## 4 Results and Discussion
### Descriptive Analysis (EDA)
#### 4.1.5 Correlation Analysis  
To identify significant linear relationships between waiting time and various variables, Pearson correlation coefficients were calculated using two approaches:  
1. General Correlation (Table 4): Includes all trips. * Urban Bus (0.1035) and Valparaiso Commune (0.0947) exhibit positive correlations with waiting time, consistent with high demand and potential frequency constraints, particularly during peak hours. * Start Time (0.0756) suggests that later trips are slightly more prone to longer waits. * Minutes Before/After the Trip (0.0418 and 0.0348) show weak positive correlations; those who walk more may face lower service frequencies, although this relationship is minor. * Metro Valparaiso (\(-\)0.0414), Colectivo Taxi (\(-\)0.0844), Quilpue (\(-\)0.0433), and Villa Alemana (\(-\)0.0919) present negative correlations, associating these modes and communes with shorter waits, though the strength of these relationships is also weak.
2. Correlation for Prolonged Waiting Times (Table 5): Focuses on trips with waits exceeding 20 min (outliers). * Quilpue (0.1725) stands out with a moderately positive correlation, signaling a higher probability of extremely long waits, possibly due to interurban routes with reduced frequency or coordination issues. * Urban Bus (0.1044) and Trip Start Time (0.1050) remain influential, reflecting peak-demand pressure at specific times of day. * Metro Valparaiso (\(-\)0.0263) and Taxi Colectivo (\(-\)0.0986) confirm their role in limiting extreme waits. * Minutes Before/After become negative (\(-\)0.0447 and \(-\)0.0675), suggesting that in this subset of data, walking more might grant access to more regular services. * Valparaiso (\(-\)0.1598) shows a moderately negative correlation here, likely due to alternative modes (such as Funiculars or Trolleybuses) that reduce the likelihood of very long waits.  
Figure 4: Hourly trip frequency and waiting time analysis in public transportation.  
Comparing the two tables reveals both consistent patterns (urban bus emerges as a factor associated with longer waits) and differences, such as the change in Valparaiso's correlation when focusing on very prolonged waiting times. This outcome underscores the need to analyze trips with extreme delays separately.  
In the overall dataset, Valparaiso and urban bus exhibit positive correlations, while Quilpue and modes like Metro or Taxi Colectivo show negative correlations. In the subset featuring prolonged waits, Quilpue becomes more prominent due to a higher probability of severe delays, whereas Valparaiso's correlation turns negative, possibly because alternative modes reduce waiting times there.  
These observations reinforce the importance of improving service frequency and capacity on critical urban bus routes while promoting more reliable transport modes. Moreover, variability in \"walking minutes\" suggests that having frequent service can compensate for longer pedestrian journeys. Together, these findings support the subsequent spatial autocorrelation and clustering analyses, providing insights for optimizing public transport service.