# A Study of Cellular Traffic Data Prediction by Kernel ELM with Parameter Optimization
## 4 Experiment Results and Analysis  
### Performance Analysis of ELM with Different Kernel Functions  
In this section, we investigate the predictive performance of kEML for cellular flow under different kernel functions. As the prediction performance of kEML is different when the parameters of different kernel functions take different values, we used the optimal prediction of flow of different kernel functions for comparison. Three metaheuristic optimization algorithms with excellent performance were used for parameter optimization of kEML. In the process of parameter optimization, each experiment was independently repeated 10 times. Finally, the optimal results were determined. The curves of MAPE of the training set the test sets of kEML with different kernel functions are shown in Figures 4 and 5. Figure 4: The mean absolute percentage error (MAPE) curve of the training set. (**a**) Gaussian, (**b**) linear, (**c**) polynomial, (**d**) sigmoid. In Figure 4, it can be seen that when the kernel function of ELM was Gaussian and polynomial, the value of MAPE appeared to increase with the number of iterations of the optimization algorithm. However, as shown in Figure 5, all kernel functions decreased with the number of iterations of the optimization algorithm. This is because we used the minimum MAPE of the test set as a function of the fitness of the optimization algorithm to prevent model overlearning. From Figures 4 and 5, it can be intuitively seen that, in the initial stage of parameter optimization, the value of MAPE was largest when the kernel function of kELM was sigmoid. The parameters for the three optimization algorithm searches led to its MAPE being greater than 60%. Then, its MAPE decreased rapidly as parameter optimization was carried out, which meant that the sigmoid kernel function predicted the cellular flow more sensitively to the parameters in the given variable interval. In addition, the linear kernel function had relatively minimal variation of the optimal MAPE for the search of the three optimization algorithms, but its optimal MAPE was the largest for both the training and test sets. The optimal results of the four kernel functions of MVO, MFO, and PSO for optimized kELM are shown in Table 5.  
Table 5 shows in detail the optimal results of kELM under multiple optimizations of the four kernel functions by the three metaheuristic algorithms. From Table 4, we can see that when the kernel function was Gaussian, the MAPE of the MFO search was the smallest at 11.150%. When the kernel function was polynomial, the MAPE of the MVO search was the smallest at 11.495%, and the result of the MFO search was relatively large at 11.611%. This indicates that the ability of the same metaheuristic optimization algorithm to search parameters of different kELM kernel functions is different, which is the main reason we chose multiple search algorithms for parameter optimization in this study. Figure 5: The MAPE curve of the test set. (**a**) Gaussian, (**b**) linear, (**c**) polynomial, (**d**) sigmoid. kELM predicts cellular traffic in the public setting. For the test set, the Gaussian kernel function had the minimum MAPE at 11.150%, while the polynomial kernel function had the second smallest MAPE at 11.495%. The MAPE was the largest when the kernel function was linear, both for the test and training sets. In order to prevent overlearning, the fitness function of each optimization algorithm should have the minimum MAPE in the test set. Therefore, the kernel function of the minimum MAPE of the test set and the optimization algorithm were combined for cellular flow prediction of the base station in the public setting. In other words, we used the results of Gaussian kernel function of MFO-kELM optimization for the following experimental comparison. The \"time\" in Table 5 is the time consumed in the parameter optimization process.