Resumen
The upper level set (ULS) scan statistic, its theory, design and implementation, and its extension to the bivariate data are discussed. We provide the ULS-Hotspot algorithm that obtains the response rates, maintains a list of connected components at each level of the rate function, and yields the ULS-tree. The tree is grown in the immediate successor list, which provides a computationally efficient method for likelihood evaluation, visualization, and storage. An example shows how the zones are formed and the likelihood function is developed for each candidate zone. The general theory of bivariate hotspot detection is explained, including the bivariate binomial model, the multivariate exceedance approach, and the bivariate Poisson distribution. We propose the Intersection method that is simple to implement, using a univariate hotspot detection method. We study the sensitivity of the joint hotspots to the degree of association between the variables. An application for the mapping of crime hotspots in the counties of the state of Ohio is presented.