Inhomogeneous Spatial Point Processes Research Articles

Second‐order analysis of inhomogeneous spatio‐temporal point process data

Second‐order methods provide a natural starting point for the analysis of spatial point process data. In this note we extend to the spatio‐temporal setting a method proposed by Baddeley et al. [Statistica Neerlandica (2000) Vol. 54, pp. 329–350] for inhomogeneous spatial point process data, and apply the resulting estimator to data on the spatio‐temporal distribution of human Campylobacter infections in an area of north‐west England.

On Consistent Nonparametric Intensity Estimation for Inhomogeneous Spatial Point Processes

A common nonparametric approach to estimate the intensity function of an inhomogeneous spatial point process is through kernel smoothing. When conducting the smoothing, one typically uses events only in a local set around the point of interest. But the resulting estimator often is inconsistent, because the number of events in a fixed set is of order 1 for spatial point processes. In this article we propose a new covariate-based kernel smoothing method to estimate the intensity function. Our method defines the distance between any two points as the difference between their associated covariate values. Consequently, we determine the kernel weight for a given event of the process as a function of its new distance to the point of interest. Under some suitable conditions on the covariates and the spatial point process, we prove that our new estimator is consistent for the true intensity. To handle the situation with high-dimensional covariates, we also extend sliced inverse regression, a useful dimension-reduction tool in standard regression analysis, to spatial point processes. Simulations and an application to a real data example are used to demonstrate the usefulness of the proposed method.

Second-Order Analysis of Inhomogeneous Spatial Point Processes With Proportional Intensity Functions

We consider pairs of spatial point processes with intensity functions sharing a common multiplicative term. We introduce two novel approaches to estimating the pair correlation function for one of the point processes by treating the other as a baseline so as to account for the unspecified part of the intensity functions. The first approach is based on nonparametric kernel-smoothing, whereas the second approach uses a conditional likelihood estimation approach to fit a parametric model for the pair correlation function. A great advantage of the proposed methods is that they do not require the often difficult estimation of the unspecified part of the intensity functions. We establish the consistency of the resulting estimators and discuss how the parametric estimator can be applied in model diagnostics and inference on regression parameters for the intensity functions. We apply the proposed procedures to two spatial point patterns regarding the spatial distributions of birds in the U.K.'s Peak District in 1990 and 2004.

Estimating functions for inhomogeneous spatial point processes with incomplete covariate data

The R package spatstat provides a very flexible and useful framework for analysing spatial point patterns. A fundamental feature is a procedure for fitting spatial point process models depending on covariates. However, in practice one often faces incomplete observation of the covariates and this leads to parameter estimation error which is difficult to quantify. In this paper, we introduce a Monte Carlo version of the estimating function used in spatstat for fitting inhomogeneous Poisson processes and certain inhomogeneous cluster processes. For this modified estimating function, it is feasible to obtain the asymptotic distribution of the parameter estimators in the case of incomplete covariate information. This allows a study of the loss of efficiency due to the missing covariate data.

Variance Estimation for Statistics Computed from Inhomogeneous Spatial Point Processes

Summary The paper introduces a new approach to estimate the variance of statistics that are computed from an inhomogeneous spatial point process. The approach proposed is based on the assumption that the observed point process can be thinned to be a second-order stationary point process, where the thinning probability depends only on the first-order intensity function of the (unthinned) original process. The resulting variance estimator is proved to be asymptotically consistent for the target parameter under some very mild conditions. The use of the approach proposed is demonstrated in two important applications of modelling inhomogeneous spatial point processes: residual diagnostics of a fitted model and inference on the unknown regression coefficients. A simulation study and an application to a real data example are used to demonstrate the efficacy of the approach proposed.

Second‐Order Analysis of Inhomogeneous Spatial Point Processes Using Case–Control Data

Methods for the statistical analysis of stationary spatial point process data are now well established, methods for nonstationary processes less so. One of many sources of nonstationary point process data is a case-control study in environmental epidemiology. In that context, the data consist of a realization of each of two spatial point processes representing the locations, within a specified geographical region, of individual cases of a disease and of controls drawn at random from the population at risk. In this article, we extend work by Baddeley, Møller, and Waagepetersen (2000, Statistica Neerlandica54, 329-350) concerning estimation of the second-order properties of a nonstationary spatial point process. First, we show how case-control data can be used to overcome the problems encountered when using the same data to estimate both a spatially varying intensity and second-order properties. Second, we propose a semiparametric method for adjusting the estimate of intensity so as to take account of explanatory variables attached to the cases and controls. Our primary focus is estimation, but we also propose a new test for spatial clustering that we show to be competitive with existing tests. We describe an application to an ecological study in which juvenile and surviving adult trees assume the roles of controls and cases.

Inhomogeneous spatial point processes by location-dependent scaling

A new class of models for inhomogeneous spatial point processes is introduced. These locally scaled point processes are modifications of homogeneous template point processes, having the property that regions with different intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted Markov point processes are discussed in detail. It is shown that the locally scaled versions are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process. Approximations are suggested that simplify calculation of the density, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.

Inhomogeneous spatial point processes by location-dependent scaling

A new class of models for inhomogeneous spatial point processes is introduced. These locally scaled point processes are modifications of homogeneous template point processes, having the property that regions with different intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted Markov point processes are discussed in detail. It is shown that the locally scaled versions are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process. Approximations are suggested that simplify calculation of the density, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.