Abstract
Given a discrete sample of event locations, we wish to produce a probability density that models the relative probability of events occurring in a spatial domain. Standard density estimation techniques do not incorporate priors informed by spatial data. Such methods can result in assigning significant positive probability to locations where events cannot realistically occur. In particular, when modelling residential burglaries, standard density estimation can predict residential burglaries occurring where there are no residences. Incorporating the spatial data can inform the valid region for the density. When modelling very few events, additional priors can help to correctly fill in the gaps. Learning and enforcing correlation between spatial data and event data can yield better estimates from fewer events. We propose a non-local version of maximum penalized likelihood estimation based on the H1 Sobolev seminorm regularizer that computes non-local weights from spatial data to obtain more spatially accurate density estimates. We evaluate this method in application to a residential burglary dataset from San Fernando Valley with the non-local weights informed by housing data or a satellite image.
Highlights
In real-world applications, satellite images, housing data, census data and other types of geographical data become highly relevant for modelling the probability of a certain type of event
We present non-local (NL) H1 maximum penalized likelihood estimation (MPLE), which modifies the standard H1 MPLE energy to account for spatial inhomogeneities, but unlike Smith et al [14], we do so in a non-local way, which has the benefit of leveraging recent fast algorithms and the potential to generalize to other applications
Because the method assumes a relationship between the spatial data g and the density u, we generate a synthetic density that is closely related to the housing data, shown in the bottom left of figure 2. This density is given by taking a random linear combination of the first five approximated eigenvectors of the graph Laplacian and shifting and normalizing the result to yield a probability density
Summary
In real-world applications, satellite images, housing data, census data and other types of geographical data become highly relevant for modelling the probability of a certain type of event. The H1 seminorm is a common, well-understood regularizer in image processing related to Poisson’s equation, the heat equation and the Weiner filter, producing visually smooth surfaces For this reason, it is often a default choice when little is known about the data being modelled. A theory of non-local calculus was developed first by Zhou & Schölkopf in 2004 [22] and put in a continuous setting by Gilboa & Osher in 2008 [23] Such methods were originally used for image denoising [23,24], but the general framework led to methods for inpainting, reconstruction and deblurring [25,26,27,28,29].
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Schedule a callMore From: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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