Analyzing spatial point patterns subject to measurement error

Abstract

We address the issue of inference for a noisy point pattern. The unobserved true point process is modelled as a nonhomogeneous Poisson process. For modeling the underlying intensity surface we use a scaled Gaussian mixture distribution. The noise that creeps in during the measurement procedure causes random displacement of the true locations. We consider two settings. With a bounded region of interest, (i) this displacement may cause a true location within the boundary to be associated with an ‘observed’ location outside of the region and thus missed and (ii) we have the possibility in (i) but also vice versa; the displacement may bring in an observed location whose true location lies outside the region. Under (i), we can only lose points and, depending on the variability in the measurement error as well as the number of true locations close to boundary, this can cause a significant number of locations to be lost from our recorded set of data. Estimation of the intensity surface from the observed data can be misleading especially near the boundary of our domain of interest. Under (ii), the modeling problem is more difficult; points can be both lost and gained and it is challenging to characterize how we may gain points with no data on the underlying intensity outside the domain of interest. In both cases, we work within a hierarchical Bayes framework, modeling the latent point pattern using a Cox process and, given the process realization, introducing a suitable measurement error model. Hence, the specification includes the true number of points as an unknown. We discuss choice of measurement error model as well as identifiability problems which arise. Models are fitted using an markov chain Monte Carlo implementation. After validating our method against several synthetic datasets we illustrate its application for two ecological datasets.