# Population Dynamics in Genetic Programming for Dynamic Symbolic Regression
## 3 Results
### Base Experiment Results  
We initially assess the algorithms' effectiveness over the runs. In contrast to static optimization problems, a singular conclusive metric for the best solution is not provided, given the dynamic nature of the problem, where older metrics become obsolete with changes to the problem. The most relevant metric is the quality of the current best individual within the population at a specific generation, offering insight into the algorithm's optimal performance given the current state of the dynamic optimization problem. Additionally, we present the current mean quality of the population, providing an overview of the algorithm's general state and its convergence status. Figure 6 illustrates the mean population quality and best quality (measured in Pearson R\({}^{2}\)) over time for each algorithm across different problem instances. It is evident that, for all instances, the best solution over time predominantly reaches 1.0 R\({}^{2}\) exactly or very close, indicating that the algorithms consistently discover high-quality models. Notably, regular dips in the average quality coincide with epoch changes. This pattern is particularly noticeable in the Friedman-based instances (F1 and F2), where the data transition abruptly compared to the more gradual changes in the Winkler-based instances (W1, W2, and W3). For the Friedman-based instances, variations in the difficulty among the epochs are apparent, with a lower average quality indicating greater challenges in obtaining good models. In these cases, even the best solutions found do not consistently achieve a perfect 1.0 R\({}^{2}\). For the ALPS, recurring drops in the average quality align with the reseeding of layer zero. Due to this reseeding mechanism and the population growth over time, the ALPS exhibits a lower overall population quality compared to the GA. Although comparing only the top layer of the ALPS to the GA might yield similar results, we deliberately opted to analyze the total ALPS population to more accurately reflect the actual dynamics of the ALPS. The results suggest that, for the presented problem instances, the algorithms quickly adapt to changes in the training data within a few generations. This rapid adaptation could imply that the difficulty for individual epochs was generally low. However, challenges persist in the last phase for both Friedman-based instances, where the algorithms struggle to fully solve the problem. It is worth considering that these challenges may also arise from imposed limitations on the tree size, potentially rendering the problem unsolvable within the given constraints. Figure 7 depicts the average variable frequencies within the population across all runs for each algorithm and instance. This visualization offers insights into the variables retained in the population and how their relative frequencies change in response to epoch variations. The shifts in the population composition are clearly observable in all instances, aligning well with the epoch changes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**W1** & **W2** & **W3** & **F1** & **F2** \\ \hline Mutation Probability & \(15\%/10\%/5\%/1\%/0.5\%/0.1\%/0.05\%/0.01\%/0.005\%/0.001\%/0.00\%\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters for the experiment analyzing varied mutation rates. The table contains only modifications made to the base experiments. Figure 6: The figure depicts the mean and best quality of the population, measured in Pearson’s R\({}^{2}\), over time. The light-shaded lines represent individual runs, while the darker lines represent the mean values across all runs. Subplots are organized column-wise by the algorithms indicated at the top and instances noted on the right. Figure 7: The variable frequencies over time, averaged across all runs for each generation, with distinct variables represented by different colors. Subplots are arranged column-wise by algorithm and row-wise by benchmark instance.