# Parallel Hierarchical Genetic Algorithm for Scattered Data Fitting through B-Splines
## 5 Simulation Results
### Results  
In the experimental phase we performed 18 tests by the method using six functions, one size of sample points and three levels of noise. Each experiment was executed using 100 different test sets and 999 epochs for the PHGA, then the best individual by island was selected agree with its fitness, to finally choose the best of the best. To evaluate and compare the results for each function, we created validation sets consisting of 200 values uniformly distributed into their respective domain (Table 2); then they were evaluated in their respective function \(f(x_{i})\) and results were compared to the same set of values estimated by PHGA, CSSPL, FCF and LOWESS through a mean square error (MSE) given by: \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Island population size & 200 \\ Number of Islands & 10 \\ Size of elite population & 10 \\ Crossover probability & 0.85 \\ Mutation probability & 0.01 \\ Migration size & 10 \\ Migration interval & 200 epoch \\ Migration policy & Best Replace worst \\ Migration topology & Ring \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the genetic algorithm. \[MSE(\hat{f})=\frac{1}{n}\sum_{i=1}^{n}(\hat{f}(x_{i})-f(x_{i}))^{2}, \tag{14}\]  
where \(\hat{f}(x_{i})\) is the estimated function by PHGA, CSSPL, FCF and LOWESS, and the benchmark function \(f(x_{i})\). Table 4 shows the mean and standard error of the mean for MSE values obtained from each method for each test. As we can see in Table 4, in most of the cases, PHGA produced lower mean MSE values (marked in bold). This suggests that our method shows a considerable improvement over the CSSP, FCF and LOWESS methods, which is more appreciable in the box plots of the MSE values. Box plots are an effective way to compare the distribution of our results, because we can see how is the data distributed into quartiles. For each function and SNR level we generated a box plot graph using the MSE values obtained from each test using PHGA, CSSPL, FCF and LOWESS approaches. For the test with SNR = 3, showing in Figure 4, we can see that for function 1, the performance of PHGA was slightly better than CSSPL and FCF, since the distribution of the MSE obtained was similar and the median of the MSE for PHGA was lower. For function 2, PHGA was significantly better than the other methods due to the third quartile obtained with PHGA is slightly higher than the second quartile from CSSPL, which means that nearly of 75% of error obtained with PHGA tests had a lower MSE values than the 50% of error obtained using CSSPL, for FCF and LOWESS, PHGA improved considerably because of the upper adjacent value from PHGA errors was lower than the lower adjacent value from FCF and LOWESS. For function 3, we can see that the performance of PHGA and CSSPL was practically tied, and for FCF and LOWESS the third quartile from PHGA was lower than the first quartile from FCF and LOWESS. In the case of function 4, the third quartile from PHGA was lower than the median of CSSPL, for FCF and LOWESS against PHGA, it last was considerably better. In function 5, visually there was no difference between PHGA and CSSPL but PHGA was