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Too Fine to be Good? Issues of Granularity, Uniformity and Error in Spatial Crime Analysis

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Abstract

Objectives

Crime counts are sensitive to granularity choice. There is an increasing interest in analyzing crime at very fine granularities, such as street segments, with one of the reasons being that coarse granularities mask hot spots of crime. However, if granularities are too fine, counts may become unstable and unrepresentative. In this paper, we develop a method for determining a granularity that provides a compromise between these two criteria.

Methods

Our method starts by estimating internal uniformity and robustness to error for different granularities, then deciding on the granularity offering the best balance between the two. Internal uniformity is measured as the proportion of areal units that pass a test of complete spatial randomness for their internal crime distribution. Robustness to error is measured based on the average of the estimated coefficient of variation for each crime count.

Results

Our method was tested for burglaries, robberies and homicides in the city of Belo Horizonte, Brazil. Estimated “optimal” granularities were coarser than street segments but finer than neighborhoods. The proportion of units concentrating 50% of all crime was between 11% and 23%.

Conclusions

By balancing internal uniformity and robustness to error, our method is capable of producing more reliable crime maps. Our methodology shows that finer is not necessarily better in the micro-analysis of crime, and that units coarser than street segments might be better for this type of study. Finally, the observed crime clustering in our study was less intense than the expected from the law of crime concentration.

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Notes

  1. Some studies differentiate between granularity and resolution (Fonseca et al. 2002; Degbelo and Kuhn 2012), but we consider this distinction to be out of scope for this paper.

  2. An alternative terminology, more common in geostatistics, is that of signal and noise, as in Atkinson et al (2007).

  3. Silverman’s rule of thumb, for one dimensional cases, states that the optimal bandwidth h to minimize mean integrated squared error, assuming an underlying Gaussian distribution, is \(h = \left( {\frac{{4\hat{\sigma }^{5} }}{3n}} \right)^{{\frac{1}{5}}} \cong 1.06\hat{\sigma }n^{{ - \frac{1}{5}}}\), with n being the number of points considered and \(\hat{\sigma }\) the standard deviation of the points’ locations.

  4. This principle can be related to the more general accuracy-precision distinction, as in Kuhn (2012).

  5. Additionally, at least for the case studies considered, the difference between the two methods is small for the purposes of finding a granularity.

  6. While robustness is dependent on k, in practice the estimated granularity is not very sensitive to k. See the Appendix for a sensitivity analysis.

  7. It is worth noting that for granularities as fine as 25 meters (typically encompassing no more than 4 addresses), few points are likely to be found per quadrat sampled. This should not impact the tests, since only samples with more than one point will be considered for the purposes of testing uniformity and robustness. Since burglaries are registered per address, samples with only one address within the quadrat will (correctly) be resolved by the test as being uniform.

  8. This of course depends on georeferenced crime data being actually recorded and published, either by the police or by victimization surveys. While not a universal practice, some police departments do release their geocoded crime data, and crime maps have often been published (in varying granularities) by the police or third parties, such as news agencies or specialized websites.

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Authors and Affiliations

  1. INPE National Institute for Space Research, Av. dos Astronautas, 1.758 - Jardim da Granja, São José Dos Campos, São Paulo, 12227-010, Brazil

    Rafael G. Ramos

  2. Department of Sociology, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

    Bráulio F. A. Silva

  3. Department of Geography, University of California Santa Barbara, Santa Barbara, CA, USA

    Keith C. Clarke

  4. Department of Statistics, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

    Marcos Prates

Authors
  1. Rafael G. Ramos
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  2. Bráulio F. A. Silva
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  3. Keith C. Clarke
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  4. Marcos Prates
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Correspondence to Rafael G. Ramos.

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Appendix

Appendix

This appendix expands on the results presented in the Results section, adding more details that were not included there for brevity and simplicity.

In Fig. 7, the graphs for internal uniformity and robustness to error varying with granularity are shown for all three types of crime considered: burglaries, robberies, and homicides (the simplified graph was shown in Fig. 4). All four variants for estimating internal uniformity are plotted, as well as the six variants for estimating robustness to error.

Fig. 7
figure 7

Internal uniformity and robustness to error estimated for three types of crimes: residential burglary, street robbery, and homicides

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As can be seen from Fig. 7, the patterns displayed are similar for the three types of crime: internal uniformity decreasing as granularity becomes coarser and robustness to error increasing, though the granularity ranges in which that occurs vary. For robustness to error, all six variants yielded similar values; for internal uniformity, though, a slight but noticeable difference can be observed between nearest-neighbor and quadrat count approaches. Nevertheless, both exhibit the same general pattern of decreasing approximately at the same rate as granularity becomes coarser.

In Fig. 8, for each type of crime, plots for all three criteria proposed for estimating the optimal uniformity are shown. The optimal granularity according to each criterion is listed in Table 3 for each type of crime, as well as the mean values and standard deviations.

Fig. 8
figure 8

Optimal granularity estimated by three different criteria (balance of gains, product and sum criteria) for the three different types of crimes (burglary, robbery and homicides)

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Table 3 Optimal granularity for each type of crime, estimated with each criterion
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As can be seen from Table 3, for a given type of crime, the optimal granularity estimated by each criterion is quite similar, while the mean optimal granularity differs significantly for each type of crime.

Finally, Fig. 9 shows how this study’s methodology is not particularly sensitive to the value chosen for k in the robustness to error metric. The tradeoff analysis is shown for burglaries using three different values of k: while the curves may be different, the estimated optimal granularities are similar (on the order of 300 m) for the different values of k.

Fig. 9
figure 9

Tradeoff analysis for burglaries using three different values of k for the robustness metric (k  = 2, k  = 3, and k = 4)

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Ramos, R.G., Silva, B.F.A., Clarke, K.C. et al. Too Fine to be Good? Issues of Granularity, Uniformity and Error in Spatial Crime Analysis. J Quant Criminol 37, 419–443 (2021). https://doi.org/10.1007/s10940-020-09474-6

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