Figure 5.6 Simple illustration of the modifiable areal unit problem

figures

code

I’ve remade this one in colour, because I think it’s a bit easier to see this way.

Some libraries:

Code

library(sf)
library(dplyr)
library(tmap)
library(MASS)

We need to make a grid of values, with a clear gradient in one direction. The code for making geometries in R is ugly, so I’ve hidden it.

Code

make_square <- function(x = 0, y = 0, d = 1) {
  xs <- x + c(-1, 1, 1, -1) * d / 2
  ys <- y + c(-1, -1, 1, 1) * d / 2
  xs <- c(xs, xs[1])
  ys <- c(ys, ys[1])
  st_polygon(list(matrix(c(xs, ys), ncol = 2)))
}

grid <- expand.grid(1:10, 1:10)

polys <- list()
for (r in 1:nrow(grid)) {
  polys <- c(polys, make_square(x = grid[r, 1], y = grid[r, 2]))
}

poly_sf <- polys %>% 
  lapply(list) %>%
  lapply(st_polygon) %>%
  st_sfc() %>%
  st_sf(x = grid[, 1], y = grid[, 2])
n <- dim(poly_sf)[1]
poly_sf$val <- poly_sf$x + runif(n, -0.5, 0.5)

Anyway… we have a 10 by 10 grid of squares with values that increase from left to right:

Code

brks <- .5 + (0:100) / 10

tm_shape(poly_sf) +
  tm_polygons(col = "val", palette = "Spectral", breaks = brks, lwd = 0.5) +
  tm_layout(frame = FALSE, legend.show = FALSE)

Now we aggregate into columns and rows, taking the mean value of our variable in each case.

Code

column_sf <- poly_sf %>%
  group_by(x) %>%
  summarise(val = mean(val)) 

rows_sf <- poly_sf %>%
  group_by(y) %>%
  summarise(val = mean(val))

And now we can make maps of the results. The column-wise aggregation emphasizes the gradient, while the row-wise aggregation erases it completely, since every row has a similar set of values ranging from low to high, and when these are combined each row ends up pretty much the same.

Code

m1 <- tm_shape(column_sf) +
  tm_polygons(col = "val", palette = "Spectral", breaks = brks, lwd = 0.5) +
  tm_layout(frame = FALSE, legend.show = FALSE)

m2 <- tm_shape(rows_sf) +
  tm_polygons(col = "val", palette = "Spectral", breaks = brks, lwd = 0.5) +
  tm_layout(frame = FALSE, legend.show = FALSE)

tmap_arrange(list(m1, m2), nrow = 1)

Code

# License (MIT)
#
# Copyright (c) 2023 David O'Sullivan
#
# Permission is hereby granted, free of charge, to any person
# obtaining a copy of this software and associated documentation
# files (the "Software"), to deal in the Software without restriction,
# including without limitation the rights to use, copy, modify, merge,
# publish, distribute, sublicense, and/or sell copies of the Software,
# and to  permit persons to whom the Software is furnished to do so,
# subject to the following conditions:
#
# The above copyright notice and this permission notice shall be included
# in all copies or substantial portions of the Software.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
# LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
# FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
# DEALINGS IN THE SOFTWARE.