## 3 Experiments and Results
Additionally, both the crossover rate and mutation rate should be non-zero values. In this paper, multiple runs were conducted with different starting populations and only the best results are reported. Figures 2 and 3 illustrate the results of the GA experiments for two crossover operations. In both Figures 2 and 3, the top solid line is the best-fitness population member and the dotted line is the average population members' fitness. While the top line appears straight, minor improvements in the best population member fitness occur. For Figure 2, the best population member fitness value improves from 0.823015 in the 1 st generation to 0.823325 in the 200 th generation. In Figure 3, these numbers are 0.822516 and 0.823437, respectively. The average fitness values for the posterior distribution sample extracted at the 192 nd generation from Figure 2 and the 194 th generation for Figure 3 were 0.631558 and 0.714597, respectively. The higher average fitness value for the sample extracted from Figure 3 suggests that arithmetic crossover results are better and will provide a greater percentage of IW improvement (PIWI) from Equation (15). The reader may also visually observe the gap between the average population fitness values and the best fitness population member values. This gap is lower in Figure 3, indicating that arithmetic crossover results are better than single-point crossover results. Table 2 illustrates the descriptive statistics of results obtained from the two crossover operators. Bayesian regression results taken from a text [9] are also reported for comparison. As expected, the PIWIs are higher for the arithmetic crossover, with all PIWIs being higher than 46%. This results in a tighter 95% confidence interval of the GA regression parameters for the arithmetic crossover operator. In all cases, the arithmetic crossover operator PIWIs are higher than those for the single-point crossover operator. A point of interest for the reader may be to note the starting average fitness values for both crossover operations at generation one. Both operators start at an average fitness value of around 0.4. Since the initial GA population is seeded with values close to the OLS regression parameters, the starting point illustrates that there is enough diversity in the population for the GA operators to still evolve the population to a higher overall fitness. Figure 3: The arithmetic crossover results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **Mean** & **Std. Dev.** & **2.5\%** & **Median** & **97.5\%** & **PIWI** \\ \hline \multicolumn{6}{c}{Single-Point Crossover Results} \\ \hline \(\beta_{1}\) & \(1.548\) & \(0.66\) & \(-0.702\) & \(1.712\) & \(1.712\) & \(61.98\%\) \\ \hline \(\beta_{2}\) & \(0.073\) & \(0.64\) & \(-0.635\) & \(0.015\) & \(2.279\) & \(54.10\%\) \\ \hline \(\beta_{3}\) & \(0.984\) & \(1.18\) & \(-2.907\) & \(1.069\) & \(2.416\) & \(16.16\%\) \\ \hline \multicolumn{6}{c}{Arithmetic Crossover Results} \\ \hline \(\beta_{1}\) & \(1.491\) & \(0.71\) & \(0.568\) & \(1.604\) & \(1.604\) & \(83.68\%\) \\ \hline \(\beta_{2}\) & \(0.003\) & \(0.62\) & \(-1.203\) & \(0.015\) & \(1.450\) & \(58.21\%\) \\ \hline \(\beta_{3}\) & \(2.089\) & \(0.93\) & \(-1.036\) & \(2.331\) & \(2.331\) & \(46.97\%\) \\ \hline \multicolumn{6}{c}{Bayesian Model Results (Taken from [9])} \\ \hline \(\beta_{1}\) & \(1.610\) & \(0.18\) & \(1.272\) & \(1.609\) & \(1.968\) & \\ \hline \(\beta_{2}\) & \(0.014\) & \(0.01\) & \(0.007\) & \(0.014\) & \(0.022\) & \\ \hline \(\beta_{3}\) & \(2.356\) & \(1.19\) & \(-0.039\) & \(2.372\) & \(4.635\) & \\ \hline \end{tabular}
\end{table} 
Table 2: Posterior distribution summaries. Figure 2: The single-point crossover results. Figure 3: The arithmetic crossover results. When viewing the arithmetic crossover results with the Bayesian model results, the Bayesian model confidence bounds are tighter, partly due to the likelihood distribution assumptions that Bayesian models make.