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.
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Notes
Nearest neighbor measures are typically based on Euclidean distances.
k-means is one of the clustering methods used in CrimeStat 2 for hot-spot detection.
These properties will be explored in the next section.
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).
The FCP can be solved using iterative approach that stops when the objective function converges (Kaufman and Rousseeuw, 1990).
NCSS (http://www.ncss.com) limits the sample size for both the MCP and FCP to 1000 observations.
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.
Similar to the MCP approach, silhouette values can be used to identify the strength/quality of the derived clusters.
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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
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DOI: https://doi.org/10.1007/s10940-005-9003-6