# Improving Victimization Risk Estimation: A Geographically Weighted Regression Approach

## Abstract

**:**

## 1. Introduction

#### 1.1. Literature on Crime Standardization and the Estimation of Victimization Risk

#### 1.2. Literature on Geographically Weighted Regression

## 2. Materials and Methods

#### 2.1. Problem Specification

- Crime counts are often a fraction $f$ of the victimization rates, being also subject to other types of error ${\mathit{\epsilon}}_{C}$ (e.g., missing data, geocoding errors, multiple reports of the same crime, and other less systematic forms of error affecting the relation between victimization and crime counts):$$C=fV+{\mathit{\epsilon}}_{C}$$
- Population data may not be a perfect measure of the actual pool of potential victims of the crime we are considering:$${P}^{*}-P={\mathit{\epsilon}}_{P}\ne 0$$
- Actual victimization rates may not be exactly the expected ones, but fluctuate around it:$$V-E\left[V\right]=\mathit{\u03f5}\ne 0$$

#### 2.2. Proposed Solution

#### 2.3. Validating the Method via a Simulation Study

#### 2.4. Application: Residential Burglaries in the City of Belo Horizonte, Brazil

## 3. Results

#### 3.1. Results for the Validation Study

#### 3.1.1. Simulation Study with One Reference Population

#### 3.1.2. Simulation Study with Two Reference Population

#### 3.2. Results for the Application Study

## 4. Discussion

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

**Table A1.**Results of the simulation study with one population, showing the fitness scores for estimated risk calculated using three different methods, and using different parameters for generating maps of true risk R and reference population P. The unit for the range parameters is in number of map cells, while the units for sill and nugget correspond to the square of the unit for risk (i.e., the unit for its variance: crimes

^{2}/targets

^{2}). Bold was used to highlighted the parameters that area varying from row to row.

Parameter | Fit for Estimated R | ||||||
---|---|---|---|---|---|---|---|

range_{R} | range_{P} | sill_{P} | nugget_{P} | ${\mathbf{\epsilon}}_{\mathbf{C}}$ | Fit (naïve) | Fit (GWRisk) | Fit (Bayes) |

50 | 7 | 16,500 | 1250 | 15% | 0.17 | 0.66 | 0.46 |

50 | 7 | 16,500 | 2500 | 15% | 0.11 | 0.65 | 0.37 |

50 | 7 | 16,500 | 5000 | 15% | 0.13 | 0.71 | 0.43 |

50 | 7 | 16,500 | 10,000 | 15% | 0.17 | 0.71 | 0.48 |

75 | 7 | 16,500 | 1250 | 15% | 0.13 | 0.65 | 0.39 |

25 | 7 | 16,500 | 1250 | 15% | 0.17 | 0.68 | 0.46 |

10 | 7 | 16,500 | 1250 | 15% | 0.17 | 0.57 | 0.50 |

5 | 7 | 16,500 | 1250 | 15% | 0.23 | 0.39 | 0.50 |

50 | 3.5 | 16,500 | 1250 | 15% | 0.12 | 0.73 | 0.42 |

50 | 14 | 16,500 | 1250 | 15% | 0.16 | 0.60 | 0.44 |

50 | 28 | 16,500 | 1250 | 15% | 0.26 | 0.59 | 0.55 |

50 | 56 | 16,500 | 1250 | 15% | 0.40 | 0.57 | 0.60 |

50 | 7 | 10,000 | 1250 | 15% | 0.18 | 0.66 | 0.52 |

50 | 7 | 20,000 | 1250 | 15% | 0.14 | 0.69 | 0.44 |

50 | 7 | 40,000 | 1250 | 15% | 0.12 | 0.71 | 0.34 |

50 | 7 | 80,000 | 1250 | 15% | 0.09 | 0.68 | 0.28 |

50 | 7 | 16,500 | 1250 | 5% | 0.32 | 0.68 | 0.76 |

50 | 7 | 16,500 | 1250 | 25% | 0.08 | 0.63 | 0.20 |

50 | 7 | 16,500 | 1250 | 50% | 0.03 | 0.65 | 0.06 |

50 | 7 | 16,500 | 1250 | 100% | 0.01 | 0.54 | 0.02 |

## Appendix C

**Figure A1.**An example case showing maps for simulated reference population, (true) victimization risk and crime counts, as well as the estimated victimization risks using each of the three methods considered: GWRisk, naïve estimation, and the Empirical Bayes Estimator method. Parameters for this case are listed in Table A1 (Appendix B), 15th entry. Very high values were removed for naïve and Empirical Bayesian to allow comparison between lower values. (See Figure 5 for uncapped figures). Notice that spurious peaks still exist even in this version.

**Figure A2.**Estimated victimization risks of burglary for single-family houses and residential apartments, calculated using each of the three methods tested. Very high values were removed for naïve and Empirical Bayesian to allow comparison between lower values. (See Figure 6 for uncapped figures). Notice that (probably spurious) peaks still exist even in this version.

## References

- Biderman, A.D.; Reiss, A.J., Jr. On exploring the “dark figure" of crime. Ann. Am. Acad. Political. Soc. Sci.
**1967**, 374, 1–15. [Google Scholar] [CrossRef][Green Version] - Radzinowicz, L.; King, J.F. The Growth of Crime: The International Experience; Basic Books: New York, NY, USA, 1977; pp. 3–9. [Google Scholar]
- Payne, J.L.; Hutton, F. Mapping Common Crime. In The Palgrave Handbook of Australian and New Zealand Criminology, Crime and Justice; Palgrave Macmillan: Cham, Switzerland, 2017; pp. 113–129. [Google Scholar] [CrossRef]
- Langton, L.; Planty, M.; Lynch, J.P. Second major redesign of the national crime victimization survey (ncvs). Criminol. Pub. Pol’y
**2017**, 16, 1049. [Google Scholar] [CrossRef] - Williams, D.; Edwards, S.; Giambo, P.; Kena, G. Cost Effective Mail Survey Design. In Proceedings of the Federal Committee on Statistical Methodology Research and Policy Conference, Washington, DC, USA, 1–3 December 2018. [Google Scholar]
- Ratcliffe, J. Crime mapping: Spatial and temporal challenges. In Handbook of Quantitative Criminology; Springer: New York, NY, USA, 2010; pp. 5–24. [Google Scholar] [CrossRef]
- Boggs, S.L. Urban crime patterns. Am. Sociol. Rev.
**1965**, 899–908. [Google Scholar] [CrossRef] - Solymosi, R.; Ashby, M.; Cohen, T.; Sidebottom, A. Alternative denominators in transport crime rates. SocArXiv
**2017**. [Google Scholar] [CrossRef] - Rengert, G.F. Burglary in Philadelphia: A critique of an opportunity structure model. In Environmental Criminology; Brantingham, P.J., Brantingham, P.L., Eds.; Sage: Beverly Hills, CA, USA, 1981; pp. 189–201. [Google Scholar]
- Stipak, B. Alternatives to population-based crime rates. Int. J. Comp. Appl. Crim. Justice
**1988**, 12, 247–260. [Google Scholar] [CrossRef] - Pettiway, L.E. Measures of opportunity and the calculation of the arson rate: The connection between operationalization and association. J. Quant. Criminol.
**1985**, 1, 241–268. [Google Scholar] [CrossRef] - Kounadi, O.; Ristea, A.; Leitner, M.; Langford, C. Population at risk: Using areal interpolation and Twitter messages to create population models for burglaries and robberies. Cartogr. Geogr. Inf. Sci.
**2018**, 45, 205–220. [Google Scholar] [CrossRef] - Malleson, N.; Andresen, M.A. The impact of using social media data in crime rate calculations: Shifting hot spots and changing spatial patterns. Cartogr. Geogr. Inf. Sci.
**2015**, 42, 112–121. [Google Scholar] [CrossRef] - Andresen, M.A.; Jenion, G.W.; Reid, A.A. An evaluation of ambient population estimates for use in crime analysis. Crime Mapp. J. Res. Pract.
**2012**, 4, 7–30. [Google Scholar] - Chainey, S.; Desyllas, J. Modelling pedestrian movement to measure on-street crime risk. In Movement-Aware Applications for Sustainable Mobility: Technologies and Approaches; Wachowicz, M., Ed.; IGI Global: Hershey, PA, USA, 2010; pp. 243–263. [Google Scholar] [CrossRef]
- Andresen, M.A. Crime measures and the spatial analysis of criminal activity. Br. J. Criminol.
**2006**, 46, 258–285. [Google Scholar] [CrossRef] - Eck, J.E.; Weisburd, D.L. Crime places in crime theory. In Crime and Place: Crime Prevention Studies; Hebrew University of Jerusalem Legal Research Paper: Jerusalem, Israel, 2015; Volume 4, pp. 1–33. Available online: https://ssrn.com/abstract=2629856 (accessed on 15 May 2021).
- Weisburd, D.; Groff, E.R.; Yang, S.M. The Criminology of Place: Street Segments and Our Understanding of the Crime Problem; Oxford University Press: Oxford, UK, 2012. [Google Scholar]
- Sherman, L.W.; Gartin, P.R.; Buerger, M.E. Hot spots of predatory crime: Routine activities and the criminology of place. Criminology
**1989**, 27, 27–56. [Google Scholar] [CrossRef] - Kafadar, K. Smoothing geographical data, particularly rates of disease. Stat. Med.
**1996**, 15, 2539–2560. [Google Scholar] [CrossRef] - Anselin, L.; Kim, Y.W.; Syabri, I. Web-based analytical tools for the exploration of spatial data. J. Geogr. Syst.
**2014**, 6, 197–218. [Google Scholar] [CrossRef] - Beato Filho, C.C.; Assunção, R.M.; Silva, B.F.A.D.; Marinho, F.C.; Reis, I.A.; Almeida, M.C.D.M. Conglomerados de homicídios e o tráfico de drogas em Belo Horizonte, Minas Gerais, Brasil, de 1995 a 1999. Cadernos de Saúde Pública
**2001**, 17, 1163–1171. [Google Scholar] [CrossRef] - Santos, A.E.; Rodrigues, A.L.; Lopes, D.L. Aplicações de Estimadores Bayesianos Empíricos para Análise Espacial de Taxas de Mortalidade. In Proceedings of the Simpósio Brasilerio de Geoinformática, 7 (GeoInfo), Campos do Jordão, Brazil, 20–23 November 2005; pp. 300–309. [Google Scholar]
- Liu, H.; Zhu, X. Exploring the influence of neighborhood characteristics on burglary risks: A Bayesian random effects modeling approach. ISPRS Int. J. Geo-Inf.
**2016**, 5, 102. [Google Scholar] [CrossRef][Green Version] - Zhu, L.; Gorman, D.M.; Horel, S. Hierarchical Bayesian spatial models for alcohol availability, drug" hot spots" and violent crime. Int. J. Health Geogr.
**2006**, 5, 54. [Google Scholar] [CrossRef][Green Version] - Song, C.; He, Y.; Bo, Y.; Wang, J.; Ren, Z.; Yang, H. Risk assessment and mapping of hand, foot, and mouth disease at the county level in mainland China using spatiotemporal zero-inflated Bayesian hierarchical models. Int. J. Environ. Res. Public Health
**2018**, 15, 1476. [Google Scholar] [CrossRef][Green Version] - Lai, Y.S.; Zhou, X.N.; Pan, Z.H.; Utzinger, J.; Vounatsou, P. Risk mapping of clonorchiasis in the People’s Republic of China: A systematic review and Bayesian geostatistical analysis. PLoS Negl. Trop. Dis.
**2017**, 11. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tzala, E.; Best, N. Bayesian latent variable modelling of multivariate spatio-temporal variation in cancer mortality. Stat. Methods Med Res.
**2008**, 17, 97–118. [Google Scholar] [CrossRef] - Bailey, T.C.; Cordeiro, R.; Lourenço, R.W. Semiparametric modeling of the spatial distribution of occupational accident risk in the casual labor market, Piracicaba, Southeast Brazil. Risk Anal. Int. J.
**2007**, 27, 421–431. [Google Scholar] [CrossRef] [PubMed] - Kelsall, J.E.; Diggle, P.J. Spatial variation in risk of disease: A nonparametric binary regression approach. J. R. Stat. Soc. Ser. C
**1998**, 47, 559–573. [Google Scholar] [CrossRef] - Brunsdon, C.; Fotheringham, A.S.; Charlton, M.E. Geographically weighted regression: A method for exploring spatial nonstationarity. Geogr. Anal.
**1996**, 28, 281–298. [Google Scholar] [CrossRef] - Fotheringham, A.S.; Charlton, M.E.; Brunsdon, C. Geographically weighted regression: A natural evolution of the expansion method for spatial data analysis. Environ. Plan. A
**1998**, 30, 1905–1927. [Google Scholar] [CrossRef] - Bitter, C.; Mulligan, G.F.; Dall’erba, S. Incorporating spatial variation in housing attribute prices: A comparison of geographically weighted regression and the spatial expansion method. J. Geogr. Syst.
**2007**, 9, 7–27. [Google Scholar] [CrossRef][Green Version] - Chang, L.F.; Lin, C.H.; Su, M.D. Application of geographic weighted regression to establish flood-damage functions reflecting spatial variation. Water SA
**2008**, 34, 209–216. [Google Scholar] [CrossRef][Green Version] - Cardozo, O.D.; García-Palomares, J.C.; Gutiérrez, J. Application of geographically weighted regression to the direct forecasting of transit ridership at station-level. Appl. Geogr.
**2012**, 34, 548–558. [Google Scholar] [CrossRef] - Huang, B.; Wu, B.; Barry, M. Geographically and temporally weighted regression for modeling spatio-temporal variation in house prices. Int. J. Geogr. Inf. Sci.
**2010**, 24, 383–401. [Google Scholar] [CrossRef] - Fotheringham, A.S.; Yang, W.; Kang, W. Multiscale geographically weighted regression (MGWR). Ann. Am. Assoc. Geogr.
**2017**, 107, 1247–1265. [Google Scholar] [CrossRef] - Wu, C.; Ren, F.; Hu, W.; Du, Q. Multiscale geographically and temporally weighted regression: Exploring the spatiotemporal determinants of housing prices. Int. J. Geogr. Inf. Sci.
**2019**, 33, 489–511. [Google Scholar] [CrossRef] - Tobler, W.R. A computer movie simulating urban growth in the Detroit region. Econ. Geogr.
**1970**, 46, 234–240. [Google Scholar] [CrossRef] - Comber, A.; Brunsdon, C.; Charlton, M.; Dong, G.; Harris, R.; Lu, B.; Lü, Y.; Murakami, D.; Nakaya, T.; Wang, Y.; et al. The GWR route map: A guide to the informed application of Geographically Weighted Regression. arXiv
**2020**, arXiv:2004.06070. [Google Scholar] - Farber, S.; Páez, A. A systematic investigation of cross-validation in GWR model estimation: Empirical analysis and Monte Carlo simulations. J. Geogr. Syst.
**2007**, 9, 371–396. [Google Scholar] [CrossRef] - Chiles, J.P.; Delfiner, P. Geostatistics: Modeling Spatial Uncertainty; John Wiley & Sons: Hoboken, NJ, USA, 2009; Volume 487. [Google Scholar] [CrossRef]
- Oliver, M.A.; Webster, R. Basic Steps in Geostatistics: The Variogram and Kriging; Springer International Publishing: New York, NY, USA, 2015. [Google Scholar] [CrossRef]
- Schabenberger, O.; Gotway, C.A. Statistical Methods for Spatial Data Analysis; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar] [CrossRef]
- Ramos, R.G.; Silva, B.F.; Clarke, K.C.; Prates, M. Too Fine to be Good? Issues of Granularity, Uniformity and Error in Spatial Crime Analysis. J. Quant. Criminol.
**2020**, 1–25. [Google Scholar] [CrossRef]

**Figure 2.**Map of residential burglaries in Belo Horizonte, Brazil, from 2008 to 2014. Grid used consists of uniform square cells of 278.5 m per side.

**Figure 5.**An example case showing maps for simulated reference population, (true) victimization risk and crime counts, as well as the estimated victimization risks using each of the three methods considered: GWRisk, naïve estimation, and the Empirical Bayes Estimator method. Parameters for this case are listed in Table A1 (Appendix B), 15th entry.

**Figure 6.**Estimated victimization risks of burglary for single-family houses and residential apartments, calculated using each of the three methods tested. See Figure A2 in Appendix C for naïve and Empirical Bayes maps with very high values removed.

**Table 1.**Summary of simulation study with one population, showing the mean values of the fitness scores for each method, as well as their standard deviations and coefficient of variations.

Fit for Estimated R | Mean | Std. Dev. | Coef. Var. |
---|---|---|---|

Fit (naïve) | 0.16 | 0.08 | 52% |

Fit (GWR) | 0.61 | 0.08 | 14% |

Fit (Bayes) | 0.42 | 0.16 | 39% |

**Table 2.**Summary of the simulation study with two populations, showing the mean values of the fitness scores for each method, as well as their standard deviations and coefficient of variations.

Mean | Std. Dev. | Coef. Var | |
---|---|---|---|

Fit for estimated R1 | |||

Fit (naïve) | 0.01 | 0.02 | 147% |

Fit (GWR) | 0.67 | 0.07 | 11% |

Fit (Bayes) | 0.02 | 0.03 | 148% |

Fit for estimated R2 | |||

Fit (naïve) | 0.01 | 0.00 | 47% |

Fit (GWR) | 0.28 | 0.08 | 29% |

Fit (Bayes) | 0.01 | 0.01 | 48% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ramos, R.G.
Improving Victimization Risk Estimation: A Geographically Weighted Regression Approach. *ISPRS Int. J. Geo-Inf.* **2021**, *10*, 364.
https://doi.org/10.3390/ijgi10060364

**AMA Style**

Ramos RG.
Improving Victimization Risk Estimation: A Geographically Weighted Regression Approach. *ISPRS International Journal of Geo-Information*. 2021; 10(6):364.
https://doi.org/10.3390/ijgi10060364

**Chicago/Turabian Style**

Ramos, Rafael G.
2021. "Improving Victimization Risk Estimation: A Geographically Weighted Regression Approach" *ISPRS International Journal of Geo-Information* 10, no. 6: 364.
https://doi.org/10.3390/ijgi10060364