Incorporating Spatial Information into Density Estimates and Street Gang Models

The spatial features within a region influence many processes in human activity. Mountains, lakes, oceans, rivers, freeways, population densities, housing densities, and road networks are examples of geographical factors that impact spatial behaviors. Separated into two parts, the work presented here incorporates this information into both density estimation methods and models of street gang rivalries and territories.Part I discusses methods for producing a probability density estimate given a set of discrete event data. Common methods of density estimation, such as Kernel Density Estimation, do not incorporate geographical information. Using these methods could result in non-negligible portions of the support of the density in unrealistic geographic locations. For example, crime density estimation models that do not take geographic information into account may predict events in unlikely places such as oceans, mountains, etc. To obtain more geographically accurate density estimates, a set of Maximum Penalized Likelihood Estimation methods based on Total Variation norm and H1 Sobolev semi-norm regularizers in conjunction with a priori high resolution spatial data is proposed. These methods are applied to a residential burglary data set of the San Fernando Valley using geographic features obtained from satellite images of the region and housing density information.Part II addresses the behaviors and rivalries of street gangs and how the spatial characteristics of the region affect the dynamics of the system. Gangs typically claim a specific territory as their own, and they tend to have a set space, a location they use as a center for their activities within the territory. The spatial distribution of gangs influences the rivalries that develop within the area. One stochastic model and one deterministic model are proposed, providing different types of outputs. Both models incorporate important geographical features from the region that would inhibit movement, such as rivers and large highways. In the stochastic method, an agent-based model simulates the creation of street gang rivalries. The movement dynamics of agents are coupled to an evolving network of gang rivalries, which is determined by previous interactions among agents in the system. Basic gang data, geographic information, and behavioral dynamics suggested by the criminology literature are integrated into the model. The deterministic method, derived from a stochastic approach, modifies a system of partial differential equations from a model for coyotes. Territorial animals and street gangs often exhibit similar behavioral characteristics. Both groups have a home base and mark their territories to distinguish claimed regions. To analyze the two methods, the Hollenbeck policing division of the Los Angeles Police Department is used as a case study