# A Novel Fuzzy Kernel Extreme Learning Machine Algorithm in Classification Problems
## 2 Methods
### Kernel-Based ELM  
The ELM was introduced to train single hidden layer (HL) feed-forward neural networks (SLFNs), the most widely used artificial neural network structure. A traditional SLFN consists of three layers--the input, hidden, and output layers. The goal of an ELM is to minimize both training errors and the output weight norm simultaneously. To achieve this, an ELM randomly selects input weights and biases, and then computes output neurons using the Moore-Penrose generalized inverse [22]. The weakness of the classical ELM model is that involving randomly assigned weights gives rise to the computation of different accuracies in various trials. However, to eliminate this inconvenience, the proposed kernel-based ELM executes tasks with a modification of random weight assignment between the input and hidden layers [23]. A brief overview of the KELM model will follow. The structure of the KELM is shown in Figure 2.  
The mathematical framework of an ELM can be summarized as in Equation (1) [24]:  
\[\sum_{i=1}^{N}\beta_{i}f\left(w_{i}x_{i}+b_{i}\right)=t_{j},\quad i=1,2,3, \ldots,K \tag{1}\]  
where \(t\) is the output vector, \(x\) is the input feature vector, \(w\) is the weight vector between the input layer (IL) and the hidden layer (HL), \(b\) is the bias, \(N\) is hidden nodes, and \(K\) is the training sample. Assuming optimal conditions, the difference between the target value (t) and the model output (o) is expected to be zero error [23].  
In this case, Equation (1) can be rewritten as Equation (2):  
\[\sum_{i=1}^{N}\beta_{i}f\left(w_{i}x_{i}+b_{i}\right)=o_{j},\quad i=1,\ldots,K \tag{2}\]  
As a result, the formula expressed by Equation (2) can be shortened to be represented by Equation (3).  
\[H\beta=T \tag{3}\]  
Figure 2: A structural representation of the kernel extreme learning machine (KELM).  
Since Equation (3) is a linear problem, it is obtained by the Moore-Penrose inverse known as \(\mathrm{H}^{\dagger}\).  
\[\beta=\mathrm{H}^{\dagger}\mathrm{T} \tag{4}\]  
In a KELM, the kernel structure maps the data from the input layer to the hidden layer and computes the \(\mathrm{H}^{\dagger}\) matrix by applying the orthogonal projection procedure. Applying [PERSON]'s condition to define the kernel matrix of KELM operates as follows [25]:  
\[\Omega_{\mathrm{N}}=\mathrm{H}_{\mathrm{N}}\mathrm{H}_{\mathrm{N}}^{\mathrm{T} }:\Omega_{\mathrm{N}_{\mathrm{i}\mathrm{j}}}=\mathrm{h}(\mathrm{x}_{\mathrm{i }})-\mathrm{h}\big{(}\mathrm{x}_{\mathrm{j}}\big{)}=\mathrm{K}\big{(}\mathrm{x }_{\mathrm{i}},\mathrm{x}_{\mathrm{j}}\big{)} \tag{5}\]  
where \(\mathrm{H}_{\mathrm{N}}\) is the hidden layer's output matrix, \(\Omega_{\mathrm{N}_{\mathrm{i}\mathrm{j}}}\) is the kernel matrix, and \(\mathrm{K}\big{(}\mathrm{x}_{\mathrm{i}},\mathrm{x}_{\mathrm{j}}\big{)}\) is the kernel function. The output of the KELM model is calculated using Equations (6) and (7).  
\[\mathrm{f}\big{(}\mathrm{x}_{\mathrm{P}}\big{)}=\begin{bmatrix}\mathrm{K} \big{(}\mathrm{x}_{\mathrm{P}},\mathrm{x}_{\mathrm{1}}\big{)}\\ \vdots\\ \mathrm{K}\big{(}\mathrm{x}_{\mathrm{P}},\mathrm{x}_{\mathrm{N}}\big{)}\end{bmatrix}^ {\mathrm{T}}\alpha_{\mathrm{N}} \tag{6}\]  
\[\alpha_{\mathrm{N}}=\bigg{(}\frac{\mathrm{I}_{\mathrm{N}}}{\mathrm{C}}+\Omega_ {\mathrm{N}}\bigg{)}^{-\mathrm{T}}\mathrm{T}_{\mathrm{N}} \tag{7}\]  
where \(\Omega_{\mathrm{N}}\) is the kernel function matrix of X [25].