# Population Dynamics in Genetic Programming for Dynamic Symbolic Regression
## 3 Results
### Base Experiment Results
Similar to the quality dynamics, we can readily observe the reseeding pattern in the ALPS, which aims to normalize the overall frequencies by evenly distributing newly generated individuals in layer zero. However, the impact of reseeding typically normalizes after a few generations. Beginning with the W1 instance, we observe that all the variables are initially equally distributed in the population, adjusting over time in response to the dynamic nature of the benchmark. While variable x1 maintains a relatively constant frequency, x2 and x3 undergo cycles corresponding to the changing relevance dictated by the benchmark's underlying function and epoch shifts (as defined earlier in Figure 4). The variable frequency changes exhibit different speeds, reflecting the gradual or abrupt nature of the state changes over epochs. However, these changes occur steadily and slowly, spanning multiple generations rather than happening instantaneously. The overall patterns for instances W2 and W3 closely resemble W1, albeit with slightly more fluctuations due to the increased number of variables. In cases where multiple variables of a single term are tuned down during epoch progression, their occurrences in the population decrease concurrently. In the F1 instance, we observe a mix of relevant variables. For the initial three epochs, the variables operate independently, as defined in Figure 5. The subsequent epochs introduce combinations of two terms, and the last epoch displays an equal distribution of the variables when all the terms are relevant. A similar observation holds for the F2 instance but with two variables per term. Similar to the variable frequencies over time, Figure 8 illustrates the impacts (as defined in Section 2.3), representing the usefulness of the variables in predicting the target. A notable observation is that the impacts do not all start at the same initial value, reflecting instances where the generating function does not use all the variables in the first epoch. Figure 8: The variable impacts of the populations 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. Another significant difference from variable frequency is the rapid change in impacts, reflecting how quickly solutions change their behavior with new problem data after an epoch change. For example, in benchmark W1, the drop in variable impact at generation 210 and around 540 is abrupt, while the shift from generation 310 to 350 occurs more gradually, leading to slower changes in the variable impacts. In the Friedman benchmarks, where all the epoch changes are abrupt, this characteristic is even more evident. However, due to the nature of the underlying function generating the target value for the benchmark instances, it is observed that the impacts of the variables differ, even when all are relevant for the model. This discrepancy arises when different terms or subtrees contributing to the target value have varying value ranges, contributing to the final target to different degrees. Consequently, benchmarks with a higher number of features, especially F2 and F3, become challenging to interpret due to the variables having different maximum impacts. Even in the separated figures for the variable frequency and impacts, there is a clear indication of a close relationship between the two. This aligns with the intuitive understanding of these measurements, where variable impacts assess the immediate usefulness of variables, while the variable frequency reflects the consequence of the model's fitness, with the fitness closely related to identifying variables with a high impact. To better analyze the relationship between these two measurements, we focus on a single problem instance where we can plot the variable frequency and variable impact of individual variables for closer inspection. Figure 9 illustrates the relation between the frequency and impact over time for the instance W1.