Spatial Interpolation

From precincts to districts

Last Time

1. Gerrymandering packs and cracks voters.
2. The mean-median score detects skew.
3. The efficiency gap compares wasted votes.
4. An ensemble shows what fair maps look like.

Key Ideas for Today

1. Interpolation splits by area.
2. Votes are extensive.
3. SR precincts have shapes.
4. Two approaches estimate new-map votes.

Concept Acquisition

1. Area-weighted interpolation moves values from source polygons to target polygons in proportion to overlapping area, assuming each value is spread evenly over its source.
2. Counts like votes are spatially extensive (split with area); rates and shares are intensive (averaged).
3. SV precincts are drawn by counties each election and may not match a geographic area; SR precincts are stable statewide units with both vote counts and shapefiles.
4. The project’s new-map votes can be estimated by area (Approach A) or through census blocks weighted by registered voters (Approach B).

Tool Acquisition

1. st_interpolate_aw() and its extensive argument.
2. st_intersection() to cut one layer by another.

Concept Application

1. Find the total of a target polygon from area shares (the pink-polygon warm-up).
2. Sketch the sf data frame behind the project’s 2024 map: its rows and its columns.
3. Choose Approach A or B for the project’s new-map analysis.

Your Turn

Question: What is the total value of the pink polygon on the right?

01:30

Collect answers. Annotate the right figure: shade the part of each piece inside the pink box and write its share of that piece’s area. Any answer needs those shares and the assumption that each value is spread evenly over its piece.

Agenda

1. Spatial Interpolation
2. Project: Gerrymandering
   1. Old map analysis
   2. New map analysis

Spatial Interpolation

Area-weighted Interpolation

Estimates a value for each target polygon by giving it, from each source polygon it overlaps, the share of the source’s value equal to the share of the source’s area that falls inside the target. Assumes each value is spread evenly over its source.

st_interpolate_aw(x, to, extensive)

A toy map

library(sf)
precincts_sf <- st_sf(votes = c(40, 30, 60),
  geometry = st_sfc(
    st_polygon(list(rbind(c(0,1), c(1,1), c(.5,.5), c(0,1)))),
    st_polygon(list(rbind(c(0,0), c(.5,0), c(.5,.5), c(0,1), c(0,0)))),
    st_polygon(list(rbind(c(.5,0), c(1,0), c(1,1), c(.5,.5), c(.5,0))))),
  crs = 3310)
districts_sf <- st_sf(district = c("North", "South"),
  geometry = st_sfc(
    st_polygon(list(rbind(c(0,.5), c(1,.5), c(1,1), c(0,1), c(0,.5)))),
    st_polygon(list(rbind(c(0,0), c(1,0), c(1,.5), c(0,.5), c(0,0))))),
  crs = 3310)

Live-code: build both sf objects and print each one.

A toy map

plot(st_geometry(precincts_sf), lwd = 3)
plot(st_geometry(districts_sf), border = c("steelblue", "deeppink"),
     lwd = 5, lty = 2, add = TRUE)
text(st_coordinates(st_centroid(st_geometry(precincts_sf))),
     labels = precincts_sf$votes, cex = 2)

Live-code the plot one layer at a time; check it against the sketch.

By hand

library(dplyr)
pieces <- precincts_sf |>
  mutate(precinct_area = st_area(geometry)) |>
  st_intersection(districts_sf) |>
  mutate(piece_area = st_area(geometry),
         piece_votes = votes * as.numeric(piece_area / precinct_area))

st_intersection() cuts one layer by another: each piece keeps the columns of both.

Live-code the pipeline one step at a time, printing after each. The 40 precinct touches South only at its tip, so its South piece has zero area and gets 0 votes: touching isn’t overlapping.

By hand

pieces |>
  group_by(district) |>
  summarize(votes = sum(piece_votes))

Live-code the totals: North 70, South 60, adding back to 130.

Or, in one step

interp <- st_interpolate_aw(precincts_sf["votes"], districts_sf, extensive = TRUE)
interp

Live-code it and compare with the by-hand totals; then rerun with extensive = FALSE.

Votes are counts: they’re spatially extensive (split them with the area). A rate or a share is intensive.

The result keeps the target geometries but drops district; add it back with interp$district <- districts_sf$district (rows come back in the same order).

Guess the Metaphor

Spatial interpolation is like a tea valley: when a new path cuts through the bushes (a new district line through the precincts), the harvest on each side is its share of every bush it crosses, as long as the leaves grow evenly over each bush (votes are spread evenly over each precinct).

Project: Gerrymandering

Project: Gerrymandering

Research question:

Will the adoption of the new district map lead to an increase in partisan advantage relative to 2024? If so, by how much?

Your answer will take the form of a dashboard.

Prop 50 Dashboard

Your final deliverable is a dashboard with six pages:

1. Results (opens first): both maps, and seats, mean-median score, and efficiency gap for 2024 and the re-run
2. About the Data
3. Data Cleaning
4. Exploratory Data Analysis
5. Measuring Partisan Advantage
6. Re-running the 2024 Election

Old Map Analysis

Question: Sketch the sf data frame needed to make this dashboard. Be sure to note:

1. Number of rows
2. Names and types of columns

03:00

A tale of two precinct types

SV Precincts

1. Created by counties each election
2. Two counties might use the same precinct name
3. Unique ID: SVPREC x FIPS
4. May not correspond to geographic area Example:
   • SVPREC: 30010 holds the vote by mail
   • SVPREC: 30010A holds the in-person votes

   This is why we won’t have shapefiles for every precinct

A tale of two precinct types

SR Precincts

1. Created by the statewide data base to have a stable precinct unit from election to election.
2. One SR precinct is generally an aggregation of smaller SV precincts.

We can get both votes and shapefiles for SR precincts.

How to construct the old map df

Ingredients:

1. Statement of Vote, by SR precinct
2. 2024 district shapefile

How to construct the old map df

Ingredients:

1. Statement of Vote, by SR precinct
2. 2024 district shapefile

• California has a primary system where the two candidates with the most votes advance to the general election regardless of party.
• How to deal with this effect in your analysis is up to you.

Old Map New Map

Approach A: Area-weighted Interpolation

Assume voters are spread evenly across each SR precinct, then use st_interpolate_aw() with extensive = TRUE to move each precinct’s votes into the new districts by area.

How to construct the new map df

Approach A

Ingredients:

1. Statement of Vote, by SR precinct
2. SR precinct shapefiles
3. Proposed map shapefiles

But wait… we have more data

A tale of two precinct types and a block

Census Block

1. Defined by the census bureau as the smallest geographic unit on which they gather data.
2. Often a city block but always geographical.
3. We don’t have voting data but we do have # registered voters.

How can this be used?

Approach B: Proportional allocation

How to construct the new map df

Approach A

Ingredients:

1. Statement of Vote, by SR precinct
2. SR precinct shapefiles
3. Proposed map shapefiles

Approach B

Ingredients:

1. Statement of Vote, by SR precinct
2. SR precinct to 2020 Block proportional mapping
3. Block to new district mapping

Both: the proposed map shapefile, to draw the Results map.

Key Ideas for Today

1. Interpolation splits by area.
2. Votes are extensive.
3. SR precincts have shapes.
4. Two approaches estimate new-map votes.

For Next Time

1. Sign up for Quiz 5 (11/2-11/3).
2. Finish PS 5, due 10/31.
3. Project Studio 4 is the week of 11/2.