## 5 Elm\(-\)SVM Model  
In hysteresis modeling, hysteresis operators such as stop and play operators play a significant role. However, these operators and models based on these operators, particularly the Preisach and Prandtl models, are suitable only for congruent hysteresis. This paper uses DS operators that enjoy deterioration in their hysteresis loops to propose a new intelligent model for congruent and non\(-\)congruent hysteresis. This new model is a composite of ELM and LS\(-\)SVM concepts, and it is called the ELM\(-\)SVM model in this paper. The ELM\(-\)SVM model is an extension of the Generalized Prandtl Neural Network (GPNN) previously proposed by [PERSON] et al. [22]. Compared to the GPNN model in which the hidden DS neurons are directly connected to the output layer, in the ELM\(-\)SVM model, between the hidden layer containing DS neurons and the output layer is where the LS\(-\)SVM is situated. There are two adjustable parameters for each DS neuron in the first hidden layer (\(r_{i}\), \(\beta_{i}\)). Noteworthy is the fact that \(\beta-\)values adjust the ELM\(-\)SVM model's overall deterioration rate so that if all the \(\beta_{i}\) have high values, the deteriorating capability will diminish. The training process of this model consists of two stages. By inspiring ELM concept, the free parameters of DS, which are \(r_{i}\) and \(\beta_{i}\) for \(i=1,\ 2,\ldots,n\) are randomly set, in the first step:  
\[0<r_{i}<r_{max} \tag{12}\]  
\[0<\beta_{i}<\beta_{max} \tag{13}\]  
where \(r_{max}\) and \(\beta_{max}\) are the maximum values for \(r_{i}\) and \(\beta_{i}\), respectively, and \(i=1,\ 2,\ \ldots,\ n\) in which \(n\) is the number of DS neurons. Preassigning internal parameters of DS neurons facilitates the learning process and reduces the computational cost the same as in ELMs. Additionally, the existence of an LS\(-\)SVM as an additional hidden layer in this model improves the capabilities of GPNN since the used LS\(-\)SVM in the second hidden layer is a nonlinear mapping. In the second step, an LS\(-\)SVM has been selected to train one\(-\)to\(-\)one mapping caused by DS operators. Compared to the neural networks, LS\(-\)SVMs enjoy error functions with an unrivaled global minimum according to their parameters \(\mathbf{\alpha}\) and \(b\). The LS\(-\)SVM would be trained by using Equation (8) with the preassigned hyperparameters \(\sigma\) and \(\gamma\). The LS\(-\)SVM's high generalization capability depends on properly tuning these hyperparameters. The Nelder\(-\)Mead simplex algorithm [30] is used for tuning the hyperparameters. In other to prevent overfitting, the cross\(-\)validation approach has been adopted in this paper. It is an evaluation method that determines to what extent statistical analysis results on a dataset can be generalized and independent of the training data. \(k-\)fold cross\(-\)validation error has been considered an objective function of the tuning process. This approach divides the shuffled training dataset into \(k\) smaller datasets or folds. The model is trained on \(k-1\) folds each time, and then the resulting model is evaluated on the remaining part of the dataset using a performance measure. The training and evaluation steps are repeated \(k\) times, and the average performance measure is reported as cross\(-\)validation performance. For the rate\(-\)dependent hystereses, an extra memory has been considered for the ELM\(-\)SVM model in this paper. It enhances the modeling accuracy of the presented model for rate\(-\)dependent hystereses. Details of the ELM\(-\)SVM model for both rate\(-\)independent and rate\(-\)dependent hystereses can be seen in Figures 3 and 4, respectively.