Analysis of multivariate spatial point patterns

This project focuses on three directions for multivariate spatial point processes. The first direction is to develop model-free tests. There are several questions that one may have to answer before fitting any model to the data. Consider a forest with different tree species. Are the patterns of locations of individual species independent and do they have the same distribution? For the latter, some tests have been developed recently, under the assumption that the patterns are independent, which is natural when comparing patterns on, say, a cancer and a healthy tissue. However, for patterns of two different cells on the same tissue, the independence assumption may be violated. The proposal suggests a solution by using some newly proposed higher-order cross summary characteristics; the idea is to estimate the null distributions of test statistics constructed from these new characteristics by generating bootstrap samples obtained by simulating an auxiliary independent point process or an auxiliary random variable. Asymptotic approaches, based on nonparametric estimation of the variances of the test statistics, will also be developed.

The second direction is to study random graphs that are constructed from multivariate point processes. Random graphs, either topological or constructed from univariate point processes so that vertex locations and edge physical lengths are relevant, have been well studied. The project suggests to construct random graphs for multivariate point processes so that points are connected if they meet some geometrical conditions and some conditions on their types. For example, the so-called six degrees of separation, stating that any two persons in the world are connected by a path of at most six persons knowing each other, as an example of the small-word phenomenon, will be revisited to study how long the path is if the persons have to be of different nationalities. Technically, this project attempts to establish the degree distribution of the typical vertex, characterise the spread by the asymptotic length of the shortest path between a randomly chosen pair of connected vertices, and find the limit of the mean clustering coefficient and the percolation threshold for the existence of a giant component in such a graph.

The third direction is to study multivariate marked point processes, e.g. a forest with different tree species marked by their diameters. Existing summary characteristics for marked point processes do not take the inter-type mark correlation into consideration. This project proposes several new summary characteristics and studies the properties of their estimators