39 Model Setup
Bit of a nothing burger section, just making sure working definition of the two processes are clear.
In the previous section, we observed that LGCPs and CSCPs can produce remarkably similar second-order structure when their parameters are suitably chosen. We now formalise this comparison and investigate the extent to which the two models can be distinguished.
To isolate the role of second-order structure, we restrict attention to stationary and isotropic models with exponential correlation function \[ \rho(r) = \exp\left(-\frac{r}{s}\right), \] where \(s > 0\) is a scale parameter.
Throughout, we consider the shifted single-component CSCP with intensity \[ \Lambda(u) = \mu + Z(u)^2, \] where \(Z(u)\) is a mean-zero Gaussian random field.
For both the CSCP and LGCP, we adopt the parameterisation \((\lambda, \phi, s)\), where:
- \(\lambda\) is the first-order intensity,
- \(\phi = g(0) - 1\) measures clustering strength,
- \(s\) is a scale parameter governing the decay of dependence.
This parameterisation is particularly convenient, as \(\lambda\) and \(\phi\) have a common interpretation across both models, allowing for a direct comparison of their second-order structure.
Under this parameterisation, the CSCP satisfies:
First-order intensity: \[ \lambda = \mu + \sigma^2, \]
Pair correlation function: \[ g(r) = 1 + \phi \exp\left(-\frac{2r}{s}\right). \]
For comparison, we also consider the LGCP with intensity \[ \Lambda(u) = \exp(Z(u)), \] where \(Z(u)\) is a Gaussian random field with mean \(\mu\) and exponential correlation function \[ \rho(r) = \exp\left(-\frac{r}{s}\right). \]
Under the parameterisation \((\lambda, \phi, s)\), the LGCP has pair correlation function \[ g(r) = \exp\left(\log(1+\phi)\, \exp\left(-\frac{r}{s}\right)\right). \]