# A Study of Cellular Traffic Data Prediction by Kernel ELM with Parameter Optimization
## 2. Accident Description and Prediction Method
### Prediction Methods  
#### 2.2.1 Kernel ELM  
The ELM is a new SLFN that overcomes the disadvantages of traditional neural networks, such as slow training speed, easily falling into a local minimum point, and sensitive learning rate selection. The connection weight between the input layer, the hidden layer, and threshold values of neurons in the hidden layer of ELM are all random. There is no need for adjustment in the training process. The weights are obtained through a matrix operation, which means fast learning speed and strong generalization ability. Although ELM has excellent learning speed and generalization ability, its learning ability needs to be improved. [PERSON] et al. borrowed the idea of feature mapping of SVM to improve ELM, called kELM, which not only inherits the advantages of ELM but also has the same learning performance as SVM. Let \(T=\{(x_{1},y_{1}),\cdots,(x_{l},y_{l})\}\), \(x\in R^{n}\), \(y\in R^{m}\). The ELM model with \(l\) hidden nodes and \(g(\cdot)\) as the activation function is as follows:  
\[f\Big{(}x_{j}\Big{)}=\sum\
olimits_{i=1}^{N}\beta_{i}g\Big{(}w_{i}\cdot x_{j} +b_{i}\Big{)} \tag{1}\]  
where \(w_{i}=[w_{i1},\cdots,w_{in}]\) is the weight of the \(i\)-th input neuron and hidden neurons, \(\beta_{i}=[\beta_{i1},\cdots,\beta_{in}]\) is the weight between the \(i\)-th hidden layer neuron and the output layer, and \(b\) is the bias. Let  
\[G=\left[ \begin{array}{ccc}g(w_{1}\cdot x_{1}+b_{1})&\cdots&g(w_{l}\cdot x_{ 1}+b_{l})\\ \vdots&\cdots&\vdots\\ g(w_{1}\cdot x_{n}+b_{1})&\cdots&g(w_{l}\cdot x_{n}+b_{l})\\ \end{array} \right]_{n\times d}\beta=\left[ \begin{array}{c}\beta_{1}^{T}\\ \vdots\\ \beta_{l}^{T}\\ \end{array} \right]_{D\times m},F=\left[ \begin{array}{c}f_{1}^{T}\\ \vdots\\ f_{l}^{T}\\ \end{array} \right]_{n\times m}\]  
Equation (1) can be written as follows:  
\[G\beta=F \tag{2}\]  
In the BP neural network, Equation (2) is solved by multiple iterations using the error backpropagation method. However, Equation (2) is solved by matrix operations in the ELM as shown in Equation (3):  
\[\beta=G^{+}F \tag{3}\]  
where \(G^{+}=\left(G^{T}G\right)^{-1}G^{T}\). The mechanism tells us that the core factor of fast ELM learning is that the assignment of bias \(b\) and weight \(w\) between the input layer and hidden layer is a random number. Meanwhile, it is not modified during the model learning process, and the weight \(\beta\) is obtained through matrix operations. The solution obtained in Equation (3) is just one solution, but it is impossible to avoid errors in practical situations. In applications, the solution of ELM model can also be written as follows:  
\[\|G\beta-F\|=\min_{\beta}\|G\hat{\beta}-F\| \tag{4}\]  
In Equation (4), the obtained \(\beta\) is an approximate solution of the weight matrix, which, to a certain extent, improves model generalization. However, its learning ability still needs to be improved. [PERSON] et al. improved the ELM through feature mapping, called kELM, and its mathematical model is as follows:  
\[ \begin{array}{l}\text{Minimize}:L_{kELM}=0.5\|\beta\|^{2}+0.5C\sum_{i=1}^{n }\|\varepsilon_{i}\|^{2}\\ \hskip 14.226378 pts.t.\hskip 2.845276 ptg(x_{i})\beta=f_{i}-\varepsilon_{i}\end{array}  \tag{5}\]  
where \(\varepsilon_{i}=[\varepsilon_{i1},\cdots,\varepsilon_{im}]\) is the training error vector, and \(C\) is a parameter set by the user. According to Karush-Kuhn-Tucker (KKT) conditions, Equation (5) can be transformed into its dual problem as follows:  
\[L^{\prime}_{kELM}=0.5\|\beta\|^{2}+0.5C\sum\
olimits_{i=1}^{n}\|\varepsilon_ {i}\|^{2}\hskip 5.690551 pt-\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}\alpha_{i, j}\Big{(}g(x_{i})\beta_{j}-f_{i,j}+\varepsilon_{i,j}\Big{)} \tag{6}\]  
where \(\alpha_{i,j}\) is the Lagrangian operator. The solution of Equation (6) is as follows:  
\[\beta=G^{T}\Big{(}\frac{I}{C}+GG^{T}\Big{)}^{-1}F \tag{7}\]  
Therefore, the output of kELM model is as follows:  
\[\hat{y}(x)=g(x)\beta=g(x)G^{T}\Big{(}\frac{I}{C}+GG^{T}\Big{)}^{-1}F \tag{8}\]  
where, \(\hat{y}\) is the model output.