In order to determine the significance of spatial distributions of point events and their change over time reference units are needed. For events in the plane regular grids or polygonal (administrative) units frequently serve as spatial reference units. They can be used, for example, for the calculation of density or distribution measures.
For events which are network-bound, the planar approach is of little value. Leaks in a gas network only occur along the pipes, road accidents only happen on roads … Consequently reference units for the investigation of network-bound point events need to be defined in the network. For this several approaches exist. A very common one is to split the network graph in equally long segments.
Although very often used e.g. in road accident analysis (see the information-rich GIS platform of Upper Austria, DORIS , for an example), this approach has at least two drawbacks: If the segments are too short, the number of incidents is probably too small; and if they are too long, patterns or hot spots might be smoothed out. Additionally a purely edge based definition of reference units/segments doesn’t reflect the complex situations at road junctions, where a majority of accidents tend to occur.
For the analysis of bicycle accidents (comparable rare event: ~ 3,000 events in 10 years in a network with > 1,000 km total length) we* have developed an approach for the definition of network-based reference units which makes use of Voronoi diagrams. The basic idea is to generate random points and calculate network-based Voronoi diagrams as proposed by Okabe & Sugihara (2012). The distribution of random points is related to the density of the road network. This means, more points are generated where the network’s density is high resulting in smaller reference units and vice versa. Of course, the number and distribution of the random points (and the resulting Voronoi diagrams) has a strong influence on the results of any further analysis. Thus the workflow – see sketch on the left – for the calculation was coded in Python (with ArcGIS Model Builder as GUI) in order to allow for rapid re-calculations of the reference units.
ArcGIS was used for the generation of the random points. The Voronoi diagrams were calculated with an incredible ArcGIS AddIn called SANET . In ArcGIS random points can only be calculated in the plane with a polygon as constraining feature. To overcome this limitation and place random points exclusively along the network a buffer with a distance smaller than the snapping tolerance was generated.
As can be seen in the example result on the left, the size of the reference units is directly related to the network density. In the city center more random points (red dots) are placed and thus smaller reference units are calculated. This reflects the distribution of incidents, in our case bicycle accidents, quite well.
In the next step the frequency of accidents per reference unit is calculated and compared to a random distribution. We hope to draw valid conclusions about the significance of hot spots from this approach. The explanatory power likely depends on the number (and size) of the reference units. The example on the left is based on 100 random points distributed according to the network density in a 5×5 grid. Let’s see how it fits for the following analyses …
* Thanks to Christoph Traun for his initial ideas!
Reference: OKABE, A. & SUGIHARA, K. 2012. Spatial Analysis along Networks – Statistical and Computational Methods, Chichester, John Wiley & Sons.