43 🚧 Looking Beyond Second-Order Structure
Second-order summaries are insufficient to distinguish LGCP and CSCP in practice.
We now investigate what additional structure may allow these models to be separated.
The central question of this section is:
Given that second-order structure alone is not sufficient to distinguish between LGCP and CSCP models, what additional features of the point pattern can be used to identify the underlying generating process?
43.1 Plan
Second-order structure captures only pairwise interactions between points.
To distinguish between models, we must therefore consider features that encode additional information. In particular, we explore:
- the marginal distribution of the intensity field \(\Lambda(u)\);
- higher-order interactions (third- and fourth-order structure);
- likelihood-based and conditional approaches.
43.2 Marginal distribution of \(\Lambda(u)\)
A natural first avenue is to examine the marginal distribution of the random intensity field \(\Lambda(u)\).
At any fixed location \(u \in W\), the LGCP has
\[ \Lambda(u) = \exp(Z(u)), \qquad Z(u) \sim \mathcal{N}(\mu, \sigma^2), \]
so that
\[ \Lambda(u) \sim \mathrm{logNormal}(\mu, \sigma^2). \]
In contrast, under the CSCP formulation
\[ \Lambda(u) = \mu + Z(u)^2, \qquad Z(u) \sim \mathcal{N}(0, \sigma^2), \]
we have
\[ \Lambda(u) \sim \mu + \sigma^2 \chi^2_1, \]
that is, a shifted and scaled chi-square distribution.
These distributions are fundamentally different. The LGCP produces a log-normal intensity field with heavy right tails, while the CSCP produces a more concentrated but still skewed distribution.
Crucially, these differences are not reflected in the pair correlation function, and therefore cannot be detected by second-order methods.
However, they may be detectable through summaries that reflect local variation in intensity, such as quadrat counts or local aggregation statistics.
43.3 Higher-order structure
Another possibility is to consider higher-order interaction structure.
While the pair correlation function captures only second-order (pairwise) dependence, higher-order summaries involve interactions between three or more points, and therefore encode additional structural information about the process.
Since the LGCP is driven by a Gaussian random field, whereas the CSCP is based on a squared Gaussian field, their higher-order moment structures differ fundamentally. These differences are not visible at the level of second-order summaries, but may be revealed through third- or higher-order statistics.
In principle, this provides a route to distinguishing between the models.
I feel like I have been told in general the PCF estimation is somewhat unreliable.
Does this mean we expect third/fourth-order estimation to be even more unreliable?
43.4 Likelihood-based inference
A further approach is to consider likelihood-based methods.
Unlike minimum contrast estimation, which relies only on second-order summaries, likelihood-based inference incorporates the full distributional structure of the model. In principle, this provides access to information beyond the PCF, and may allow the two models to be distinguished.
However, for Cox processes the likelihood is typically intractable, and must be approximated using techniques such as composite likelihood or numerical methods.
43.5 In the proceeding chapters
To begin, we look at the marginal distribution.