31  Defining the CSCP

The aim of the following chapters is to establish a clear and usable definition of the single-component Chi-Square Cox process (CSCP).

Unlike the LGCP, which has a standard and widely accepted formulation, there is no (or at least I / we cannot find) a canonical definition of the CSCP in the literature. And so before getting started on investigating the properties, we obviously need a definition.

We consider three natural candidates:

• the non-central formulation,
• the central formulation,
• the shifted (or external mean) formulation.

Each of these results from slightly different ways of constructing the underlying random intensity field and, as we will see, lead to distinct relationships between the parameters and the resulting first- and second-order structure.

The purpose of the following chapters is therefore:

• to define each formulation precisely,
• to derive its basic properties (in particular the intensity and pair correlation function),
• to examine how the parameters control clustering behaviour, and
• to compare the formulations in terms of interpretability and suitability for statistical inference.

This analysis will allow us to identify a formulation that is both mathematically natural and practically useful, which we will adopt as the canonical definition of the CSCP in the remainder of the thesis.