Demonstration on sample data
First, let’s load a few libraries we’ll be using in this vignette.
Data input
The uci package comes with a sample data for demonstration and test purposes. The data is a small sample of the spatial distribution of the population, jobs and schools around the city center of Belo Horizonte, Brazil. This sample data set is a good illustration of the type of data input required by uci.
Here is how the sample data looks like:
data_dir <- system.file("extdata", package = "uci")
grid <- readRDS(file.path(data_dir, "grid_bho.rds"))
head(grid)
#> Simple feature collection with 6 features and 4 fields
#> Geometry type: POLYGON
#> Dimension: XY
#> Bounding box: xmin: -43.96438 ymin: -19.97414 xmax: -43.93284 ymax: -19.96717
#> Geodetic CRS: WGS 84
#> id population jobs schools geometry
#> 1 89a881a5a2bffff 439 180 0 POLYGON ((-43.9431 -19.9741...
#> 2 89a881a5a2fffff 266 134 0 POLYGON ((-43.94612 -19.972...
#> 3 89a881a5a67ffff 1069 143 0 POLYGON ((-43.94001 -19.972...
#> 4 89a881a5a6bffff 245 61 0 POLYGON ((-43.9339 -19.9728...
#> 5 89a881a5a6fffff 298 11 0 POLYGON ((-43.93691 -19.971...
#> 6 89a881a5b03ffff 555 1071 0 POLYGON ((-43.96136 -19.970...The data is an object of class "sf" "data.frame" with
spatial polygons covering our study area and a few columns indicating
the number of activities (e.g. jobs, schools, population) in each
polygon. Our particular sample data is based on a spatial hexagonal grid
(H3 index). While there are advantages of using regular spatial grids to
calculate spatial statistics, uci also works with
non-regular geometries, such as census tracts, enumeration areas or
municipalities.
We can visualize the spatial distribution of jobs using
ggplot2:
Calculating UCI
In the example below, we calculate how mono/polycentric our study area is considering its spatial distribution of jobs considering Euclidian distances between polygons.
df <- uci(
sf_object = grid,
var_name = 'jobs',
dist_type = 'euclidean'
)
df
#> UCI location_coef spatial_separation spatial_separation_max
#> 1 0.2612105 0.5278007 3880.114 7681.927By default, the uci() function uses Euclidean distances
between polygons. However, Euclidean distances can lead to misleading
results in the case of areas with a concave shape (like a bay). In these
cases, it is strongly recommended to use
dist_type = 'spatial_link'. These spatial link distances
consider Euclidean distances along the links of spatial neighbour links.
This is a bit more costly computationally speaking, but it does return
more realistic results.
Mind you, though, that when using "spatial_link"
distances, there should be no disconnected island of polygon (or group
of polygons). In some cases, users might need to manually edit their
spatial data to include a polygon representing a road or bridge
connecting disconnected parts of the area.
df <- uci(
sf_object = grid,
var_name = 'jobs',
dist_type = 'spatial_link'
)
df
#> UCI location_coef spatial_separation spatial_separation_max
#> 1 0.2609465 0.5278007 4281.819 8468.847Where there are no bridges or roads connecting isolated polygons, we
can remove them by passing the arugment
remove_islands = TRUE. Let’s see this in another example
data set using the city of Rio de Janeiro, Brazil.
The grid_rio.rds data set covers the municipality of
Rio, with a concave outline around the Guanabara and Sepetiba bays and
several islands, including single isolated cells. The data comes from
the Access to
Opportunities Project (AOP, Ipea, 2019), aggregated to H3 resolution
8.
ggplot(data = grid_rio) +
geom_sf(aes(fill = jobs), color = NA) +
scale_fill_viridis_c() +
theme_void()
As you can see, the case of Rio include a few islands disconnected
from mainland. Because of these islands, we can only consider the full
grid in the calculation of UCI if we use
dist_type = 'euclidean'.
df <- uci(
sf_object = grid_rio,
var_name = 'jobs',
dist_type = 'euclidean'
)
#> Warning: The study area has a concave shape or a large gap inside it (e.g. a bay or a
#> lake). Euclidean distances between centroids can cross areas outside it and
#> bias UCI.
#> ℹ Consider `dist_type = "spatial_link"`.
df
#> UCI location_coef spatial_separation spatial_separation_max
#> 1 0.4201327 0.693971 17632.32 44684.47The warning is expected: Rio’s coastline is concave, so Euclidean distances between centroids can cross the bays.
Alternatively, you use dist_type = 'spatial_link' and
pass remove_islands = TRUE. In this case,
uci() drops the islands calculates UCI only for the largest
group of neighbouring polygons. For large areas, users can speed up the
process of calculating ‘spatial_link’ distances by passing
parallel = FALSE.
df <- uci(
sf_object = grid_rio,
var_name = 'jobs',
dist_type = 'spatial_link',
remove_islands = TRUE,
parallel = FALSE
)
#> Removed 44 polygons (255 jobs) outside the largest contiguous area of
#> `sf_object`.
df
#> UCI location_coef spatial_separation spatial_separation_max
#> 1 0.4045099 0.6894746 19331.93 46773.76To see which polygons are dropped before calculating UCI, use the
function remove_islands(). It returns the sf
object with only the largest group of neighbouring polygons, which you
can map and then pass to uci().
Formal definition of UCI
The Urban Centrality Index (UCI) is given by the product of the Location Coefficient (LC) and the Proximity Index (PI). With this formulation, UCI calculates urban centrality simultaneously accounting for both the concentration of activities and the proximity between activities.
\[
\begin{aligned}
UCI = LC * PI
\end{aligned}
\] The Location Coefficient (LC) was introduced
by Florence (1948) to measure the unequal
distribution factor of jobs within an urban area. It works as a
nonspatial inequality measure (similarly to Gini, for example). It
reflects, for example, the extent to which the number of activities are
concentrated in few polygons. The range of the LC is 0 to
(1 - 1/n). If LC equals zero, then economic activity is
evenly distributed, while values close to (1 - 1/n)
indicate that employment is concentrated in a few areas.
\[ \begin{aligned} LC = \frac{1}{2} \sum_{1}^{n}∣S_{i} − \frac{1}{n}|\\ S_{i} = \frac{E_{i}}{E} \end{aligned} \]
where Si is the share of employment in area i (Ei)
relative to the total employment (E) of the city; E is the total number
of jobs in a city; and n = number of areas.
Meanwhile, the Proximity Index (PI) is a transformation that Pereira et al. (2013) proposed for the Venables index, originally developed by Midelfart-Knarvik et al. (2002) to examine changes in the spatial distribution of economic activity. The Venables index (aka spatial separation index) is defined as:
\[
\begin{aligned}
V = S' * D * S
\end{aligned}
\] where S is a column vector of Si;
and D is a distance matrix whose entry dij is
the distance between the centroids of areas i and j.
When all employment activity is concentrated in just one spatial
unit, the minimum value of V is reached; that is, zero (no
matter where this spatial unit is located). However, the index has no
maximum value and therefore cannot be compared across different
spatial settings. To overcome this limitation, it is necessary to
calculate the maximum attainable value of V.
The Proximity Index (PI) solves the normalization
issue with V and changes its interpretation to suit our
needs. The interpretation of PI is the opposite of V, with
its theoretical range being (0, 1). Values of PI closer to
1 mean that employment is clustered in one single center.
If PI is 0, employment is as spatially separated as
possible. In other words, activities are distributed in a way that
maximizes the distances between them.
\[ \begin{aligned} P = 1 - \frac{V}{V_{max}} \end{aligned} \]
The normalization by Vmax is what makes the comparison
of urban areas of different shapes and sizes possible. However, the
estimation of Vmax is not trivial, because it has no
closed-form solution. In a very simple square grid, Vmax is
obtained when each corner has one-fourth of the total employment. In a
region forming a perfect circle, the maximum value of V occurs when all
employment is evenly distributed along the external edge.
Finding Vmax
By default (max_method = "exact"), uci()
searches for the distribution of activities among the polygons on the
border of the study area that maximises V. With Euclidean
distances between centroids this problem is concave, so the search (a
simple multiplicative update of the weights, stopped when the optimality
conditions hold) reaches the global maximum. The result is then refined
with the self distance of each polygon included, so that
Vmax is a stationary point of the same quantity as
V, and checked against the polygons outside the border.
With "spatial_link" distances the same procedure gives a
stationary point.
df_exact <- uci(
sf_object = grid,
var_name = 'jobs',
showProgress = FALSE
)
head(df_exact)
#> UCI location_coef spatial_separation spatial_separation_max
#> 1 0.2612105 0.5278007 3880.114 7681.927With dist_type = "spatial_link", users can set
parallel = TRUE to compute shortest-path distances using
several threads.
Two other approaches are kept so that results of previous versions
can be reproduced. In the original paper, Pereira
et al. (2013) proposed a simple heuristic approach to calculate
Vmax (the “opposite of maximum proximity”) as a homogeneous
distribution of values along the edge of the study area
(max_method = "heuristic"). The bootstrap approach
(max_method = "bootstrap") simulates 20000 random
distributions of activities along the border: it simulates that all jobs
are concentrated in up to 2, 3, 4, 5 … 51 polygons along the border, and
for each number of selected polygons it shuffles 400 random positions of
those polygons along the border, keeping the largest V
found. Both are lower bounds of Vmax, so they overestimate
V / Vmax and underestimate UCI, by an amount that depends
on the shape of the study area.