# Geographically Weighted Flow Cross K-Function for Network-Constrained Flow Data
## 4 Case Study: Identifying Competition Patterns between Taxi and Ride-Hailing Services
### Results of Local Version  
The local Geographically Weighted Flow Cross K-Function was used to identify where the strong spatial associations occur. Figure 9 shows the percentage of taxi/ride-hailing flows that exhibit strong spatial associations with nearby ride-hailing/taxi flows. Both percentages increase with the increase in scale. When the scale is small, only a few spatial flows of taxis and ride-hailing exhibit strong spatial associations, which might suggest that competition usually exists between services of the same type. The percentage of taxi flows is less sensitive to the scale than ride-hailing flows. Even at a scale of 2000 m, only a fraction (0.08) of the taxi flows is spatially associated with nearby ride-hailing flows, which may indicate that the spatial range of exclusive markets of taxis is relatively large. This suggests that taxis may be competing more with themselves than with ride-hailing. To further reveal where the competition occurs, the spatial flows of taxis and ride-hailing that exhibit strong spatial associations at a relatively small scale (800 m) are displayed in Figure 10. The local Geographically Weighted Flow Cross K-Function detected some spatial flows (1384 taxi flows and 9599 ride-hailing flows) that were spatially associated with nearby spatial flows of the other type. The detected spatial flows can be seen as 'outliers', because these spatial flows do not follow the global trend. Both detected taxi and ride-hailing flows are mainly located in Huli and Siming, meaning that, compared with other districts, competition between taxi services and ride-hailing services is fiercer in the Huli and Siming districts. There are also some detected spatial flows located in the Jimei and Haicang districts. For example, many taxis face competition from ride-hailing around Xiamen North Railway Station. Figure 11 shows the top 200 flows of each type with the highest values of the local Geographically Weighted Flow Cross K-Function at the scale of 800 m. As shown in Figure 9, we roughly identified the origin/destination regions of these flows (the ellipse represents the origin regions and the rectangle represents the destination regions) for the analysis of the detected spatial associations. Origin points in regions 1 and 2 are in residential areas, and destination points in regions 3 and 4 are in workplaces (Guanyinshan International Business Center). These spatial flows may suggest that taxi services are facing competition in the daily commuting market from ride-hailing services. Origin points in regions 5 and 6 are in residential areas, and destination points in regions 7 and 8 are in workplaces, while destination points in regions 8 and 9 are near transportation hubs (Xiamen Railway Station  
Figure 9: The percentage of flows that exhibit strong spatial associations. results of experiments using simulated datasets demonstrated that the proposed method can detect predefined bivariate patterns correctly. The case study showed that the method could identify competition patterns between taxi and ride-hailing services effectively. It should be noted that the proposed method is only designed to detect spatial associations, and it cannot capture the dynamic patterns of spatial associations. In future work, the Geographically and Temporally Weighted Cross K-Function of spatial flows should be developed to detect the dynamic patterns of spatial associations. Moreover, the time cost of the proposed method is high because the calculation of the network-constrained mean distance is applied as the flow distance measurement, and the Monte Carlo simulation method is used to evaluate the statistical significance of detected spatial associations. A parallel version of the Geographically Weighted Flow Cross K-Function should be developed in the future. Conceptualization, W.Z. and H.X. ; methodology, W.L. ; validation, J.Z., Z.T. and H.X. ; formal analysis, Z.T. ; investigation, W.Z. ; resources, Z.T. ; data curation, W.L. ; writing--original draft preparation, W.Z. ; writing--review and editing, H.X.