Owing to the importance of choosing the proper kernel function in modeling accuracy, in the first step, different structures of employed kernel-depend approaches were analyzed through various kernel functions. Table 5 lists the obtained results of employed kernel-depend approaches with different kernels for models F(IV) and S(IV). Moreover, optimal parameters of employed models are listed in Tables 6–8, respectively.
The results of kernel-depend methods with various kernel functions
| Model . | Method . | Kernel type . | Performance criteria . | |||
|---|---|---|---|---|---|---|
| Testing . | ||||||
| R . | SI . | BIAS . | RMSE . | |||
| F(IV) | KELM-GWO | Linear | 0.609 | 0.049 | 0.000473 | 0.038 |
| Polynomial | 0.927 | 0.023 | −0.005946 | 0.018 | ||
| RBF | 0.490 | 0.067 | −0.010038 | 0.052 | ||
| Wavelet | 0.725 | 0.040 | −0.001569 | 0.031 | ||
| GPR | Exponential | 0.911 | 0.023 | −0.002162 | 0.018 | |
| Squared Exponential | 0.899 | 0.025 | −0.003053 | 0.019 | ||
| Matern 3/2 | 0.900 | 0.025 | −0.002817 | 0.019 | ||
| Matern 5/2 | 0.899 | 0.025 | −0.003014 | 0.019 | ||
| Rational Quadratic | 0.899 | 0.025 | −0.003053 | 0.019 | ||
| SVM | Linear | 0.785 | 0.037 | 0.005165 | 0.029 | |
| Polynomial | 0.620 | 0.045 | −0.002548 | 0.035 | ||
| RBF | 0.896 | 0.027 | −0.001713 | 0.021 | ||
| Sigmoid | 0.403 | 1.01 | −0.131587 | 0.785 | ||
| S(IV) | KELM-GWO | Linear | 0.085 | 0.328 | 0.019855 | 0.254 |
| Polynomial | 0.365 | 0.341 | 0.035280 | 0.266 | ||
| RBF | 0.839 | 0.185 | −0.027257 | 0.144 | ||
| Wavelet | 0.839 | 0.185 | −0.027258 | 0.144 | ||
| GPR | Exponential | 0.944 | 0.102 | 0.009989 | 0.079 | |
| Squared Exponential | 0.877 | 0.156 | 0.000228 | 0.121 | ||
| Matern 3/2 | 0.931 | 0.117 | 0.005591 | 0.090 | ||
| Matern 5/2 | 0.917 | 0.130 | 0.003291 | 0.100 | ||
| Rational Quadratic | 0.877 | 0.156 | 0.000228 | 0.121 | ||
| SVM | Linear | 0.158 | 0.310 | −0.026804 | 0.239 | |
| Polynomial | 0.559 | 1.33 | 0.257247 | 1.026 | ||
| RBF | 0.940 | 0.106 | −0.000572 | 0.022 | ||
| Sigmoid | 0.064 | 0.315 | −0.002886 | 0.243 | ||
| Model . | Method . | Kernel type . | Performance criteria . | |||
|---|---|---|---|---|---|---|
| Testing . | ||||||
| R . | SI . | BIAS . | RMSE . | |||
| F(IV) | KELM-GWO | Linear | 0.609 | 0.049 | 0.000473 | 0.038 |
| Polynomial | 0.927 | 0.023 | −0.005946 | 0.018 | ||
| RBF | 0.490 | 0.067 | −0.010038 | 0.052 | ||
| Wavelet | 0.725 | 0.040 | −0.001569 | 0.031 | ||
| GPR | Exponential | 0.911 | 0.023 | −0.002162 | 0.018 | |
| Squared Exponential | 0.899 | 0.025 | −0.003053 | 0.019 | ||
| Matern 3/2 | 0.900 | 0.025 | −0.002817 | 0.019 | ||
| Matern 5/2 | 0.899 | 0.025 | −0.003014 | 0.019 | ||
| Rational Quadratic | 0.899 | 0.025 | −0.003053 | 0.019 | ||
| SVM | Linear | 0.785 | 0.037 | 0.005165 | 0.029 | |
| Polynomial | 0.620 | 0.045 | −0.002548 | 0.035 | ||
| RBF | 0.896 | 0.027 | −0.001713 | 0.021 | ||
| Sigmoid | 0.403 | 1.01 | −0.131587 | 0.785 | ||
| S(IV) | KELM-GWO | Linear | 0.085 | 0.328 | 0.019855 | 0.254 |
| Polynomial | 0.365 | 0.341 | 0.035280 | 0.266 | ||
| RBF | 0.839 | 0.185 | −0.027257 | 0.144 | ||
| Wavelet | 0.839 | 0.185 | −0.027258 | 0.144 | ||
| GPR | Exponential | 0.944 | 0.102 | 0.009989 | 0.079 | |
| Squared Exponential | 0.877 | 0.156 | 0.000228 | 0.121 | ||
| Matern 3/2 | 0.931 | 0.117 | 0.005591 | 0.090 | ||
| Matern 5/2 | 0.917 | 0.130 | 0.003291 | 0.100 | ||
| Rational Quadratic | 0.877 | 0.156 | 0.000228 | 0.121 | ||
| SVM | Linear | 0.158 | 0.310 | −0.026804 | 0.239 | |
| Polynomial | 0.559 | 1.33 | 0.257247 | 1.026 | ||
| RBF | 0.940 | 0.106 | −0.000572 | 0.022 | ||
| Sigmoid | 0.064 | 0.315 | −0.002886 | 0.243 | ||
Italic values indicate superior results.
Optimal parameters of employed KELM-GWO model
| Method . | Model . | Kernel type . | Parameter . | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Ρ . | c . | d . | γ . | ω . | α . | β . | |||
| KELM-GWO | F(IV) | Linear | 9.2969 | – | – | – | – | – | – |
| Polynomial | 1 | 4.6017 | 5.1627 | – | – | – | – | ||
| RBF | 201 | – | 10 | – | – | – | |||
| Wavelet | 160.8304 | – | – | – | 7.8560 | 5.3051 | 8.6632 | ||
| S(IV) | Linear | 1.6319 | – | – | – | – | – | – | |
| Polynomial | – | – | – | – | |||||
| RBF | 201 | – | – | 0.1037 | – | – | – | ||
| Wavelet | 201 | – | – | – | 0.1036 | 10 | 0.2307 | ||
| Method . | Model . | Kernel type . | Parameter . | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Ρ . | c . | d . | γ . | ω . | α . | β . | |||
| KELM-GWO | F(IV) | Linear | 9.2969 | – | – | – | – | – | – |
| Polynomial | 1 | 4.6017 | 5.1627 | – | – | – | – | ||
| RBF | 201 | – | 10 | – | – | – | |||
| Wavelet | 160.8304 | – | – | – | 7.8560 | 5.3051 | 8.6632 | ||
| S(IV) | Linear | 1.6319 | – | – | – | – | – | – | |
| Polynomial | – | – | – | – | |||||
| RBF | 201 | – | – | 0.1037 | – | – | – | ||
| Wavelet | 201 | – | – | – | 0.1036 | 10 | 0.2307 | ||
Optimal palrameters of employed GPR mode
| Method . | Model . | Kernel type . | Parameter . | ||
|---|---|---|---|---|---|
| Lσ . | Fσ . | Log likelihood . | |||
| GPR | F(IV) | Exponential | 10.6417 | 0.0700 | 168.7399 |
| Squared Exponential | 1.9743 | 0.0793 | 170.9388 | ||
| Matern 3/2 | 4.3681 | 0.0799 | 170.8195 | ||
| Matern 5/2 | 3.1036 | 0.0792 | 171.2564 | ||
| Rational Quadratic | 1.9743 | 0.0793 | 171.2564 | ||
| S(IV) | Exponential | 0.2721 | 0.4724 | 57.1633 | |
| Squared Exponential | 0.1200 | 0.5359 | 90.8263 | ||
| Matern 3/2 | 0.1618 | 0.5011 | 75.6284 | ||
| Matern 5/2 | 0.1439 | 0.5154 | 82.3447 | ||
| Rational Quadratic | 0.1200 | 0.5359 | 90.8263 | ||
| Method . | Model . | Kernel type . | Parameter . | ||
|---|---|---|---|---|---|
| Lσ . | Fσ . | Log likelihood . | |||
| GPR | F(IV) | Exponential | 10.6417 | 0.0700 | 168.7399 |
| Squared Exponential | 1.9743 | 0.0793 | 170.9388 | ||
| Matern 3/2 | 4.3681 | 0.0799 | 170.8195 | ||
| Matern 5/2 | 3.1036 | 0.0792 | 171.2564 | ||
| Rational Quadratic | 1.9743 | 0.0793 | 171.2564 | ||
| S(IV) | Exponential | 0.2721 | 0.4724 | 57.1633 | |
| Squared Exponential | 0.1200 | 0.5359 | 90.8263 | ||
| Matern 3/2 | 0.1618 | 0.5011 | 75.6284 | ||
| Matern 5/2 | 0.1439 | 0.5154 | 82.3447 | ||
| Rational Quadratic | 0.1200 | 0.5359 | 90.8263 | ||
Italic values indicate superior results.
Optimal parameters of employed SVM model
| Method . | Model . | Kernel type . | Parameter . | ||||
|---|---|---|---|---|---|---|---|
| c . | d . | γ . | C . | ε . | |||
| SVM | F(IV) | Linear | – | – | – | 1 | 0.001 |
| Polynomial | 3 | 3 | 50 | 1 | 0.001 | ||
| RBF | – | – | 200 | 1 | 0.001 | ||
| Sigmoid | 2 | – | 200 | 14 | 0.4 | ||
| S(IV) | Linear | – | – | – | 12 | 0.001 | |
| Polynomial | 1 | 5 | 150 | 0 | 0.201 | ||
| RBF | – | – | 90 | 3 | 0.001 | ||
| Sigmoid | 2 | – | – | 1 | 0.001 | ||
| Method . | Model . | Kernel type . | Parameter . | ||||
|---|---|---|---|---|---|---|---|
| c . | d . | γ . | C . | ε . | |||
| SVM | F(IV) | Linear | – | – | – | 1 | 0.001 |
| Polynomial | 3 | 3 | 50 | 1 | 0.001 | ||
| RBF | – | – | 200 | 1 | 0.001 | ||
| Sigmoid | 2 | – | 200 | 14 | 0.4 | ||
| S(IV) | Linear | – | – | – | 12 | 0.001 | |
| Polynomial | 1 | 5 | 150 | 0 | 0.201 | ||
| RBF | – | – | 90 | 3 | 0.001 | ||
| Sigmoid | 2 | – | – | 1 | 0.001 | ||