Cody Arlie ReedResearch + data
All projectsLocating sharp racial boundaries

Method and interactive dashboard

Locating sharp racial boundaries

The questionWhere do sharp boundaries between racialized groups occur within a city?

  • Metric construction
  • Spatial gradients
  • Visualization

ApproachI combine the strength and relative direction of two groups’ population-share gradients with their local co-presence.

Overview

What I developed

The Sharp Boundary Index locates boundaries between racialized groups, such as Black and White residents. It identifies where one group’s share of the local population rises sharply as another’s falls.

Explore the Chicago dashboard

How the measure works · Synthetic population fields

Group A’s shareSynthetic spatial surface: Group A’s share. Darker burgundy indicates higher values.0–1 of the total population
Group B’s shareSynthetic spatial surface: Group B’s share. Darker burgundy indicates higher values.0–1 of the total population
Boundary scoreSynthetic spatial surface: Boundary score. Darker burgundy indicates higher values.0–1 relative to this example’s maximum
Two overlapping population fields and a constant background population. The boundary score is highest where both groups are present and their shares change sharply in opposite directions.

Constructing the measure

Use spatial gradients to locate boundaries between two racialized groups.

Image edge detection identifies sharp changes in brightness or color. Taking inspiration from this approach, I developed the SBI to locate boundaries between racialized groups. The measure accounts for both the strength and relative direction of racial gradients.

I smooth Census population counts on a common grid using a Gaussian kernel with σ = 1 km. From these population fields, I calculate each group’s local share and its spatial gradient. The Black–White measure combines two quantities:

01Opposing gradients

The negative dot product is positive when the two group gradients point in opposing directions. It increases as those opposing changes become steeper.

02Relative presence of both groups

The co-presence weight is highest when the two groups have equal local population shares. It decreases as one group becomes less common relative to the other.

See a racial gradient in Chicago

The formula
pₐ = dₐ / d∇pₐ = (∇dₐ − pₐ ∇d) / dC = 4 dₐ dᵦ / (dₐ + dᵦ)²SBI = C × max(0, −∇pₐ · ∇pᵦ)

d is smoothed total population density; dₐ and dᵦ are the two groups’ smoothed densities. The gradient ∇p describes the direction and rate of change in a group’s share of the total population. The dot product compares the two gradients’ directions and magnitudes.

Compared with segregation

Where is the boundary, and which groups does it separate?

A segregation index summarizes how unevenly groups are distributed across a city. The SBI locates sharp transitions between racialized groups: places where one group becomes more common as another becomes less common.

Chicago dashboard

Compare racial boundaries with travel communities.

The dashboard compares racial boundaries, population shares, and travel communities across the Chicago area. Its boundary measure compares population-density gradients across six racial pairs.

Explore the Chicago dashboard