34  The Shifted Model

The centered construction considered previously, while analytically convenient, is highly restrictive. In particular, the pair correlation function is fixed at \(g(0) = 3\), regardless of the choice of covariance function. This implies that the model lacks any mechanism for controlling clustering strength, and so cannot represent processes with weaker or stronger clustering behaviour.

To address this limitation, we consider a simple modification of the model by introducing a constant baseline intensity. Specifically, we define

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

where \(\mu \ge 0\) is a constant, and \(Z(u)\) is a mean-zero Gaussian random field.

As we will see, this modification restores the flexibility in the clustering structure, while still maintaining a simple and interpretable form for the second-order properties.

34.1 First-order intensity

Under this model, the first-order intensity is

\[ \lambda(u) = \mathbb{E}[\Lambda(u)] = \mu + C(u,u). \]

34.2 Second-order intensity

Similarly, the second-order intensity is given by

\[ \lambda^{(2)}(u,v) = \mathbb{E}[\Lambda(u)\Lambda(v)] = \lambda(u)\lambda(v) + 2C(u,v)^2. \]

34.3 Pair correlation function

The pair correlation function becomes

\[ g(u,v) = 1 + \frac{2C(u,v)^2}{\lambda(u)\lambda(v)}. \]

34.4 Stationary and isotropic case

Under stationarity and isotropy, we have

\[ \lambda = \mu + \sigma^2, \quad C(r) = \sigma^2 \rho(r), \]

and hence

\[ g(r) = 1 + \frac{2\sigma^4 \rho(r)^2}{(\mu + \sigma^2)^2}. \]

Evaluating at \(r = 0\), we obtain

\[ g(0) = 1 + \frac{2\sigma^4}{(\mu + \sigma^2)^2}. \]

Unlike the centered case, the clustering strength is no longer fixed. Instead, it depends on the relative magnitude of \(\mu\) and \(\sigma^2\), and can vary continuously within the range

\[ 1 \le g(0) \le 3. \]

So, essentially introducing the baseline intensity \(\mu\) restores flexibility in the clustering behaviour of the model. Smaller values of \(\mu\) lead to stronger clustering, while larger values of \(\mu\) reduce clustering towards complete spatial randomness.

This can be interpreted as a dilution effect: the deterministic component \(\mu\) reduces the relative influence of the stochastic fluctuations induced by \(Z(u)^2\).

34.5 Closing remarks

The shifted model provides a natural extension of the centered construction, retaining its analytical simplicity while overcoming its key limitation. In contrast to the centered case, the clustering strength is no longer fixed, but instead governed by model parameters, making the process suitable for statistical modelling.

For this reason, we will adopt the shifted construction as the canonical form of the single-component CSCP in the remainder of this work type shii type shii.