testing data are shown in Figure 10. Figure 10A shows the multi-element geochemical map of the testing data classified using the RBF-GSVM, and Figure 10B shows the corresponding map when using the PF-GSVM. Although both kernels can detect multi-element geochemical anomalies with the GSVM model, the PF kernel (Figure 10B) obviously has more accuracy. Figure 11 exhibits the validation of the GSVM model with the testing data; it can be observed that the results of the GSVM model for both kernels have precision of 100% and impressive accuracies (98.33% and 99.17%), making them the ideal choice for classifying the testing data. Here, the sensitivity of the RBF-GSVM is lower than that of the PF-GSVM because the number of predicted FN values has increased. The GSVM with RBF kernel labels more background samples as anomalies than the GSVM with PF kernel. The ROCs presented in Figure 12 show the comparison of the prediction abilities of the trained models in this work. The PF-GSVM has greater accuracy than that of the RBF-GSVM model as the AUC values of the testing data classification (class 2 = 0.973, class 1 = 0.961) with the PF-SVM (Figure 12B) are greater than the AUC values (class 2 = 0.969, class 1 = 0.957) obtained with the RBF-SVM (Figure 12A).