Skip to main content
SpringerLink
Log in

On The Application of Fuzzy Clustering for Crime Hot Spot Detection

  • Published:
Journal of Quantitative Criminology Aims and scope Submit manuscript

 

One of the fundamental challenges in crime mapping and analysis is pattern recognition. Efforts and methods to detect crime hot-spots, or geographic areas of elevated criminal activity, are wide ranging. For aggregate data, such as total crime events in a census tract(s), measures of spatial autocorrelation have proven useful. For disaggregate data (i.e. individual crime events), kernel density smoothing and non-hierarchical cluster analysis (e.g. k-means), are widely used. Non-hierarchical techniques are particularly effective in delineating geographic space into areas of higher or lower crime concentrations, because each observation is assigned to one and only one cluster. The resulting set of partitions provides clear-cut spatial boundaries that can be used for hot-spot evaluation and interpretation. However, the strength of non-hierarchical methods can also be viewed as a weakness. Although the hard-clustering of observations into a set of discrete clusters is helpful, there are many cases where ambiguity exists in the data. In such cases, a more generalized approach for hot-spot detection would be helpful. The purpose of this paper is to explore the use of a generalized partitioning method known as fuzzy clustering for hot-spot detection. Functional and visual comparisons of fuzzy clustering and two hard-clustering approaches (medoid and k-means), across a range of cluster values are analyzed. The empirical results suggest that a fuzzy clustering approach is better equipped to handle intermediate cases and spatial outliers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.

Similar content being viewed by others

Notes

  1. Additional discussions of crime hot-spots can be found in Ratcliffe and McCullagh (2001), Craglia et al. (2000, 2001), and Ackerman and Murray (2004).

  2. Nearest neighbor measures are typically based on Euclidean distances.

  3. k-means is one of the clustering methods used in CrimeStat 2 for hot-spot detection.

  4. These properties will be explored in the next section.

  5. There are a few additional quirks to this model. Each pair of observations i,j is encountered twice because j,i also occurs. As a result, the sum must be divided by two (Kaufman and Rousseeuw, 1990).

  6. The FCP can be solved using iterative approach that stops when the objective function converges (Kaufman and Rousseeuw, 1990).

  7. NCSS (http://www.ncss.com) limits the sample size for both the MCP and FCP to 1000 observations.

  8. k=8 is clearly not the optimal solution for the fuzzy cluster analysis. However, this choice will help readers make better and more informed visual comparisons between the FCP, k-means and MCP results.

  9. Similar to the MCP approach, silhouette values can be used to identify the strength/quality of the derived clusters.

References

  • Ackerman W. V., Murray A. T. (2004). Assessing spatial patterns of crime in Lima, Ohio. Cities 21(5):423–437

    Article  Google Scholar 

  • Aldenderfer M., Blashfield R. (1984). Cluster Analysis. Sage Publications, Beverly Hills

    Google Scholar 

  • Arabie P., Hubert L. (1996). An Overview of Combinatorial Data Analysis: Clustering and Classification. World Scientific Publishers, Singapore

    Google Scholar 

  • Bailey T. C., Gatrell A. C. (1995). Interactive Spatial Data Analysis. Longman, Harlow

    Google Scholar 

  • Belbin L. (1987). The use of non-hierarchical allocation methods for clustering large sets of data. Aust. Comput. J. 19:32–41

    Google Scholar 

  • Bezdek J. C. (1981). Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York

    Google Scholar 

  • Boreiko D. (2003). EMU and accession countries: Fuzzy cluster analysis of membership. Int. J. Finance Econ. 8(4): 309–325

    Article  Google Scholar 

  • Bowers K. J., Johnson S. D., Pease K. (2004). Prospective hot-spotting: The future of crime mapping?. Br. J. Criminol. 44:641–658

    Article  Google Scholar 

  • Brantingham P. J., Brantingham P. L. (1981) Environmental criminology. Sage Publications, Beverly Hills

    Google Scholar 

  • Brimicombe, A. J. (2003). A variable resolution approach to cluster discovery in spatial data mining. In Kumar, V. et al. (eds.), ICCSA 2003

  • Chainey S., and Cameron J. (2000). Understanding Hot Spots. Presentation prepared for the 2000 Crime Mapping Research Center Conference. San Diego, CA

  • Chen G. H., Wu X. S., Wang D. Q., Qin J., Wu S. L., Zhou Q. L., Xie E., Cheng R., Xu Q., Liu B., Zhang X. Y., Olowofeso O. (2004). Cluster analysis of 12 Chinese native chicken population using microsatellite markers. Asian-australas. J. Anim. Sci. 17(8):1047–1052

    Google Scholar 

  • Cohen L. E., Felson M. (1979). Social change and crime rate trends: A routine activity approach. Am. Sociol. Rev. 44:588–607

    Article  Google Scholar 

  • Craglia M., Haining R., Wiles P. (2000). A comparative evaluation of approaches to urban crime pattern analysis. Urban Stud. 37(4):711–729

    Article  Google Scholar 

  • Craglia M., Haining R., Signoretta P. (2001). Modelling high-intensity crime areas in English cities. Urban Stud. 38(11):1921–1941

    Article  Google Scholar 

  • Estivill-Castro V., and Murray A. T. (2000). Hybrid optimization for clustering in data mining. CLAIO 2000 on CD-ROM, IMSIO, Mexico

  • Fisher W. (1958). On grouping for maximum homogeneity. J. Am. Stat. Assoc. 53:789–798

    Article  Google Scholar 

  • Gatrell A.C., Rowlingson B.S. (1994). Spatial point process modeling in a geographical information system environment. In: Fotheringham S., Rogerson P. (eds). Spatial Analysis and GIS. Taylor and Francis, London

    Google Scholar 

  • Gatrell A. C., Baley T. C., Diggle P. J., Rowlingson B. S. (1996). Spatial point pattern analysis and its application in geographical epidemiology. Trans. Inst. Br. Geogr. 21:256–274

    Article  Google Scholar 

  • Gordon A. D. (1996). How Many Clusters? An investigation of five procedures for detecting nested cluster structure. In Forer, P., Yeh A., and He, J. (eds.), Proceedings of 9th International Symposium on Spatial Data Handling. Beijing International Geographical Union

  • Gordon A.D. (1999). Classification. Chapman and Hall, New York

    Google Scholar 

  • Gorr W., Harries R. (2003). Introduction to crime forecasting. Int. J. Forecast. 19:551–555

    Article  Google Scholar 

  • Graham R. (1972). An efficient algorithm for determining the convex hull of a finite point set. Info. Proc. Letters 1:132–133

    Article  Google Scholar 

  • Greenburg S., Rohe W. (1984). Neighborhood design and crime. J. Am. Plann. Assoc. 50:48–61

    Article  Google Scholar 

  • Grubesic T. H., and Murray A. T. (2001). Detecting Hot Spots Using Cluster Analysis and GIS. Fifth Annual International Crime Mapping Research Conference. Dallas, TX

  • Grubesic T. H., Murray A. T. (2002). Imperfect Spatial Information: Implications for Crime Mapping and Analysis. Sixth Annual International Crime Mapping Research Conference. Denver, CO

    Google Scholar 

  • Hakimi S. L. (1964). Optimum locations of switching centers and the absolute centers and medians of a graph. Oper. Res. 12:450–459

    Article  Google Scholar 

  • Hansen P., Jaumard B. (1997). Cluster analysis and mathematical programming. Math. Program. 79:191–215

    Google Scholar 

  • Harries K. (1999). Mapping Crime: Principle and Practice. National Institute of Justice (NCJ 178919), Washington, DC

    Google Scholar 

  • Hintze J. (2001). NCSS and PASS. Number Cruncher Statistical Systems. Kaysville, Utah

  • Hoppner F., Klawonn F., Kruse R., Runkler T. (1999). Fuzzy Cluster Analysis: Methods for Classification, Data Analysis and Image Recognition. John Wiley, West Sussex

    Google Scholar 

  • Jarvis R. A. (1973). On the identification of the convex hull of a finite set of points in the plane. Info. Proc. Letters. 2:18–21

    Article  Google Scholar 

  • Jefferis E. S., and Mamalian C. A. (1998). Crime Mapping Research Center’s Hot Spot Project. The Second Annual Crime Mapping Research Conference, December 1998. Arlington, VA

  • Kaufman L., Rousseeuw P. (1990). Finding Groups in Data: An Introduction to Cluster Analysis. John Wiley, New York

    Google Scholar 

  • Lawson A. B. (2001). Statistical Methods in Spatial Epidemiology. John Wiley and Sons, Chichester

    Google Scholar 

  • Levine N. (1999). CrimeStat: A Spatial Statistics Program for the Analysis of Crime Incident Locations, version 1.0. Ned Levine and Associates/National Institute of Justice, Washington DC

  • Levine N. (2001). CrimeStat: A Spatial Statistics Program for the Analysis of Crime Incident Locations, version 2.0. Ned Levine and Associates/National Institute of Justice, Washington DC

  • Liu Z. J., and George R. (2003). Fuzzy cluster analysis of spatio-temporal data. Computer and Information Sciences – ISCIS 2003 – Lecture Notes in Computer Science. 2869: 984–991

  • MacQueen J. (1967). Some methods for classification and analysis of multivariate observations. In Le Cam L. and Neyman, J. (eds.), Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. I. University of California Press, Berkeley

  • Milligan G. W., Mahajan V. (1980). A note on procedures for testing the quality of a clustering of a set of objects. Decision Sci. 11:669–677

    Article  Google Scholar 

  • Milligan G. W., Cooper M. C. (1985). An examination of procedures for determining the number of clusters in a dataset. Psychometrika 50(2):159–179

    Article  Google Scholar 

  • Murray A. T. (1999). Spatial analysis using clustering methods: Evaluating central point and median approaches. J. Geogr. Syst. 1:367–383

    Article  Google Scholar 

  • Murray A. T. (2000). Spatial characteristics and comparisons of interaction and median clustering models. Geogr. Anal. 32:1–19

    Google Scholar 

  • Murray A. T., V. Estivill-Castro. (1998). Cluster Discovery Techniques for Exploratory Spatial Data Analysis. International Int. J. Geogr. Inf. Sci. 12:431–443

    Article  Google Scholar 

  • Murray A. T., Grubesic T. H. (2002). Identifying non-hierarchical spatial clusters. Int. J. Ind. Eng. Theory Appl. Practice 9(1): 86–95

    Google Scholar 

  • Openshaw, S. (1984). The modifiable areal unit problem. Concepts and Techniques in Modern Geography, Vol. 38, Norwick, Geo Books

  • Preparata F. R., Shamos M. I. (1985). Computational Geometry: An Introduction. Springer-Verlag, New York

    Google Scholar 

  • Ratcliffe J. H., McCullagh M. J. (1999). Hotbeds of crime and the search for spatial accuracy. Geogr. Syst. 1(4):385–398

    Article  Google Scholar 

  • Ratcliffe J.H., McCullagh M.J. (2001). Chasing ghosts: Police perception of high-crime areas. Br. J. Criminol. 41:330–341

    Article  Google Scholar 

  • Rousseeuw P. (1987). Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math. 20:53–65

    Article  Google Scholar 

  • Rousseeuw P., Leroy A. (1987). Robust Regression and Outlier Detection. John Wiley, New York

    Google Scholar 

  • Unwin D. J. (1996). GIS, Spatial Analysis and Spatial Statistics. Progr. Hum. Geogr. 20(4):540–551

    Article  Google Scholar 

  • Vinod H. (1969). Integer programming and the theory of grouping. J. Am. Stat. Assoc. 64:506–517

    Article  Google Scholar 

  • Zhang J.Q. (2004). Risk assessment of drought disaster in the maize-growing region of Songliao Plain, China. Agric. Ecosyst. Environ. 102(2):133–153

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Geography, University of Cincinnati, Cincinnati, OH, 45221-0131, USA

    Tony H. Grubesic

Authors
  1. Tony H. Grubesic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tony H. Grubesic.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grubesic, T.H. On The Application of Fuzzy Clustering for Crime Hot Spot Detection. J Quant Criminol 22, 77–105 (2006). https://doi.org/10.1007/s10940-005-9003-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10940-005-9003-6

Keywords

Search

Navigation