Inaccuracy, uncertainty and the space-time permutation scan statistic

Abstract

The space-time permutation scan statistic (STPSS) is designed to identify hot (and cool) spots of space-time interaction within patterns of spatio-temporal events. While the method has been adopted widely in practice, there has been little consideration of the effect inaccurate and/or incomplete input data may have on its results. Given the pervasiveness of inaccuracy, uncertainty and incompleteness within spatio-temporal datasets and the popularity of the method, this issue warrants further investigation. Here, a series of simulation experiments using both synthetic and real-world data are carried out to better understand how deficiencies in the spatial and temporal accuracy as well as the completeness of the input data may affect results of the STPSS. The findings, while specific to the parameters employed here, reveal a surprising robustness of the method's results in the face of these deficiencies. As expected, the experiments illustrate that greater degradation of input data quality leads to greater variability in the results. Additionally, they show that weaker signals of space-time interaction are those most affected by the introduced deficiencies. However, in stark contrast to previous investigations into the impact of these input data problems on global tests of space-time interaction, this local metric is revealed to be only minimally affected by the degree of inaccuracy and incompleteness introduced in these experiments.

Figure 1. Intensity of the three simulated event patterns.

Each panel shows a different perspective of a space-time cube for the three patterns. The left most column corresponds to the high intensity cluster pattern (where cluster events are concentrated in a smaller area) and the right-most column corresponds to the lowest intensity pattern. The top row shows an areal view of the space time cubes (i.e. a conventional map), the middle row shows a front view of the cubes, while the bottom row shows a side view. Lighter areas indicate a higher intensity of events.

Figure 2. The distribution of temporal ranges within which burglaries and thefts are known to have occurred in Mesa, AZ for the period 2004–2009.

Random draws from this distribution were used to offset the temporal coordinates of the original data.

Figure 3. Sample of burglary events occurring in Mesa, Arizona during 2008 employed in the analysis.

Additional geographic identifiers have been omitted from the map to preserve privacy.

Figure 4. Plots of MLCs identified with the STPSS.

The spatial footprint of the MLCs for the original datasets are shown in red and the secondary cluster with the next lowest -value is shown in green. MLCs from perturbed versions of the same dataset are shown in black. The intensity of the original clusters decreases from the top down while the intensity of perturbation increases from the left to the right. This layout is followed in subsequent graphics.

Figure 5. Plots of the duration of MLCs identified with the STPSS.

The duration of the MLCs for the original datasets are denoted using horizontal red lines, secondary clusters are shown using green lines. MLCs from perturbed versions of the same dataset are shown as black vertical lines.

Figure 6. Pie charts showing the proportion of MLCs in the set of perturbed patterns located in the vicinity of Clusters 1 and 2.

Figure 7. Percentage of patterns where Cluster 1 was reported as the MLC across the different intensity/perturbation combinations.

Figure 8. Number of perturbed patterns where Clusters 1(in red) and 2 (in green) were identified as “likely clusters” by the STPSS.

Figure 9. Pseudo -values determined by the STPSS for likely clusters identified in the vicinity of the original Cluster 1 (red solid line) and 2 (green dashed line) in each perturbed version of the original datasets.

Figure 10. Plots of MLCs identified within the Mesa crime data using the STPSS.

The spatial footprint of the MLC for the original dataset is shown in red. MLCs from perturbed versions of the same dataset are shown in black.

Figure 11. Plots of the duration of MLCs identified within the Mesa crime data using the STPSS.

The duration of the MLC for the original dataset is denoted using horizontal red lines. MLCs from perturbed versions of the same dataset are shown as black vertical lines.

Figure 12. Percentage of patterns where the original MLC was reported as the MLC across the different perturbation levels.

Figure 13. Pseudo -values determined by the STPSS for likely clusters identified in the vicinity of the original cluster 1 (red solid line) in each perturbed version of the original datasets.