# Population Dynamics in Genetic Programming for Dynamic Symbolic Regression
## 3 Results
### Mutation Rates Results  
When evaluating the impact of algorithm parameters, the usual focus is on identifying the best solution and assessing the convergence speed. However, this approach is not directly applicable to dynamic problems where multiple convergence phases and multiple best solutions occur within a single run. Instead of solely reporting the overall best solution of individual runs, we adopt a more general analysis by examining the average quality over time. This involves calculating the mean of the best individual per generation and the mean of the average population's quality per generation. The resulting insights into the algorithm's overall performance across multiple epochs are illustrated in Figure 12. Lower average qualities in this context indicate suboptimal algorithm performance. The results reveal that for the GA, mutation rates between 0.5% and 15% are effective, while the average quality drops significantly when the mutation rates are lower than 0.1%. This decline can be attributed to the fact that variables may disappear from the population and become unavailable when needed in later epochs. The GA heavily relies on mutation to maintain diversity and reintroduce lost variables. It is noteworthy that high mutation  
Figure 12: The figure displays the mean quality of the current best solution per generation over all generations (depicted in red) and the mean quality of the population per generation over all generations (depicted in blue) for different mutation rates on the x-axis. Each individual dot represents the average qualities of a single run, with the overlaid box indicating the distribution over all runs. Subplots are arranged column-wise by algorithms and row-wise by instances. rates (5-15%) lead to a reduction in the average quality of the population. This decline is attributed to the increased disruptive impact associated with higher mutation rates. In contrast, the ALPS leverages the reseeding mechanism as a potent tool to reintroduce lost diversity, reducing the dependence on mutation. However, it is essential to highlight that this success with the ALPS without mutation may be contingent on the application of local optimization to all numeric parameters of symbolic models using gradient descent. Without this local optimization, algorithm convergence would likely be slow or nonexistent, emphasizing the significance of mechanisms for changing numeric parameters in the absence of mutation. As a next step, we aim to conduct a more detailed analysis of the impact of mutation rates on both the GA and ALPS. This involves focusing on a specific variable within a benchmark, closely examining the variable frequency of the population over time for each individual run. Additionally, we will distinguish cases in which a variable becomes extinct, i.e., it drops out of the population at some point. This loss of relevant building blocks and the subsequent ability to reintroduce them into the population is typically associated with mutation; therefore, we hypothesize that lower mutation rates may lead to more frequent instances of variables becoming extinct, which could explain the inferior results observed at lower mutation rates. Regarding the extinction of variables, we classify three distinct cases of extinction:  
Never: The variable is consistently present in the population at all times. Momentarily: The variable drops out of the population at a specific time but is later reintroduced, typically through mutation. Permanently: The variable drops out of the population at a specific time but is not reintroduced, remaining absent for the rest of the run. Figure 13 shows the variable frequencies for the individual runs of variable x3 from benchmark W1 across various mutation rates. The color code indicates whether the variable was never extinct (consistently present), momentarily extinct, or permanently extinct. Please note that the entire run is color-coded based on the extinction case of the variable, rather than the specific time frame in which the extinction occurred.