# A Novel Fuzzy Kernel Extreme Learning Machine Algorithm in Classification Problems
## 2 Methods
### Fuzzy Wavelet Mexican Hat Wavelet Kernel Extreme Learning Machine (FWMHKELM) Algorithm  
In this section, we will introduce our proposed basic hybrid algorithm. The work cited in [21] showed the equivalence of the FW structure and the ELM structure. However, edge detection or robustness to noise are important steps in facial expression classification. Therefore, in this study, the FW structure is integrated with a KELM. A Mexican hat wavelet is selected as the kernel. The MH wavelet is defined as the second derivative of the Gaussian function (Equation (11)). This derivative property helps to detect more precisely the distinct change points in the signal or data. In classification and pattern recognition problems, it is important to capture sharp transitions. The MH wavelet kernel was chosen for this study due to its better edge detection; its real-valued, positive semi-definite robustness to noise; and its better performance over a wide spectrum. The hybrid model has a hierarchical and sequential fusion structure. The operation of the model follows a hierarchical structure--First, the input data are disambiguated by fuzzification (FIS). Then, the frequency components of the data are analyzed by applying the wavelet transform. Finally, classification is performed with the MHKELM. This process is also sequential in nature, as it proceeds in successive steps. We have mentioned that the MH wavelet function is an acceptable ELM kernel [25]. For this case, [PERSON] et al. proved that the Fourier transform of the MH wavelet is nonnegative  
Figure 3: A structural representation of the fuzzy wavelet NN [14]. according to the translation-invariant theorem [25]. This proof is based on the theorem that the Fourier transform is nonnegative and is given [26]; we will not repeat it in this paper. When the MH wavelet is used as the mother wavelet, the MH wavelet kernel function is derived as in Equations (11) and (12):  
\[\psi(x)=\Big{(}1-x^{2}\Big{)}\mathrm{exp}\bigg{(}-\frac{x^{2}}{2}\bigg{)} \tag{11}\]  
\[K\big{(}x_{i},x_{j}\big{)}=\prod_{d=1}^{D}\left[1-\bigg{(}\frac{x_{di}-x_{di}}{a }\bigg{)}^{2}\right]\mathrm{exp}\left[-\frac{1}{2}\bigg{(}\frac{x_{di}-x_{di}}{ a}\bigg{)}^{2}\right] \tag{12}\]  
Also, the Fourier transform of the MH wavelet is shown in Equation (13):  
\[(\omega)=(2\pi)^{D/2}a^{D}\prod_{d=1}^{D}\mathrm{exp}\bigg{(}-\frac{1}{2}a^{2} w_{d}^{2}\bigg{)}\left\{\int_{-\infty}^{\infty}\mathrm{exp}\bigg{[}-\frac{1}{2}(x_{d}+ ja\omega_{d})^{2}\bigg{]}dx_{d}-\int_{-\infty}^{\infty}x_{d}^{2}\mathrm{exp} \bigg{[}-\frac{1}{2}(x_{d}+ja\omega_{d})^{2}\bigg{]}dx_{d}\right\} \tag{13}\]  
Equation (14) is obtained when this is solved by a series of integral partial integrations. \[F(\omega)=(2\pi)^{D}a^{3D}\mathrm{exp}\Big{(}-\frac{a}{2}\sum_{d=1}^{D}\omega_ {d}^{2}\Big{)}\prod_{d=1}^{D}\omega_{d}^{2}. \tag{14}\]  
The focus of the paper is how to combine the MH wavelet, mentioned above as being an adaptive ELM kernel, and the ELM method into the fuzzy wavelet structure. In this context, it will be argued that the fuzzy wavelet (FW) structure is equivalent to the MH wavelet kernel-based extreme learning machine structure and, finally, the steps of the proposed algorithm will be presented. The fuzzy wavelet (FW) structure is equivalent to the kernel-based extreme learning machine structure.