# Local Crossover: A New Genetic Operator for Grammatical Evolution
## 2 Materials and Methods
### Experimental Results
Also, a statistical comparison was performed using the two machine learning methods and the enhanced ones that use the new crossover operator. This comparison was performed using the classification datasets, and it is shown in Figure 11. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**DATASET** & **BFGS** & **GEN** & **RULE** & **NNC** & **RULE_CROSS** & **NNC_CROSS** \\ \hline ABALONE & 6.38 & 7.17 & 7.36 & 5.05 & 5.32 & 4.63 \\ \hline AIRFOIL} & 0.003 & 0.001 & 0.003 & 0.003 & 0.002 & 0.002 \\ \hline BK & 0.36 & 0.26 & 0.02 & 2.32 & 0.037 & 0.15 \\ \hline BL & 1.09 & 2.23 & 2.53 & 0.021 & 0.023 & 0.40 \\ \hline BASEBALL & 119.63 & 64.60 & 65.64 & 59.85 & 61.35 & 58.75 \\ \hline CONCRETE & 0.023 & 0.001 & 0.013 & 0.008 & 0.009 & 0.005 \\ \hline DEE & 2.36 & 0.47 & 0.43 & 0.26 & 0.32 & 0.23 \\ \hline FY & 0.19 & 0.65 & 0.041 & 0.058 & 0.046 & 0.049 \\ \hline HO & 0.62 & 0.37 & 0.019 & 0.017 & 0.019 & 0.014 \\ \hline HOUSING & 97.38 & 35.97 & 47.99 & 26.35 & 26.74 & 19.10 \\ \hline LASER & 0.03 & 0.084 & 0.055 & 0.024 & 0.032 & 0.019 \\ \hline LW & 0.26 & 0.54 & 0.012 & 0.011 & 0.013 & 0.017 \\ \hline MORTGAGE & 8.23 & 0.40 & 0.20 & 0.30 & 0.13 & 0.21 \\ \hline MUNDIAL & 0.05 & 1.22 & 0.038 & 4.47 & 0.049 & 0.76 \\ \hline PL & 0.11 & 0.03 & 0.056 & 0.045 & 0.035 & 0.036 \\ \hline QUAKE & 0.09 & 0.12 & 1.13 & 0.045 & 0.73 & 0.046 \\ \hline REALETATE & 128.94 & 81.19 & 104.74 & 76.78 & 92.49 & 69.77 \\ \hline SN & 0.16 & 0.20 & 0.025 & 0.026 & 0.026 & 0.024 \\ \hline TREASURY & 9.91 & 0.44 & 0.15 & 0.47 & 0.12 & 0.30 \\ \hline TZ & 0.21 & 0.097 & 0.036 & 5.04 & 0.035 & 0.061 \\ \hline VE & 1.92 & 2.43 & 0.028 & 6.61 & 0.043 & 0.084 \\ \hline
**AVERAGE** & **17.99** & **9.45** & **10.98** & **8.94** & **8.93** & **7.35** \\ \hline \end{tabular}
\end{table} 
Table 3: This table shows the experimental results when the regression datasets were used and the numbers stand for the average regression error as measured on the corresponding test set. Figure 5: Number of datasets that improved in the rule machine learning model using the proposed method. The vertical axis represents the number of datasets and the horizontal axis represents the percentage of reduction in error. Figure 6: Number of datasets that improved in the NNC machine learning model using the proposed method. The vertical axis represents the number of datasets and the horizontal axis represents the percentage of reduction in error. Figure 7: Box plot for the comparison between the original rule construction model and the improved one that utilizes the new crossover operator. Moreover, an additional test was executed in order to estimate the effectiveness of the new crossover rate parameter denoted as \(p_{\sigma}\). In this experiment, the rule construction machine learning model was applied on the classification datasets using a series of values for \(p_{\sigma}\), and the results are shown in Table 4.  
Looking at the table of results, one can see a significant decrease in the average classification error when the application rate of the genetic operator increases from 2.5% to 5%. However, the rate of reduction of the average error decreases significantly when the application rate increases to 7.5%. This finding reinforces the idea of implementing the new genetic operator at a rate of 5%. Figure 8: Box plot used to compare the original rule construction model and the improved one that utilizes the new crossover operator. Figure 9: Box plot used to compare the original rule construction model and the improved one that utilizes the new crossover operator for the regression datasets. Another experiment was performed in order to estimate the importance of the parameter \(N_{cr}\), which controls the number of chromosomes participating in the new crossover operator. In this experiment, the neural network construction method was tested on the classification datasets using a series of values for the parameter \(N_{cr}\), while the parameter \(p_{cr}\) was fixed to 2.5%.