Kest(X)
pcf(X)
# Inhomogeneous versions
Kinhom(X)
pcfinhom(X)11 Second-Order Estimation
11.1 Motivation
The PCF \(g(r)\) and the \(K\)-function \(K(r)\) provide fundamental summaries of spatial interaction.
In practice, these quantities are unknown and must be estimated from the data.
This section introduces the main classes of estimators used in practice, along with their key ideas and trade-offs.
11.2 Observed data
We observe a point pattern:
\[ X = \{x_1, \dots, x_n\} \]
within a bounded window \(W \subset \mathbb{R}^2\).
All second-order estimators are based on pairwise distances:
\[ d_{i,j} = \|x_i - x_j \|, \quad i \neq j \]
11.3 Overview of estimator types
There are two main approaches to second-order estimation:
Cumulative estimators
- Based on counting pairs of distances up to distance \(r\)
- Example: \(K\)-function
Differential (density) estimators
- Based on estimating the interaction at distance \(r\)
- Example: PCF \(g(r)\)
Note
All second-order estimators are built from pairwise distances, but differ in how they summarise them:
- cumulative (integrated)
- or local (smoothed)
11.4 Estimating the K-function
11.4.1 Basic Estimator
The standard estimator of the \(K\)-function is:
\[ \hat{K}(r) = \frac{|W|}{n(n-1)} \sum_{i \neq j} \mathbf{1}\{d_{ij} \le r\} \, w_{ij}^{-1}, \]
where \(w_{ij}\) are edge correction weights.
This estimator:
- counts all pairs within distance \(r\),
- rescales by intensity,
- corrects for missing neighbors near the boundary
Note
Absolute slop tbh. Such a poor explanation.
11.4.2 Variants
Different estimators arise through different choices of \(w_{ij}\):
- Border correction (simple, but biased)
- Isotropic correction (commonly used)
- Translation correction (often more stable)
11.5 Estimating the PCF
The PCF is a density-type quantity, and requires smoothing.
11.5.1 Kernel estimator
A common estimator is:
\[ \hat{g}(r) = \frac{1}{2\pi r} \cdot \frac{|W|}{n(n-1)} \sum_{i \neq j} k_h(d_{ij} - r) \, w_{ij}^{-1}, \]
where:
- \(k_h\) is a kernel with bandwidth \(h\)
- \(w_{ij}\) are edge correction weights
This estimator:
- places a smooth “bump” around each pairwise distance
- aggregates contributions near \(r\)
- produces a continuous estimate of interaction
Note
Slop city again.
11.5.2 Alternative view
The PCF can be interpreted as the derivative of the \(K\)-function:
\[ g(r) = \frac{1}{2\pi r} \frac{d}{dr} K(r) \]
so estimating can be seen as:
differentiating \(\hat{K}(r)\), or
directly smoothing pair distances.
11.6 Intensity Estimation
Both \(K\) and \(g\) estimators depend on the intensity \(\lambda\).
In practice, this is replaced by:
\[ \hat{\lambda} = \frac{n}{|W|} \]
For inhomogeneous processes, more complex estimators are required.
11.7 Inhomogeneous extensions
When the intensity varies across space, standard estimators are biased.
To address this, inhomogeneous versions are used.
11.7.1 Inhomogeneous K-function
\[ \hat{K}_{\text{inhom}}(r) = \sum_{i \neq j} \frac{\mathbf{1}\{d_{ij} \le r\}}{\hat{\lambda}(x_i)\hat{\lambda}(x_j)} \, w_{ij}^{-1} \]
11.7.2 Inhomogeneous PCF
Similarly, the PCF can be adjusted using:
\[ \frac{1}{\hat{\lambda}(x_i)\hat{\lambda}(x_j)} \]
These estimators attempt to remove variation due to intensity, isolating interaction.
11.8 Edge effects
A fundamental issue in second-order estimation is boundary bias.
Points near the edge of \(W\):
- have unobserved neighbors outside the window
- contribute fewer pairs
This leads to systemic underestimation unless corrected.
Some common strategies for correcting this are:
- Weighting pairs (\(w_{ij}\))
- Ignoring edge points (border correction)
11.9 Bandwidth selection (PCF)
The PCF estimator depends critically on the bandwidth hyper-parameter \(h\):
small \(h\) results in noisy estimates
large \(h\) results in over smoothing
There is no universally optimal choice, and results can be sensitive to \(h\).
11.10 Practical implementation
In spatstat, common estimators include:
This section needs a LOT of work.