# A Novel Fuzzy Kernel Extreme Learning Machine Algorithm in Classification Problems
## 2 Methods
### Fuzzy Wavelet Mexican Hat Wavelet Kernel Extreme Learning Machine (FWMHKELM) Algorithm
A TSK model with k inputs and r outputs is defined as in Equation (15) [21]:  
\[R^{i}:\text{If x}_{1}\text{ is A}_{1}{}^{i};\text{ x}_{2}\text{ is A}_{2}{}^{i};\text{ }\ldots;\text{ x}_{k}\text{ is A}_{k}{}^{i}\text{ Then }\Big{(}y_{1},\gamma_{1}^{i}\Big{)},\ldots,\Big{(}y_{r},\gamma_{r1}^{i} \Big{)} \tag{15}\]  
where  
\(\bullet\)\(x=\big{[}x_{1},x_{2},\ldots,x_{nk}\big{]}^{T}\) is the input vector;  
\(y=\big{[}y_{1},y_{2},\ldots,r\big{]}^{T}\) the output vector;  
\(A_{i}^{i}(j=1,2,\ldots,n;\text{ i}=1,2,\ldots,\text{c})\) is the fuzzy membership function for rule i;  
\(\bullet\)\(\gamma_{i}=\big{[}\gamma_{1}{}^{i},\gamma_{2}{}^{i},\ldots,\gamma_{m}{}^{i} \big{]}\) is calculated with the help of Equation (16) [18]:  
\[\gamma_{k}{}^{i}=q_{ik,0}+q_{ik,1}x_{1}+\cdots+q_{ik,n}x_{n} \tag{16}\]  
When an FW model is called an FIS model, Equations (15) and (16) are expressed as Equations (17) and (18):  
\[R^{i}:\text{ If x}_{1}\text{ is A}_{1}{}^{i};\text{ x}_{2}\text{ is A}_{2}{}^{i};\text{ }\ldots;\text{ x}_{k}\text{ is A}_{k}{}^{i}\text{ Then }\Big{(}y_{1},\text{s}\beta_{1}^{i}\Big{)},\ldots,\Big{(}y_{r},\text{s}\beta_{ r}^{i}\Big{)} \tag{17}\]  
where \(\text{s}\beta_{n}^{i}\) is a combination of the wavelet function and is calculated as in Equation (18). \[\text{s}\beta_{n}^{i}=q_{in,0}+q_{in,1}\Psi_{i1}(x_{1})+\cdots+q_{in,k}\Psi_{ ik}(x_{k}) \tag{18}\]  
In Equation (18), s denotes the FW structure and \(\Psi_{i}^{i}(j=1,2,\ldots,k;\text{ i}=1,2,\ldots,\text{c})\) denotes the wavelet function of the jth input of the ith rule. In Equations (17) and (18), the THEN part considers one coefficient for each of the inputs; the number of coefficients for all c rules is \((n+1)\times c\). Furthermore, the output of the FWNN is shown in Equation (10) and can again be restated as in Equation (19). \[\hat{y}=\frac{\sum_{i=1}^{L}\left[\prod_{j=1}^{n}A_{i}^{j}(x_{j})\right]y_{i}}{ \sum_{i=1}^{L}\left[\prod_{j=1}^{n}A_{i}^{j}(x_{j})\right]}=\sum_{i=1}^{L}s \beta_{i}\Omega_{ij}G_{FW}(x;c_{i},\alpha_{i}) \tag{19}\]  
When Equation (19) and (1) are compared, it can be seen that the KELM and FW models are equivalent. Moreover, both models have similar constraints on the number of hidden nodes (N) and the number of rules (c), a power that fires each rule and an activation function that enables communication between each hidden node. FWMHKELM structure--the FW structure is defined as follows:  
\[\psi(x)=\Big{(}1-x^{2}\Big{)}\mathrm{exp}\bigg{(}-\frac{x^{2}}{2}\bigg{)} \tag{11}\]0  
Depending on whether n is odd or even, \(\beta_{n}^{i}\) is represented by Equations (21) and (22):  
\[\psi(x)=\Big{(}1-x^{2}\Big{)}\mathrm{exp}\bigg{(}-\frac{x^{2}}{2}\bigg{)} \tag{11}\]1  
\[\psi(x)=\Big{(}1-x^{2}\Big{)}\mathrm{exp}\bigg{(}-\frac{x^{2}}{2}\bigg{)} \tag{11}\]2  
In Equation (20), the sigmoid function is used in the IF part of the rule, while the Mexican line wavelet kernel is used in the THEN part. The functions are shown in Equations (23) and (24):  
\[\psi(x)=\Big{(}1-x^{2}\Big{)}\mathrm{exp}\bigg{(}-\frac{x^{2}}{2}\bigg{)} \tag{11}\]3  
where t is the center and \(\alpha\) is the slope point at \(x=t\). \[\psi(x)=\Big{(}1-x^{2}\Big{)}\mathrm{exp}\bigg{(}-\frac{x^{2}}{2}\bigg{)} \tag{11}\]4  
In an arbitrary \((x_{i},t_{i})\) training dataset, \(x_{i}=[x_{i1},x_{i2},\ldots,x_{in}]^{T}\) is the input vector, \(t_{i}=[t_{i1},t_{i2},\ldots,t_{in}]^{T}\) is the output vector, and c is the number of fuzzy rules. The mathematical model of FW can be defined as Equations (25) and (26):  
\[\psi(x)=\Big{(}1-x^{2}\Big{)}\mathrm{exp}\bigg{(}-\frac{x^{2}}{2}\bigg{)} \tag{11}\]5  
\[\psi(x)=\Big{(}1-x^{2}\Big{)}\mathrm{exp}\bigg{(}-\frac{x^{2}}{2}\bigg{)} \tag{11}\]6  
where \(\Omega\) is the MH wavelet kernel (MHW-KELM) matrix and it can be defined as Equation (27):  
\[\psi(x)=\Big{(}1-x^{2}\Big{)}\mathrm{exp}\bigg{(}-\frac{x^{2}}{2}\bigg{)} \tag{11}\]7  
Equation (25) can be compactly written as follows:  
\[\psi(x)=\Big{(}1-x^{2}\Big{)}\mathrm{exp}\bigg{(}-\frac{x^{2}}{2}\bigg{)} \tag{11}\]8  
where \(\Omega\) is the hidden matrix.