35  Parameterizations

I’m not considering the central model here, as it is too restrictive to be practical, and I want to preserve my sanity.

35.1 Why consider alternatives?

The natural parameterisation of the CSCP in terms of the Gaussian random field parameters \((\mu, \sigma^2)\) is mathematically convenient, but not very interpretable.

In particular: - \(\mu\) and \(\sigma^2\) both influence the first-order intensity, - and, in the non-central formulation, both parameters also influence the second-order structure.

As a result, it is difficult to disentangle the effects of these parameters on the intensity and clustering behaviour of the process.

To address this, it is useful to consider alternative parameterisations in terms of quantities with more direct interpretation, such as: - the mean intensity \(\lambda\), and
- a parameter controlling clustering strength (e.g. \(g(0)\), or \(\phi = g(0) - 1\)).

These parameterisations will also allow for more direct comparison with other Cox processes, such as the LGCP, which are naturally expressed in terms of \((\lambda, \phi, s)\).

35.2 Shifted Model

Recall, under this definition the random intensity is

\[ \Lambda(u) = \mu + Z(u)^2 \]

where \(Z \sim N(0, \sigma^2)\), and

\[ \mathrm{Cov}(Z(u), Z(v)) = \sigma^2\rho(u, v) = \sigma^2\rho(r) \].

The parameterizations we have investigated so far are as follows:

Parameters	Definitions	First-order	PCF	Constraints
\((\mu,\sigma^2)\)	\((\mu,\sigma^2)\)	\(\mu + \sigma^2\)	\(1 + \frac{2\sigma^4}{(\mu + \sigma^2)^2}\rho(r)^2\)	\(0 \le \mu,\ 0 \le \sigma^2\)
\((\lambda, \eta)\)	\((\mu + \sigma^2, \frac{\mu}{\sigma^2})\)	\(\lambda\)	\(1 + \frac{2}{(1 + \eta)^2}\rho(r)^2\)	\(0 \le \lambda,\ 0 \le \eta\)
\((\lambda, \delta)\)	\((\mu + \sigma^2, \frac{1}{(1 + \mu / \sigma^2)^2})\)	\(\lambda\)	\(1 + 2\delta\rho(r)^2\)	\(0 \le \lambda,\ 0 \le \delta \le 1\)
\((\lambda, g_0)\)	\((\mu + \sigma^2, 1 + \frac{2\sigma^4}{(\mu + \sigma^2)^2})\)	\(\lambda\)	\(1 + (g_0 - 1)\rho(r)^2\)	\(0 \le \lambda,\ 1 \le g_0 \le 3\)
\((\lambda, \psi)\)	\((\mu + \sigma^2, \frac{2\sigma^4}{(\mu + \sigma^2)^2})\)	\(\lambda\)	\(1 + \psi\rho(r)^2\)	\(0 \le \lambda,\ 0 \le \psi \le 2\)

35.3 Non-Central Model

Recall, under this definition the random intensity is

\[ \Lambda(u) = Z(u)^2 \]

where \(Z \sim N(\mu, \sigma^2)\), and

\[ \mathrm{Cov}(Z(u), Z(v)) = \sigma^2\rho(u, v) = \sigma^2\rho(r) \].

The parameterizations investigated so far are as follows:

Parameters	Definitions	First-order	PCF
\((\mu,\sigma^2)\)	\((\mu, \sigma^2)\)	\(\mu^2 + \sigma^2\)	\(1 + \frac{2\sigma^4\rho(r)^2 + 4\mu^2\sigma^2\rho(r)}{(\mu^2 + \sigma^2)^2}\)
\((\lambda, \eta)\)	\((\mu^2 + \sigma^2, \frac{\mu^2}{\sigma^2})\)	\(\lambda\)	\(1 + \frac{2\rho(r)^2 + 4\eta\rho(r)}{(1 + \eta)^2}\)
\((\lambda, \delta)\)	\((\mu^2 + \sigma^2,\ \frac{\sigma^4}{(\mu^2 + \sigma^2)^2})\)	\(\lambda\)	\(1 + 2\delta\rho(r)^2 + 4(\sqrt{\delta} - \delta)\rho(r)\)
\((\lambda, g_0)\)	\((\mu^2 + \sigma^2,\ 1 + \frac{2\sigma^4 + 4\mu^2\sigma^2}{(\mu^2 + \sigma^2)^2})\)	\(\lambda\)	\(1 + 2a^2\rho(r)^2 + 4a(1-a)\rho(r)\), where \(a = 1 - \sqrt{\frac{3-g_0}{2}}\)

I couldn’t fit the constraints column here, but they are just the same as above.