33 The Central Model
The expressions derived in the previous section, while general, are somewhat difficult to interpret due to the presence of mixed terms involving both the mean and covariance functions of the underlying Gaussian random field.
To obtain a more tractable and interpretable model, we now consider the special case in which the Gaussian random field is centered, i.e.
\[ m(u) = \mathbb{E}[Z(u)] = 0. \]
As we will see, this assumption leads to a substantial simplification of the second-order structure.
33.1 Definition
The centered CSCP is obtained by restricting the Gaussian random field to have zero mean. That is, we consider the Cox process with random intensity
\[ \Lambda(u) = Z(u)^2, \]
where \(Z(u)\) is a mean-zero Gaussian random field with covariance function \(C(u,v)\).
33.2 First-order intensity and PCF
Under this mean-zero assumption, the first order intensity simplifies to
\[ \lambda(u) = C(u, u) \]
and the second-order intensity becomes
\[ \lambda^{(2)}(u,v) = C(u,u)C(v,v) + 2C(u,v)^2. \]
Consequently, the PCF simplifies to
\[ g(u,v) = 1 + \frac{2C(u,v)^2}{C(u,u)C(v,v)} \]
This is a substantial simplification of the second-order structure.
In particular, we no longer see the mean being involved in the PCF. In other words, the clustering behaviour is now entirely determined by the covariance structure of the underlying field.
33.3 Stationary and Isotropic
Under the assumption of stationarity and isotropy, the model simplifies further.
the first-order intensity becomes
\[ \lambda(u) = \lambda = \sigma^2, \]
the second-order intensity becomes
\[ \lambda^{(2)}(u,v) = \sigma^4 + 2\sigma^4\rho(r)^2 \]
and the PCF becomes
\[ g(r) = 1 + 2 \rho(r)^2. \]
As with the previous construction, we observe that the pair correlation function is bounded:
\[ 1 \le g(r) \le 3, \]
since \(|\rho(r)| \le 1\).
This implies that the clustering strength of the process is inherently limited, and cannot grow arbitrarily large, in contrast to processes such as the LGCP, where \(g(0)\) can be unbounded. Thus, this construction does not overcome the boundedness of the PCF observed in the more general case.
More significantly, in the centered case, a stronger restriction holds. Evaluating at \(r = 0\), we obtain
\[ g(0) = 1 + 2\rho(0)^2 = 3, \]
since \(\rho(0) = 1\). Thus, the clustering strength at zero is fixed, regardless of the choice of covariance function.
This lack of flexibility makes the centered model highly restrictive, as it cannot represent processes with weaker or stronger clustering behaviour.
In particular, the model lacks any parameter controlling clustering strength, which severely limits its usefulness in practical modelling.
Interesting note: in the clustering-strength parameterisation \((\lambda, \phi)\) introduced later, this construction corresponds to the fixed value \(\phi = 2\).
This makes it even more clear why it is so restrictive, the model has no ability to vary clustering strength.