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For determining the best performance of SVM and selecting the best kernel function, different models were predicted via SVM using various kernels. Table 4 shows the results of statistical parameters of different kernels for model S(II) of a basin with central sill. The results of Table 4 revealed that using the model with the kernel function of RBF led to better prediction accuracy in comparison to the other kernels. According to Noori et al. (2011) the SVM model via RBF kernel is very desirable for use in prediction of hydraulic and hydrological features since: (a) unlike the linear kernel, the RBF kernel can handle the case when the relation between class labels and attributes is nonlinear, (b) the RBF kernels tend to give better performance under general smoothness assumptions, (c) it has fewer tuning parameters than the polynomial and the sigmoid kernels. Therefore, RBF kernel was selected as the core tool of SVM and was applied for the rest of the models. Implementation of SVM requires the selection of three parameters, which are constant C, ɛ and kernel parameter γ, where γ is a constant parameter of the RBF kernel. The coefficient C is a positive constant that influences a trade-off between an approximation error and the regression and must be selected by the user. ɛ has an effect on the smoothness of the SVM's response and it affects the number of support vectors, so both the complexity and the generalization capability of the network depend on its value. If epsilon is larger than the range of the target values we cannot expect a good result. If epsilon is zero, we can expect overfitting (Smola 1996). The variable parameter used with kernel function (γ) considerably affects the flexibility of function. These parameters should be selected by the user. The selection of appropriate values for the three parameters (C, ɛ, γ) has been proposed by various researchers. In the current study, according to Cherkassky & Yunqian (2002), optimization of these parameters has been done by a systematic grid search of the parameters using cross-validation on the training set. A normal range of parameter settings are investigated in this grid search. First, optimized values of C and ɛ for a specified γ were obtained and then γ was changed. Statistical parameters were used to find optimums. The statistics parameters via γ values to find SVM optimums of the testing set for the model S(II) of the basin with central sill are shown in Figure 5. In the same way, optimal parameters were obtained for all models.
Table 4

The statistical parameters of SVM method with different kernel functions; model S(II)

Kernel functionTrain
Test
RDCRMSERDCRMSE
Linear 0.885 0.818 0.138 0.882 0.722 0.149 
Polynomial 0.977 0.899 0.118 0.968 0.748 0.123 
RBF 0.994 0.987 0.026 0.993 0.985 0.029 
Sigmoid 0.574 0.108 0.203 0.343 0.095 0.273 
Kernel functionTrain
Test
RDCRMSERDCRMSE
Linear 0.885 0.818 0.138 0.882 0.722 0.149 
Polynomial 0.977 0.899 0.118 0.968 0.748 0.123 
RBF 0.994 0.987 0.026 0.993 0.985 0.029 
Sigmoid 0.574 0.108 0.203 0.343 0.095 0.273 

Bold values correspond to the superior kernel type.

Figure 5

Statistics parameters via γ values to find SVM optimums of the testing set for model S(II) of channel with central sill.

Figure 5

Statistics parameters via γ values to find SVM optimums of the testing set for model S(II) of channel with central sill.

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