The Stewart potentials of population is a spatial interaction modeling approach which aims to compute indicators based on stock values weighted by distance. These indicators have two main interests:

- they produce understandable maps by smoothing complex spatial patterns;
- they enrich the stock variables with contextual spatial information.

At the European scale, this functional semantic simplification may help to show a smoothed context-aware picture of the localized socio-economic activities.

In this vignette, we show a use case of these “potentials” on the regional GDP per capita at the European scale with three maps:

- a regional map of the GDP per capita;
- a regional map of the potential GDP per capita;
- a smoothed map of the GDP per capita.

Note that this example is based on data and mapping functions proposed in the **cartography package**.

```
library(cartography)
library(sp)
library(sf)
library(SpatialPosition)
data(nuts2006)
# Compute the GDP per capita variable
nuts3.df$gdpcap <- nuts3.df$gdppps2008 * 1000000 / nuts3.df$pop2008
# Discretize the variable
bv <- quantile(nuts3.df$gdpcap, seq(from = 0, to = 1, length.out = 9))
# Draw the map
opar <- par(mar = c(0,0,1.2,0))
# Set a color palette
pal <- carto.pal(pal1 = "wine.pal", n1 = 8)
# Draw the basemap
plot(nuts0.spdf, add = F, border = NA, bg = "#cdd2d4")
plot(world.spdf, col = "#f5f5f3ff", border = "#a9b3b4ff", add = TRUE)
# Map the regional GDP per capita
choroLayer(spdf = nuts3.spdf, df = nuts3.df,
var = "gdpcap",
legend.pos = "topright",
breaks = bv, col = pal,
border = NA,
legend.title.txt = "GDP per capita",
legend.values.rnd = -2,
add = TRUE)
plot(nuts0.spdf, add = TRUE, lwd = 0.5, border = "grey30")
plot(world.spdf, col = NA, border = "#7DA9B8", add = TRUE)
# Set a layout
layoutLayer(title = "Wealth Inequality in Europe",
sources = "Basemap: UMS RIATE, 2015 - Data: Eurostat, 2008",
author = "T. Giraud, 2015")
```

We compute the potentials of GDP for each spatial unit. The computed value takes into account the spatial distribution of the stock variable and return a sum weighted by distance, according a specific spatial interaction and fully customizable function.

```
# Create a distance matrix between units
mat <- CreateDistMatrix(knownpts = nuts3.spdf,
unknownpts = nuts3.spdf)
# Merge the data frame and the SpatialPolygonsDataFrame
nuts3.spdf@data <- nuts3.df[match(nuts3.spdf$id, nuts3.df$id),]
# Compute the potentials of population per units
# function = exponential, beta = 2, span = 75 km
poppot <- stewart(knownpts = nuts3.spdf,
unknownpts = nuts3.spdf,
matdist = mat,
varname = "pop2008",
typefct = "exponential",
beta = 2,
span = 75000,
returnclass = "sf")
# Compute the potentials of GDP per units
# function = exponential, beta = 2, span = 75 km
gdppot <- stewart(knownpts = nuts3.spdf,
unknownpts = nuts3.spdf,
matdist = mat,
varname = "gdppps2008",
typefct = "exponential",
beta = 2,
span = 75000,
returnclass = "sf")
# Create a data frame of potential GDP per capita
pot <- data.frame(id = nuts3.df$id,
gdpcap = gdppot$OUTPUT * 1000000 / poppot$OUTPUT,
stringsAsFactors = FALSE)
# Discretize the variable
bv2 <- c(min(pot$gdpcap), bv[2:8], max(pot$gdpcap))
# Draw the map
par <- par(mar = c(0,0,1.2,0))
# Draw the basemap
plot(nuts0.spdf, add = F, border = NA, bg = "#cdd2d4")
plot(world.spdf, col = "#f5f5f3ff", border = "#a9b3b4ff", add = TRUE)
# Map the regional potential of GDP per capita
choroLayer(spdf = nuts3.spdf, df = pot,
var = "gdpcap",
legend.pos = "topright",
breaks = bv2, col = pal,
border = NA,
legend.title.txt = "Potential\nGDP per capita",
legend.values.rnd = -2, add = TRUE)
plot(nuts0.spdf, add=T, lwd = 0.5, border = "grey30")
plot(world.spdf, col = NA, border = "#7DA9B8", add=T)
# Set a text to explicit the function parameters
text(x = 6271272, y = 3743765,
labels = "Distance function:\n- type = exponential\n- beta = 2\n- span = 75 km",
cex = 0.8, adj = 0, font = 3)
# Set a layout
layoutLayer(title = "Wealth Inequality in Europe",
sources = "Basemap: UMS RIATE, 2015 - Data: Eurostat, 2008",
author = "T. Giraud, 2015")
```

This map gives a smoothed picture of the spatial patterns of wealth in Europe while keeping the original spatial units as interpretive framework. Hence, the map reader can still rely on a known territorial division to develop its analyses.

In this case, the potential GDP per capita is computed on a regular grid.

```
# Compute the potentials of population on a regular grid (50km span)
# function = exponential, beta = 2, span = 75 km
poppot <- stewart(knownpts = nuts3.spdf,
varname = "pop2008",
typefct = "exponential",
span = 75000,
beta = 2,
resolution = 50000,
mask = nuts0.spdf,
returnclass = "sf")
# Compute the potentials of GDP on a regular grid (50km span)
# function = exponential, beta = 2, span = 75 km
gdppot <- stewart(knownpts = nuts3.spdf,
varname = "gdppps2008",
typefct = "exponential",
span = 75000,
beta = 2,
resolution = 50000,
mask = nuts0.spdf,
returnclass = "sf")
# Create the ratio variable
poppot$OUTPUT2 <- gdppot$OUTPUT * 1e6 / poppot$OUTPUT
# Create an isopleth layer
pot <- isopoly(x = poppot, var = "OUTPUT2",
breaks = bv,
mask = nuts0.spdf,
returnclass = "sf")
# Get breaks values
bv3 <- sort(c(unique(pot$min), max(pot$max)), decreasing = FALSE)
# Draw the map
par <- par(mar = c(0,0,1.2,0))
# Draw the basemap
plot(nuts0.spdf, add = F, border = NA, bg = "#cdd2d4")
plot(world.spdf, col = "#f5f5f3ff", border = "#a9b3b4ff", add = TRUE)
# Map the potential GDP per Capita
choroLayer(x = pot, var = "center",
legend.pos = "topright",
breaks = bv3, col = pal, add=T,
border = NA, lwd = 0.2,
legend.title.txt = "Potential\nGDP per capita",
legend.values.rnd = -2)
plot(nuts0.spdf, add=T, lwd = 0.5, border = "grey30")
plot(world.spdf, col = NA, border = "#7DA9B8", add=T)
# Set a text to explicit the function parameters
text(x = 6271272, y = 3743765,
labels = "Distance function:\n- type = exponential\n- beta = 2\n- span = 75 km",
cex = 0.8, adj = 0, font = 3)
# Set a layout
layoutLayer(title = "Wealth Inequality in Europe",
sources = "Basemap: UMS RIATE, 2015 - Data: Eurostat, 2008",
author = "T. Giraud, 2015")
```

Unlike the previous maps, this one doesn’t keep the initial territorial division to give a smoothed picture of the spatial patterns of wealth in Europe. The result is easy to read and can be considered as a bypassing of the Modifiable Areal Unit Problem (MAUP).