Fig 1.
A broad overview of the method proposed here.
Grids of varying resolution are placed over two point patterns and the difference between the two patterns can be calculated. The local S Index (Sl) (depicted on the maps) indicates the difference between the two point patterns at the cell level, and the Global S (Sg) (depicted on the graph) indicates the overall difference at that resolution.
Fig 2.
Densities of the four different crime types in the two years (2015 and 2016) used in the analysis.
Census tract boundaries have been overlaid for context, although these are not used in the analysis.
Table 1.
A description of the simulated point patterns used to test the algorithm.
The parameter a is chosen so that each point pattern contains approximately 3,000 points. Parameters b and c determine the amount of clustering; larger numbers produce more clustering.
Fig 3.
The five simulated point patterns used to test the algorithm.
P1 and P2 are the most similar (they were created using identical parameter values). The remainder (P3, P4, P5) become increasingly dissimilar.
Fig 4.
A scatter plot of the global similarity (Sg) of all cells (i.e. the mean of all individual cells’ similarity) at each resolution.
All comparisons are made to P1 (SIM2 compares P1 to P2, SIM3 compares P1 to P3, and so on). Points in the scatter plot have been aggregated into hexagonal bins to aid interpretation; red hexagons contain the largest numbers of scatter points. Lines of best fit and 99% confidence intervals are estimated using Local Polynomial Regression Fitting [39] (as implemented in the stats::loess R package).
Fig 5.
Graphs illustrating the change in average expected value at each resolution for the four different crime types.
Note that the horizontal line illustrates the cut off at which point individual cells would not have a sufficiently high expected value to draw statistically significant conclusions. The vertical lines denote some different resolutions used for illustrative purposes.
Fig 6.
The proportion of cells at each resolution that have a sufficiently large number of expected crimes in order to assess their similarity.
The vertical lines denote some different resolutions used for illustrative purposes.
Fig 7.
A scatter plot of the global similarity (Sg) of all cells (i.e. the mean of all individual cells’ similarity) at each resolution.
Lines of best fit and 99% confidence intervals are estimated using Local Polynomial Regression Fitting [39] (as implemented in the stats::loess R package). The vertical lines denote some different resolutions used for illustrative purposes.
Fig 8.
An explanation of the change in similarity with the size of the spatial unit.
Fig 9.
The similarity of residential burglary (BNER) and theft of bike (TOB) in 2015 and 2016, after ignoring cells with insufficient information to make a statistically significant comparison.