This part proposes some graphical representations helping to have an idea regarding inequalities existing in a local (or spatial) context. In this case, as regard to the average of contiguous territorial units (contiguity order 1).
7.1 Spatial deviation and theoretical redistribution
The code below takes in entry the numerator (INC) and the denominator (TH) and returns the spatial deviation indicators (relative and absolute). These indicators are afterwards associated with the input SpatialDataFrame (com.spdf) for displaying on map this deviation.
par(mar=c(2,4,0,0))
# Spatial relative deviation calculation
com$ldevrel <- sdev(spdf = com.spdf,
x = com,
spdfid = "DEPCOM",
xid = "DEPCOM",
var1 = "INC",
var2 = "TH",
order = 1,
type = "rel")
# Spatial absolute deviation calculation
com$ldevabs <- sdev(spdf = com.spdf,
x = com,
spdfid = "DEPCOM",
xid = "DEPCOM",
var1 = "INC",
var2 = "TH",
order = 1,
type = "abs")
# Spatial deviation in million Euros
com$ldevabsmil <- com$ldevabs / 1000000
# Cartography
# Plot layout
par(mfrow = c(1,1), mar = c(0,0,1.2,0))
layoutLayer(title = "Spatial deviation - Tax income per households, 2013",
sources = "Data source : DGFiP, 2016",
author = "Author : RIATE, 2016",
scale = 5,
frame = TRUE,
col = "black",
coltitle = "white",
bg = "#FFFFFF",
south = FALSE,
extent = com.spdf)
# Plot territories
plot(com.spdf, col = "grey70", border="#EDEDED",lwd=0.25,add=T)
plot(ept.spdf,border="#1A1A19",lwd=1,add=T)
# Territorial deviation (relative and absolute) cartography
propSymbolsChoroLayer(spdf = com.spdf, df = com,
var = "ldevabsmil", var2 = "ldevrel",
add = TRUE,
inches = 0.3,
col = carto.pal(pal1 = "blue.pal", n1 = 3,
pal2 = "wine.pal", n2 = 3),
breaks = c(min(com$ldevrel,na.rm=TRUE),
75,90,100,111,133,
max(com$ldevrel,na.rm=TRUE)),
border = "#f0f0f0",
lwd = 0.25,
legend.var.pos = "left", legend.var2.pos = "topleft",
legend.var.title.txt = "Redistribution (Million euros)",
legend.var2.title.txt = "Deviation to the spatial context (100 = average of the contiguous territorial units - order 1)",
legend.var.style = "e")
This map highlights local discontinuities existing in the Metropole du Grand Paris. Important local statistical gaps appear in several areas: Saint-Mande and Neuilly-sur-Seine are characterised by the highest score as regards to their respective neighbours (indexes 173 and 156). The central arrondissements of Paris (6e, 7e, 8e) appear also in favourable situation in a local context. Some local “bastions” are revealed in the periphery of Paris, such as Le Raincy (index 153),Sceaux (150),Vaucresson (143), Marnes-la-Coquette (142) or Saint-Maur-des-Fosses (140). Lagging communes are mainly located in the near periphery of Paris: the lower index (61) is observed at Clichy-sous-Bois. Clichy, Puteaux, Saint-Ouen, Bagneux and Aubervilliers are also in a lagging situation: their average income per household stand around 30-40 % below their respective neighbourhood.
Looking into local redistributions, the most part of the redistribution in absolute terms concerns the arrondissements of Paris and their contiguous communes in the north-west of Paris: the 16th arrondissement of Paris should contribute to 2 billion euros; the 15th arrondissement should receive 1,5 billion euros to ensure a local equilibrium.
But as regard to the available income mass (table below), it is Saint-Mande (260 million euros, 42 % of the total income of the commune), Neuilly-sur-Seine (1 billion euros, 36 %) and the 6th arrondissement (650 million, 35 %) which should contribute to their respective poorest neighbours. On the other side of the redistribution, Clichy-sous-bois (138 million euros, 65 %), Clichy (521 million, 62 %), Puteaux (535 million, 58 %) and Saint-Ouen (311 million, 56 %) should receive the most. Interesting is to note that the local redistribution implies a lowest share of their respective income for the wealthiest communes (around 35 %) comparing to the poorest one (which should receive more than 50 % of their current available income).
# Spatial deviation - Top 10 of the potential contributors as regards to their total amount of income
com$ldevabsPerc<- com$ldevabs / com$INC * 100
com<-com[order(com$ldevabsPerc, decreasing = TRUE), ]
com[1:10,c("ldevabsmil","ldevabsPerc")]
## ldevabsmil ldevabsPerc
## Saint-Mande 260.10243 42.35190
## Neuilly-sur-Seine 1090.39208 36.07081
## Paris 6e Arrondissement 650.36361 35.24736
## Le Raincy 112.64658 34.64702
## Paris 7e Arrondissement 1052.81249 34.55177
## Sceaux 187.23855 33.59400
## Gournay-sur-Marne 45.78709 32.77097
## Coubron 25.56596 31.16313
## Vaucresson 95.59513 29.92272
## Marnes-la-Coquette 21.91940 29.89520
# Spatial deviation - Top 10 of the potential receivers as regards to their total amount of income
com<-com[order(com$ldevabsPerc, decreasing = FALSE), ]
com[1:10,c("ldevabsmil","ldevabsPerc")]
## ldevabsmil ldevabsPerc
## Clichy-sous-Bois -138.01472 -65.19469
## Clichy -521.27310 -62.11944
## Puteaux -535.70339 -58.26476
## Saint-Ouen -311.73019 -56.39945
## Bagneux -249.46913 -49.51238
## Aubervilliers -303.14284 -47.43249
## Bonneuil-sur-Marne -84.77851 -46.04200
## Nanterre -556.15780 -45.04563
## Champigny-sur-Marne -456.19566 -44.02501
## Gennevilliers -182.72267 -40.50353
7.2 Spatial autocorrelation plot and outliers analysis
Spatial autocorrelation reveals the absence of independance between geographical observations. Thus, spatialised indicators are often subject to spatial dependances (or spatial interactions), which are even stronger than spatial localisations are closer. Autocorrelation measures (Moran, Geary, Lisa indexes) allows estimating the spatial dependance between the values of a same indicator at several locations of a given study area.
The figure displayed below consists in evaluating this spatial autocorrelation by a plot crossing the spatial deviation (Y axis) and global deviation (X axis) values. It includes also a regression line, the equation and the R-Squared of this statistical model. On top of that, colours are affected to depict the territorial level belonging, following the logic of the MTA framework.
For each territorial unit of the study area, this plot crosses the values of the global deviation on absissa axis with the values of the local deviation on ordinates axis. This chart is interesting as it reveals spatial dependancy, e.g. spatial organisation of a phenomena. It provides some interesting inputs for answering to basic questions, such as: * Are territorial units globally advantaged also advantaged at local level (and reciprocally) ? * Is it possible to detect special situations as regard to the global trend, meaning the existence of “local bastions” in favourable or lagging situations ?
par(cex.lab=1)
par(cex.axis=0.75)
par(mar=c(4,4,2,2))
# Dot colours by EPT
colours <- c("#fb6a4a","#74c476","#74a9cf",
"#045a8d","#006d2c","#c7e9c0",
"#bcbddc","#fcbba1","#3690c0",
"#d9f0a3","#9ecae1","#de2d26")
# Assign correctly colours
com<-com[order(com$DEPCOM, decreasing = FALSE), ]
com$col <- as.factor(com$LIBEPT)
com$LIBEPT <- as.factor(com$LIBEPT)
levels(com$col) <- colours
# Spatial autocorrelation
lm <- summary.lm(lm(ldevrel ~ gdevrel, com))
# Plot spatial autocorrelation
plot(com$gdevrel,com$ldevrel,
ylab = "Local deviation",
ylim = c(50,260),
xlab = "Global deviation",
xlim = c(50,260),
pch = 20,
col = as.vector(com$col))
abline((lm(com$ldevrel ~ com$gdevrel)), col = "red", lwd =1)
# Specify linear regression formula and R-Squared of the spatial autocorrelation on the plot
text(140,65, pos = 4, cex = 0.8,
labels = (paste("Local Deviation =", round(lm$coefficients["gdevrel","Estimate"], digits = 3),
"* (Global Deviation) +", round(lm$coefficients["(Intercept)","Estimate"],
digits = 3))))
text(140,60, pos = 4, cex = 0.8,
labels = (paste("R-Squared =", round(summary(lm(ldevrel~gdevrel, com ))$r.squared, digits = 2))))
abline (h = seq(40,290,10), col = "gray70", lwd = 0.25, lty = 3)
abline (h = seq(50,250,50), col = "gray0", lwd = 1, lty = 1)
abline (v = seq(40,290,10), col = "gray70", lwd = 0.25, lty = 3)
abline (v = seq(50,250,50), col = "gray0", lwd = 1, lty = 1)
# Legend for territorial level
legend("topleft",
legend = levels(com$LIBEPT),
pch = 20,
col = colours,
cex = 0.5,
pt.cex = 1,
title = "Territorial context")
The output of the linear model of spatial autocorrelation reveals that the hypothesis of independance is rejected at a probability below than 0.0001. It means that, “everything is related to everything else, but near things are related than distant things” (Tobler, 1970). However, the R-squared of the relation (0.42) suggests that this statistical relation includes outliers very far from the linear regression, which may be interesting to analyse. On the plot, it is possible to visualise then communes, which mainly belong to the Seine-Saint-Denis departement (Grand-Paris-Est/Est-everything is related to everything else, but near things are related than distant things)
More empirically, the chart can also be used to examine the situation of outliers and exceptional units out of the cloud of points. The code below takes in entry the statistical residuals of the spatial autocorrelation calculated in the previous section. Then the residuals are standardised. The code returns a plot displaying the standardised residuals according to the value of the local deviation. Exceptional values (calculated using the T-Student value) are labeled on the resulting plot.
par(cex.lab=1)
par(cex.axis=0.75)
par(mar=c(4,4,2,2))
# Standardised residual calculation
lm <- lm(ldevrel ~ gdevrel, com)
res.standard <- rstandard(lm)
#risk alpha = 0.1
alpha <- 0.1
# Calculation of the threshold using T-Student at (n-p-1) degrees of freedom
seuil.standard <- qt(1-alpha/2, nrow(com) - 1)
# Plot residuals
plot(com$ldevrel,res.standard,
xlab = "Local deviation", cex.lab = 0.8,
xlim = c(60,180),
ylab = "Standardised residuals of spatial autocorrelation", cex.lab = 0.8,
pch = 20,
col = as.vector(com$col),
cex.axis =0.8)
# Adding thresholds
abline(h=-seuil.standard, col = "red")
abline(h=+seuil.standard, col = "red")
abline(h=0, col = "red")
# Detecting exceptional values and labeling them on the plot
ab.standard <- com[res.standard < -seuil.standard | res.standard > +seuil.standard,]
for (i in 1:nrow(ab.standard)){
# Take the territorial units listing below and above the threshold
communes <- row.names(ab.standard)[i]
# Plot the residual names
text(com[communes,"ldevrel"],res.standard[communes],communes,cex =0.5, pos=4)
}
abline (v = seq(50,200,10), col = "gray70", lwd = 0.25, lty = 3)
abline (v = seq(50,200,50), col = "gray0", lwd = 1, lty = 1)
# Plot the legend (territorial zoning)
legend("topleft",
legend = levels(ab.standard$LIBEPT),
pch = 20,
col = colours,
cex = 0.5,
pt.cex = 1,
title = "Territorial context")
This plot highlight exceptional situations, all things being equal to their situation in a global context. It depicts firstly the communes which are characterised by a significant lagging situation locally as regards to their average income per household (Puteaux, Clichy, Clichy-sous-Bois, Saint-Ouen, Issy-les-Moulineaux, Suresnes, Paris 1st and 15th arrondissements). Secondly, this plot reveals also communes in a significantly wealthier than their neighbourhoods (Saint-Mande, Le Raincy, Coubron, Gournay-sur-Marne, Bry-sur-Marne, Saint-Maur-des-Fosses, Rungis, Sceaux). This plot is especially useful to highlight poles of wealth and poles of poverty in a local context.
7.3 Which value for the dist and order parameters ?
In the example above, a contiguity criteria (order = 1) has been chosen to evaluate inequalities at local level. This choice has been decided arbitrarily for a convenient purpose (it is easier to conceptualise the average deviation of territorial units touching themselves). But results might be significantly different using other criteria for measuring the spatial deviation.
In fact, the values of this deviation are dependant from some criteria: * the intrinsic inequalities existing in the study area (the same than for the other deviations). * the average size of territorial units (for spatial criteria): a territorial unit with a surface below 1 km square will certainly have neighbours at a threshold of 1000 m. It is no more the case for a territorial unit with a surface of 20 km square.
* the spatial extent of the study area: the spatial criteria will not be the same for a local case study, such as the communes of the MGP or for wider case-study (Europe for instance).
Thus, it is important for have a global overview of the the effects of the neighbourhood parameters (order and dist parameters) on the calculation of the spatial deviation. It is the aim of the plot below. It takes in entry a dataframe computing desired distance or contiguity levels. Using the local deviation function (localDev), it returns a plot showing the amount of redistribution as regard to the distance or the contiguity criteria.
par(cex.lab=1)
par(cex.axis=0.75)
par(mar=c(4,4,2,2))
# Set the targeted distances for measuring spatial deviation
dfSpa <- data.frame(dist = seq(0,35000,by=500), ineq = NA)
# Calculation of spatial deviations in absolute terms for all the distances retained
spat <- function(x){
ldevabs <- sdev(spdf = com.spdf,
x = com,
spdfid = "DEPCOM",
xid = "DEPCOM",
var1 = "INC",
var2 = "TH",
dist = x[1],
type = "abs")
ldevabspos <- ldevabs[ldevabs>0]
sum(ldevabspos)
}
dfSpa$ineq <- apply(dfSpa, 1, spat)
# Convert results in million Euros and km
dfSpa$ineq <- dfSpa$ineq / 1000000
dfSpa$dist <- dfSpa$dist / 1000
dfSpa[is.na(dfSpa)] <- 0
# Set the targeted contiguity order for measuring spatial deviation
dfOrder <- data.frame(order = seq(0,20,by=1), ineq = NA)
# Calculation of spatial deviations in absolute terms for all the contiguities retained
spat <- function(x){
ldevabs <- sdev(spdf = com.spdf,
x = com,
spdfid = "DEPCOM",
xid = "DEPCOM",
var1 = "INC",
var2 = "TH",
order = x[1],
type = "abs")
ldevabspos <- ldevabs[ldevabs>0]
sum(ldevabspos)
}
dfOrder$ineq <- apply(dfOrder, 1, spat)
# Convert results in million Euros
dfOrder$ineq <- dfOrder$ineq / 1000000
dfOrder[is.na(dfOrder)] <- 0
# Combined plot - Plot 1: Euclidian Distance
plot.new()
plot.window(xlim=c(0,22500),ylim=c(0,35), xaxs="i", yaxs="i")
axis(1, pos=0, xlim=c(0,22500))
axis(2, col = '#488b37', pos=0, ylim=c(0,35))
abline (v = seq(0,22000,1000), col = "gray30", lwd = 0.25)
abline (h = seq(0,35,5), col = "#488b37", lwd = 0.25)
lines(dfSpa$dist ~ dfSpa$ineq,type='b',col='#488b37',lwd=0.25, lty = 1, pch = 20)
title(xlab = "Mass of numerator to redistribute required to reach the equilibrium (million Euros)")
mtext("Neighbourhood: Euclidian distance (km)", col = "#488b37", side = 2,line = 2)
# Adding plot 2 : contiguity
plot.window(xlim=c(0,22500),ylim=c(0,20),xaxs="i", yaxs="i")
axis(4, col ='#7a378b')
abline (h = seq(0,20,5), col = "#7a378b", lwd = 0.25, lty = 1)
lines(dfOrder$order~dfOrder$ineq,type='b',col='#7a378b',lwd=0.25, lty = 1, pch = 20)
mtext("Neighbourhood: Contiguity order", side = 4, col = "#7a378b")
#title for the graph
title(main="Numerator redistribution as regards to spatial and contiguity parameters")
box(which = "plot")
If the contiguity order or spatial distance corresponds to 0, nothing happen: each territorial unit is compared to itself and it implies consequently no spatial deviation. Reversely, if these parameters are too important (above 31 km or 12 contiguity orders in the case of the MGP), the spatial deviation takes for the calculation all the territorial units of the study area. In this case the values of the spatial deviation corresponds exactly to the global deviation.
In the plot displayed above, we can see that a contiguity of 1 order (chosen for this vignette) corresponds more or less 3,5 kilometers in euclidian distance (approximatively 11 billion Euros of income redistribution).
Out of these mathematical considerations related to the characteristics of the study area and the size of the elementary territorial units, it is important to remind that the choice of the appropriate threshold must make sense from a thematic point of view (does a neighbourhood of 5 km mean something ? Is a distance of 20 minutes by car more appropriated?)
This map highlights the communes largely above the average of the three deviations (coloured in red). They are mainly located in the West part of Paris and of the MGP. Some isolated communes appear also locally in a favorable situation in the East Part of the MGP.
The map shows the communes in difficulty regarding the income repartition in the MGP. Interesting is to note that all the territories of the Plaine Commune, Territoire des Aéroports EPT are below the threshold of 80 % for at least the global context. Most of the communes of EPT of Est Ensemble, Grand-Paris-Est, Val de Bièvres are in the same situation.