32 The Non-Central Model
A natural starting point is to construct the process by applying squaring a Gaussian random field. This is analogous to the log-Gaussian Cox process, where the exponential transformation is used, and will result in the marginal distribution of the intensity field being exactly a Chi-square distribution.
32.1 Definition
Motivated by this, we consider the model
\[ \Lambda(u) = Z(u)^2, \]
where \(Z(u)\) is a Gaussian random field (GRF) defined on a spatial domain \(W \subset \mathbb{R}^2\).
Notice that at each location \(u\), the intensity is given by the square of a Gaussian random variable, and hence follows a (scaled?) non-central chi-square distribution with one degree of freedom.
Conditional on \(\Lambda(\cdot)\), the point process \(X\) is an inhomogeneous Poisson process with intensity function \(\Lambda(u)\).
The Gaussian random field \(Z(u)\) is characterised by its mean function
\[ m(u) = \mathbb{E}[Z(u)] \]
and covariance function
\[ C(u,v) = \mathrm{Cov}(Z(u), Z(v)), \]
for \(u, v \in W\).
In many cases, it is convenient to express dependence using the correlation function
\[ \rho(u, v) = \frac{C(u,v)}{\sqrt{C(u,u)}\sqrt{C(v,v)}}. \]
32.2 First-order intensity
We have seen previously, that for any Cox process, the first order intensity has the form
\[ \lambda(u) = \mathbb{E}[\Lambda(u)] \]
So we have
\[ \begin{align} \lambda(u) &= \mathbb{E}[\Lambda(u)]\\ &= \mathbb{E}[Z(u)^2]\\ &= \mathrm{Var}(Z(u)) + \mathbb{E}[Z(u)]^2\\ &= C(u,u) + m(u)^2 \end{align} \]
32.3 Second-order intensity
We have also seen that for any Cox process, the second-order intensity can be found as
\[ \begin{align} \lambda^{(2)}(u, v) &= \mathbb{E}[\Lambda(u) \Lambda(v)] \end{align} \]
Applying the Gaussian 4th moment identity, we arrive at
\[ \begin{align} \lambda^{(2)}(u, v) &= \mathbb{E}[\Lambda(u) \Lambda(v)]\\ &= \mathbb{E}[Z(u)^2 Z(v)^2]\\ &= (C(u, u) + m(u)^2)(C(v, v) + m(v)^2) + 2 C(u, v)^2 + 4 m(u) m(v) C(u, v)\\ &= \lambda(u)\lambda(v) + 2 C(u, v)^2 + 4 m(u) m(v) C(u, v) \end{align} \]
Need to add proof.
Let \((X, Y)\) be bivariate normal with,
\[ \begin{align*} &\mu_X = \mathbb{E}[X],\qquad \mu_Y = \mathbb{E}[Y],\\ &\sigma_X^2 = \text{Var}(X),\quad \sigma_Y^2 = \text{Var}(Y)\\ &\sigma_{XY} = \text{Cov}(X, Y) \end{align*} \]
Then
\[ \mathbb{E}[X^2 Y^2] = (\sigma_X^2 + \mu_X^2)(\sigma_Y^2 + \mu_Y^2) + 2\sigma_{XY}^2 + 4\mu_X\mu_Y\sigma_{XY} \]
The first term corresponds to the baseline Poisson component, while the remaining terms capture clustering induced by the covariance structure of the underlying field. In particular, the quadratic term \(2 C(u,v)^2\) reflects purely second-order dependence, while the linear term \(4 m(u)m(v)C(u,v)\) arises from interaction between the mean and covariance structure.
It is immediately obvious if there is no covariance between \(Z(u)\) and \(Z(v)\) (i.e. \(C(u, v) = 0\)), that the second order structure is simply the first-order intensity (\(\lambda(u)\lambda(v)\)), consistent with CSR.
Easier way (??? doesn’t really seem that much easier), using the identity:
\[ \mathbb{E}[AB] = \mathrm{Cov}(A,B) + \mathbb{E}[A]\mathbb{E}[B] \]
with \(A=\Lambda(u)\) and \(B=\Lambda(v)\), which gives
\[ \lambda^{(2)}(u,v) = \mathbb{E}[\Lambda(u)\Lambda(v)] = \mathrm{Cov}(\Lambda(u),\Lambda(v)) + \lambda(u)\lambda(v) \]
So then we just need to compute \(\mathrm{Cov}(\Lambda(u),\Lambda(v))\). And since
\[ \Lambda(u)=\sum_{i=1}^k Z_i(u)^2, \qquad \Lambda(v)=\sum_{j=1}^k Z_j(v)^2, \]
biliniearity of covariance gives
\[ \mathrm{Cov}(\Lambda(u),\Lambda(v)) = \sum_{i=1}^k\sum_{j=1}^k \mathrm{Cov}(Z_i(u)^2,Z_j(v)^2). \]
…Continues on
32.4 PCF
The PCF follows directly from the above derivations:
\[ \begin{align} g(u, v) &= \frac{\lambda^{(2)}(u, v)}{\lambda(u)\lambda(v)}\\ &= \frac{\lambda(u)\lambda(v) + 2 C(u, v)^2 + 4 m(u) m(v) C(u, v)}{\lambda(u)\lambda(v)}\\ &= 1 + \frac{2 C(u, v)^2 + 4 m(u) m(v) C(u, v)}{\lambda(u)\lambda(v)}\\ \end{align} \]
It is immediately apparent from the form of the PCF, that the mean of the GRF \(Z(u)\) has a direct effect on the clustering behavior of the process.
In contrast to the LGCP, where clustering is driven solely by the covariance structure of the underlying field, the mean of the GRF here directly influences the clustering behaviour.
32.5 K-function
[To be added.]
32.6 Stationary and isotropic case
In many practical settings, it is natural to assume that the underlying Gaussian random field is stationary and isotropic. In this case:
- the mean is constant:
\[ m(u) = m \]
- the variance is constant:
\[ C(u,u) = \sigma^2 \]
- the covariance only depends on distance:
\[ C(u, v) = C(r) = \sigma^2 \rho(r), \quad r = \|u - v\| \]
Under these assumptions, we have first-order intensity
\[ \lambda(u) = \lambda = \sigma^2 + m^2, \]
and PCF
\[ \begin{align} g(r) &= 1 + \frac{2C(r)^2 + 4m^2C(r)}{(\sigma^2 + m^2)^2}\\ &= 1 + \frac{2\sigma^4\rho(r)^2 + 4m^2\sigma^2\rho(r)}{(\sigma^2 + m^2)^2} \end{align} \]
This expression reveals that the pair correlation function is composed of both a quadratic term in \(\rho(r)\) and a linear term in \(\rho(r)\), in contrast to the centered case considered later.
32.7 Thoughts on this construction
While the non-central construction provides a valid and flexible model, the resulting expressions for the second-order structure are somewhat difficult to interpret. In particular, the pair correlation function contains both linear and quadratic terms in the correlation function \(\rho(r)\), arising from interaction between the mean and covariance structure of the underlying field.
This complicates interpretation, as it is not immediately clear how changes in the mean function \(m(u)\) and covariance function \(C(u,v)\) separately influence the clustering behaviour of the process. Moreover, the presence of mixed terms makes it difficult to isolate the contribution of second-order dependence alone.
For these reasons, it is natural to consider a simplified version of the model in which the Gaussian random field is centered, (\(m(u) = 0\)). As we will see, this leads to a significantly cleaner and more interpretable form of the second-order structure.