Asymptotic methods for spatial point processes

Data in form of point patterns are often encountered in spatial statistics, e.g. locations of a certain species of plants and locations of mines. Point pattern analysis is often carried out by assuming the given pattern as a realisation of some spatial point process model, such as the Poisson process or the Strauss process, and then applying parametric methods for model fitting and testing. Some of these parametric methods are generally applicable but many are for particular classes of models only. Methods for testing goodness-of-fit therefore will be of great practical importance.

In the literature the one-sample goodness-of-fit of a given point pattern is usually accessed by deviations of some empirical function from the corresponding function of the hypothesised model and its null distribution is approximated by parametric bootstrap procedure. This approach has been shown to be quite successful for stationary processes. However, for non-stationary processes, the rich flexibility in the parametric forms of such a model means that deviations of an empirical function from the corresponding one of the fitted model may have large variation, resulting in low power bootstrap tests. For two-sample problems, the only available methods require multiple replicates from each process but in applications one often has only a single replicate from each.

This project aims at developing asymptotic methods mainly for testing (i) one-sample and two-sample goodness-of-fit hypotheses for non-stationary processes with single replicates and (ii) two-sample goodness-of-fit hypotheses for stationary processes with single replicates.

An advantage of asymptotic methods is that the limiting null distribution of a test statistic may be model-free and sometimes is just some standard distribution such as normal or chi-squared. Thus, a fully specified fitted model is then unnecessary and the p-value is approximated by asymptotics rather than estimated by simulation.

The proposal starts with mathematical derivations of the limiting distributions of a first-order statistic (which converges weakly to an integral of the difference of two Brownian motions) and a second-order statistic (which is asymptotically chi-squared) for testing whether two independent stationary samples are generated by the same model. Then, based on the mathematics for the stationary case, the proposal outlines the development of analogous statistics for non-stationary processes, in which the distinction between parametric form and nonparametric form of the one-sample goodness-of-fit hypothesis has to be made. For two non-stationary samples, in addition to the equal distribution hypothesis, the problem of testing for a common spatial trend is also briefly mentioned.