40 🚧 Theoretical(ish) Comparison of PCFs
40.1 Preliminaries
In the previous sections, we established a common parameterisation for LGCPs and CSCPs, and observed that the two models can produce very similar second-order structure when their parameters are suitably chosen.
We now study this phenomenon directly by comparing the two families of pair correlation functions.
Under the parameterisation \((\lambda, \phi, s)\), the PCFs take the form \[ g_{CSCP}(r; \lambda, \phi, s_C) = 1 + \phi \exp\left(-\frac{2r}{s_C}\right), \] \[ g_{LGCP}(r; \lambda, \phi, s_L) = (1 + \phi)^{\exp(-r/s_L)}. \]
The central question is whether these two families of functions can be distinguished when only second-order structure is available.
To formalise this, we consider the following approximation problem:
Given a CSCP PCF with parameters \((\lambda, \phi, s_C)\), how well can it be approximated by an LGCP PCF with the same \((\lambda, \phi)\) by optimally choosing the scale parameter \(s_L\)?
If the discrepancy between the two families can be made arbitrarily small through rescaling, this would indicate a fundamental limitation in using second-order structure for model identification.
40.2 Non-equivalence of the PCF families
Before studying approximation, we first note that the two PCF families are not identical.
[Need to type up proof that the two families of curves are only identical when \(\phi = 0\), not really important though.]
40.3 Best LGCP approximation of CSCP PCF
Consider a CSCP with parameters \((\lambda, \phi, s_C)\), and an LGCP with parameters \((\lambda, \phi, s_L)\).
We are interested in the following question:
For fixed \((\lambda, \phi, s_C)\), how should we choose \(s_L\) so that the LGCP PCF is as close as possible to the CSCP PCF?
To make this precise, define the discrepancy
\[ D = \sup_{r \ge 0} \left|g_{LGCP}(r; \lambda, \phi, s_L) - g_{CSCP}(r; \lambda, \phi, s_C)\right|. \]
Our goal is to solve
\[ \inf_{s_L > 0} D, \]
and to understand how small this discrepancy can be as \(\phi\) varies over \([0,2]\).
40.4 Reduction to a scale-ratio
A useful reparameterisation of the problem, reveals that the discrepancy depends only on the relative scale between the two models, rather than their absolute values.
Having this as a proposition feels kind of goofy. Only have it structured like this as before I was trying to actually find a decent analytic bound, but gave up.
Will likely restructure / revisit this section at a later date.
40.4.1 Proposition 1
Let
\[ g_{CSCP}(r; \lambda, \phi, s_C) = 1 + \phi \exp\left(-\frac{2r}{s_C}\right), \]
\[ g_{LGCP}(r; \lambda, \phi, s_L) = (1 + \phi)^{\exp(-r/s_L)}. \]
Define
\[ x = \exp\left(-\frac{2r}{s_C}\right) \in [0,1], \quad p = \frac{s_C}{2s_L}, \quad c = \log(1+\phi). \]
Then the discrepancy can be written as
\[ D = \sup_{x \in [0,1]} \left| \exp(c x^p) - (1 + \phi x) \right|. \]
40.4.2 Proof of Proposition 1
Starting with the LGCP:
\[ \begin{align} g_{LGCP}(r; \lambda, \phi, s_L) &= (1 + \phi)^{\exp(-r/s_L)} \\ &= \exp\left(\log(1 + \phi)\exp(-r/s_L)\right) \\ &= \exp\left(c \exp(-r/s_L)\right). \end{align} \]
Now note that
\[ x = \exp\left(-\frac{2r}{s_C}\right) \quad \Rightarrow \quad \exp(-r/s_L) = x^{\,p}, \quad p = \frac{s_C}{2s_L}. \]
So
\[ g_{LGCP}(r) = \exp(c x^p). \]
For the CSCP:
\[ g_{CSCP}(r) = 1 + \phi \exp\left(-\frac{2r}{s_C}\right) = 1 + \phi x. \]
Substituting both into the discrepancy,
\[ D = \sup_{x \in [0,1]} \left| \exp(c x^p) - (1 + \phi x) \right|. \]
Thus, once \(\phi\) is fixed, the discrepancy depends on \(s_C\) and \(s_L\) only through the ratio \(p = \frac{s_C}{2s_L}\).
40.5 Optimisation problem
We now define the approximation error as
\[ E(\phi, p) := \sup_{x \in [0,1]} \left| \exp(c x^p) - 1 - \phi x \right|, \quad c = \log(1 + \phi). \]
For fixed \(\phi\), \(E(\phi, p)\) represents the maximum vertical distance between the two PCF curves across all distances.
So the best possible LGCP approximation (under this discrepancy measure) is obtained by solving
\[ p^*(\phi) \in \arg \min_{p > 0} E(\phi, p). \]
Once \(p^*(\phi)\) is found, the corresponding LGCP scale is
\[ s_L^*(\phi, s_C) = \frac{s_C}{2p^*(\phi)}. \]
40.6 Practical computation
At this point, the problem is completely explicit:
- For each \(\phi \in [0,2]\),
- we numerically solve the one-dimensional minimisation
\[ \min_{p > 0} \sup_{x \in [0,1]} \left| \exp(c x^p) - 1 - \phi x \right|. \]
This yields:
- the optimal scale ratio \(p^*(\phi)\),
- and the corresponding minimum discrepancy
\[ E^*(\phi) := E(\phi, p^*(\phi)). \]
40.7 Interpretation
This formulation makes the comparison particularly transparent:
- \(\lambda\) plays no role once matched,
- \(s_C\) only sets the overall distance scale,
- \(\phi\) controls the shape of clustering,
- and the LGCP can only adjust via a single scaling parameter.
Thus, the problem reduces to understanding how well a nonlinear curve of the form \(\exp(c x^p)\) can approximate the linear function \(1 + \phi x\) over \([0,1]\).
This provides a direct way to quantify how distinguishable the two model classes are using second-order structure alone.
40.8 Summary
We have reduced the comparison between LGCP and CSCP PCFs to a simple one-dimensional optimisation problem.
For each clustering strength \(\phi\), the optimal LGCP scale is obtained by solving
\[ p^*(\phi) \in \arg \min_{p > 0} \sup_{x \in [0,1]} \left| \exp(c x^p) - 1 - \phi x \right|, \]
and setting
\[ s_L^*(\phi, s_C) = \frac{s_C}{2p^*(\phi)}. \]
This provides a complete description of the best possible second-order approximation of a CSCP by an LGCP under matched \((\lambda, \phi)\).
The remaining question is how small this discrepancy can be in practice, and whether it is large enough to distinguish the two models.
We now investigate this empirically.