Theory

What AoE is Not

Before explaining what AoE does, it’s important to clarify what it is not:

The Problem: Border Truncation

When analyzing spatial data within political or administrative boundaries, a fundamental assumption is often violated: that the sampling region represents the ecological extent of the processes being studied.

Consider sampling species occurrences within a country. Observations near the border are influenced by conditions outside that country. A forest that spans the border, a river that crosses it, or simply the continuous nature of climate and habitat means that truncating at the border introduces systematic bias.

This is border truncation: the artificial constraint of ecological processes to administrative boundaries.

The Area of Effect

The area of effect (AoE) is the spatial extent over which observations within a support are influenced by external conditions. It is computed by expanding the support boundary outward to create a halo region.

The key insight: halos are defined as a proportion of region area, not as arbitrary buffer distances. This enables consistent cross-region comparisons without units or scale dependencies.

Core and Halo Classification

Points within the AoE are classified into two categories:

Points outside the AoE are pruned (removed). They are too distant to be meaningfully related to the support region.

Point classification by AoE. Core points (green) are inside the original support. Halo points (orange) are in the expanded region. Points outside the AoE are pruned.
Point classification by AoE. Core points (green) are inside the original support. Halo points (orange) are in the expanded region. Points outside the AoE are pruned.

The Scale Parameter

The scale parameter controls how large the halo is relative to the core. The relationship between scale and area is:

\[\text{Total AoE area} = \text{Core area} \times (1 + s)^2\]

where \(s\) is the scale parameter.

Two values have special meaning:

Why Equal Area is the Default

The default scale produces equal core and halo areas. This is not arbitrary—it reflects a principled position about spatial influence.

The Symmetry Argument

When we say a point in the halo is “influenced by” the support region, we’re making a claim about spatial relevance. The question is: how much relevance should we grant to the outside?

Equal area says: the outside matters as much as the inside.

This is the maximally symmetric choice. Any other ratio implies that either:

Without domain-specific knowledge to justify asymmetry, equal weighting is the principled default.

The Information-Theoretic View

Consider the AoE as defining a probability distribution over space: “where might conditions relevant to this support come from?”

Equal areas means equal prior probability mass inside and outside the original boundary. This is the maximum-entropy choice—it encodes no bias toward internal or external dominance.

The Geometric Inevitability

The formula \(s = \sqrt{2} - 1\) is not a tuned parameter. It’s the unique solution to the constraint “core equals halo”:

\[ (1 + s)^2 - 1 = 1 \implies s = \sqrt{2} - 1 \]

There’s something satisfying about a default that isn’t chosen but derived. It removes a degree of freedom from the analyst and replaces it with a principled constraint.

When to Override

Use scale = 1 when:

Use custom scales when:

Method: Buffer vs Stamp

The package offers two methods for computing the AoE:

Buffer Method (Default)

The buffer method expands the boundary uniformly in all directions. The buffer distance is computed to achieve the target halo area.

Advantages:

How it works:

The buffer distance \(d\) is found by solving:

\[\pi d^2 + P \cdot d = A_{\text{halo}}\]

where \(P\) is the perimeter and \(A_{\text{halo}}\) is the target halo area.

Stamp Method (Alternative)

The stamp method scales vertices outward from the centroid, preserving shape proportions.

Advantages:

Limitation:

Only guarantees containment for star-shaped polygons (where the centroid can “see” all boundary points). For highly concave shapes like country boundaries, small gaps may occur where the original is not fully contained.

Use method = "stamp" when working with convex or nearly convex regions where shape preservation is important.

Hard vs Soft Boundaries

AoE distinguishes between two types of boundaries:

Political borders (soft): Administrative lines have no ecological meaning. The AoE freely crosses them. A country border does not stop species from dispersing or climate from varying.

Sea boundaries (hard): Physical barriers like coastlines are true boundaries. The optional mask argument enforces these constraints by intersecting the AoE with a land polygon.

Hard boundaries constrain the AoE. The dashed line shows the theoretical AoE; the gray area shows the AoE after applying a land mask.
Hard boundaries constrain the AoE. The dashed line shows the theoretical AoE; the gray area shows the AoE after applying a land mask.

Multiple Supports

Real-world analyses often involve multiple administrative regions (countries, provinces, protected areas). AoE handles these naturally:

This enables cross-border analyses and studies of nested administrative structures without repeated preprocessing.

Summary

The area of effect provides a principled correction for border truncation in spatial analysis:

The result is a reproducible, interpretable method that can be consistently applied across studies.