## 6 Training and Simulation Results
### Computed a Gini Policing Index (GPI)
#### 6.2.2 Results and Interpretation  
The calculated Gini policing index for the DDQN agent's simulated patrol was 0.8729. The calculated Gini policing index for the PPO agent's simulated patrol was 0.8720. Both the DDQN and PPO agents resulted in relatively high Gini indices (close to 1). This indicates a notable inequality in how patrol visits were distributed across the Wards relative to the crime counts in this simulation. The analysis of the patrol-to-crime ratios per Ward showed that patrols for both agents were heavily concentrated in a few Wards, while other Wards with significant crime counts received very few or no visits during the simulated patrol period. PPO's Gini index (0.8720) was slightly lower than DDQN's (0.8729), suggesting that, based on this simplified metric, PPO's patrol distribution relative to crime was marginally more equitable across Wards.  
It is important to note that this Gini index calculation is based on a simplified approach using approximate Ward locations and visit frequency over a limited simulation duration. A more comprehensive fairness analysis would require using actual socioeconomic data and boundaries (like Census Tracts), precise spatial mapping of patrol paths, and calculating patrol density. However, even with this simplification, the results highlight potential fairness concerns regarding the concentration of policing resources relative to crime levels across different zones.  
Beyond reward comparisons, further analysis of the spatial distribution of crime offers operational insights. High-frequency clusters--particularly those experiencing temporal peaks between 4 p.m. and 9 p.m.--should inform initial weightings in both reward shaping and graph edge costs. Integrating historical crime intensity directly into the environment simulation allows the RL agent to confront realistic patrol demands, reducing generalization error during deployment. In our framework, these distributions derived from KDE and temporal density plots were mapped to specific node weights in the ST-GCN-informed graph, thereby linking the statistical properties of real-world crime patterns with the policy learning landscape.  
Building on Seattle's crime dataset analysis, we observed dense clusters of theft and vehicle break-ins in waterfront-adjacent zones, particularly during evening hours. These high-risk segments were overlaid onto the patrol graph structure, assigning edge weights inversely proportional to incident density. This ensured frequent revisiting of volatile regions without over-penalizing low-crime neighborhoods. Furthermore, temporal shifts (e.g., weekend spikes in nightlife zones) were encoded as dynamic reward modulations, enabling agents to reprioritize routes contextually. This coupling of spatial heatmaps with RL reward design enhanced both patrol efficiency and equitable coverage.  
Analysis of Seattle's real-world incident data reveals that crime densities are not uniformly distributed across space or time. The heatmaps generated from the KDE visualizations show hotspots clustered around downtown and waterfront corridors, particularly during evening hours (6 p.m.-12 a.m.). These spatial patterns were integrated into the RL training environment by encoding incident density as a contextual variable in the reward function. Agents trained on this data exhibited stronger convergence and better geographic coverage of high-crime zones. Additionally, time-weighted graphs were used to reflect peak activity hours, which modulated the agent's node selection preferences, encouraging attention to temporally volatile regions. This alignment between observedurban crime trends and simulation dynamics enhances realism and supports transferability of learned policies.