Point Patterns II

HES 505 Fall 2025: Session 16

Matt Williamson

Point Processes and Why You Should Care

  • They’re ubiquitous

  • Well-specified to many social and ecological theories

  • Helps build intuition for interpolation

The Poisson Distribution

  • Discrete probability distribution

  • Probability of events in a fixed interval of time or space

  • Mean (expectation) = variance = \(\lambda\)

Re-visiting CSR

  • Constant intensity

  • Points are independent

  • Density is a crude estimate of intensity

Revisting CSR

First-Order Effects

  • Intensity varies spatially

  • Points are independent

First-Order Effects

Second-Order Effects

  • Intensity of “parents” is first-order

  • Intensity of “children” is second-order

Second-Order Effects

Key Points

  • Poisson distribution: mean = variance = \(\lambda\) -> First Order

  • Homogeneous refers to the rate, not it’s outcome

  • Statistical tests ask “Could this outcome be generated by a constant rate?”

Second-Order Analysis

Distance based metrics

  • Provide an estimate of the second order effects

  • Mean nearest-neighbor distance: \[\hat{d}_{min} = \frac{\sum_{i = 1}^{m} d_{min}(\mathbf{s_i})}{n}\]

Mean Nearest Neighbor Distance

Ripley’s \(K\) Function

  • Nearest neighbor methods throw away a lot of information

  • The K function is an alternative, based on a series of circles with increasing radius

\[ \begin{equation} K(d) = \lambda^{-1}E(N_d) \end{equation} \]

  • We can test for clustering by comparing to the expectation:

\[ \begin{equation} K_{CSR}(d) = \pi d^2 \end{equation} \]

  • if \(k(d) > K_{CSR}(d)\) then there is clustering at the scale defined by \(d\)

Ripley’s \(K\) Function

  • When working with a sample the distribution of \(K\) is unknown

  • Estimate with

\[ \begin{equation} \hat{K}(d) = \hat{\lambda}^{-1}\sum_{i=1}^n\sum_{j=1}^n\frac{I(d_{ij} <d)}{n(n-1)} \end{equation} \]

where:

\[ \begin{equation} \hat{\lambda} = \frac{n}{|A|} \end{equation} \]

\(K\) for HPP

\(K\) for IPP

\(K\) for Clustered Processes

Adjustments for \(K\)

  • \(K\) is sensitive to edge effects (window size) (correction argument to Kest)

  • \(K\) assumes HPP (Kinhom function instead of Kest)

\(K\) vs \(Kinhom\) for IPP

\(K\) vs \(Kinhom\) for Clustered PP

Ripley’s \(K\) Function

  • accounting for variation in \(d\)

Take-Aways

  • Density and Intensity for first-order effects

  • Distance for second-order effects

  • Linkages for kriging and interpolation