Given discrete event data, we wish to produce a probability density that can model the relative probability of events occurring in a spatial region. 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. We propose a set of Maximum Penalized Likelihood Estimation methods based on Total Variation and H1 Sobolev norm regularizers in conjunction with a priori high resolution spatial data to obtain more geographically accurate density estimates. We apply this method 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. 1

Copula density estimation by total variation penalized likelihood with linear equality constraints

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Given discrete event data, we wish to produce a probability density that can model the relative probability of events occurring in a spatial region. Common methods of density estimation, such as Kernel Density Estimation, do not incorporate geographical information. Using these methods could result in nonnegligible 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, and so forth. We propose a set of Maximum Penalized Likelihood Estimation methods based on Total Variation and H1 Sobolev norm regularizers in conjunction with a priori high resolution spatial data to obtain more geographically accurate density estimates. We apply this method 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. 1.

Abstract Our goal is to estimate a probability density based on discrete point data via segmentation techniques. Since point data may represent certain activities, such as crime, our method can be successfully used for detecting regions of high activity. In this work we design a binary segmentation version of the well-known Maximum Penalized Likelihood Estimation (MPLE) model, as well as a minimization algorithm based on thresholding dynamics originally proposed by Merriman, Bence and Osher [20]. We also present some computational examples, including one with actual residential burglary data from the San Fernando Valley.