A Comparison of Simulated Annealing, Elliptic and Genetic Algorithms for Finding Irregularly Shaped Spatial Clusters

Methods for the detection and evaluation of the statistical significance of spatial clusters are important geographic tools in epidemiology, disease surveillance and crime analysis. Their fundamental role in the elucidation of the etiology of diseases (Lawson, 1999; Heffernan et al., 2004; Andrade et al., 2004), the availability of reliable alarms for the detection of intentional and non-intentional infectious diseases outbreaks (Duczmal and Buckeridge, 2005, 2006a; Kulldorff et al., 2005, 2006) and the analysis of spatial patterns of criminal activities (Ceccato, 2005) are current topics of intense research. The spatial scan statistic (Kulldorff, 1997) and the program SatScan (Kulldorff, 1999) are now widely used by health services to detect disease clusters with circular geometric shape. Contrasting to the naive statistic of the relative count of cases, the scan statistic is less prone to the random variations of cases in small populations. Although the circular scan approach sweeps completely the configuration space of circularly shaped clusters, in many situations we would like to recognize spatial clusters in a much more general geometric setting. Kulldorff et al. (2006) extended the SatScan approach to detect elliptic shaped clusters. It is important to note that for both circular and elliptic scans there is a need to impose size limits for the clusters; this requisite is even more demanding for the other irregularly shaped cluster detectors. Other methods, also using the scan statistic, were proposed recently to detect connected clusters of irregular shape (Duczmal et al., 2004, 2006b, 2007, Iyengar, 2004, Tango & Takahashi, 2005, Assuncao et al., 2006, Neill et al., 2005). Patil & Tallie (2004) used the relative incidence cases count for the objective function. Conley et al. (2005) proposed a genetic algorithm to explore a configuration space of multiple agglomerations of ellipses; Sahajpal et al. (2004) also used a genetic algorithm to find clusters shaped as intersections of circles of different sizes and centers. Two kinds of maps could be employed. The point data set approach assigns one point in the map for each case and for each non-case individual. This approach is interested in finding, among all the allowed geometric shape candidates defined within a specific strategy, the one that encloses the highest ratio of cases vs. non-cases, thus defining the most likely cluster. The second approach assumes that a map is divided into M regions, with total population N and C total cases. Defining the zone z as any set of connected regions, the