Voronoï polygons
One way of measuring the spread of a sample was developed by Stevens and Olsen (2004) and then suggested by Grafström, Lundström, and Schelin (2012). It is based on the Voronoï polygons and is given by
\[ B(\bf s) = \frac{1}{n}\sum_{i \in s} (v_i - 1)^2 \] where \(v_i\) is equal to the sum of the inclusion probabilities inside the \(i\)th polygons and \(\bf s\) is the vector of size \(N\) with elements equal 0 or 1. This quantity is implemented in the package BalancedSampling with the function sb(). We calculate the values of the \(v_k\) with the function sb_vk.
The closer \(B(\bf s)\) is to zero, the better is the spatial balance of the sample. Graphically, we obtain the following plot.
library(sp)
library(sampling)
library(ggvoronoi)
data("meuse")
data("meuse.area")
v <- sb_vk(pik,as.matrix(meuse[,1:2]),s)
meuse$v <- v
p <- p + geom_voronoi(data = meuse[which(s == 1),],
aes(x = x,y = y,fill = v),
outline =as.data.frame(meuse.area),
size = 0.1,
colour = "black")+
geom_point(data = meuse,
aes(x = x,y = y,size = copper),
shape = 1,
stroke = 0.3)+
geom_point(data = meuse[which(s == 1),],
aes(x = x,y = y,size = copper),
shape = 16)+
scale_fill_gradient2(midpoint = 1)
p
BalancedSampling::sb(pik,as.matrix(meuse[,1:2]),which(s == 1))
#> [1] 0.08031111
Moran index
Another way to estimate the spatial spread is developed by Tillé et al. (2018), it uses a corrected version of the traditional Moran’s \(I\) index. This estimator use spatial weights \(w_{ij}\) that indicates how a unit \(i\) is close from the unit \(j\). Such matrix is supposed to include inclusion probabilities in its computation, hence, the spatial weights matrix \(\bf W\) is generally not symmetric. The spatial balance measure is given by
\[ I_B =\frac{(\bf s-\bf \bar{s}_w)^\top \bf W (\bf s-\bf \bar{s}_w)}{\sqrt{(\bf s-\bf \bar{s}_w)^\top \bf D (\bf s-\bf \bar{s}_w) (\bf s-\bf \bar{s}_w)^\top \bf B (\bf s-\bf \bar{s}_w) }}, \] where \(\bf D\) is the diagonal matrix containing the \(w_i\), \[\bf \bar{s}_w = \bf 1 \frac{\bf s^\top \bf W \bf 1}{\bf 1^\top \bf W \bf 1},\] and \[\bf B = \bf W^\top \bf D^{-1} \bf W - \frac{\bf W^\top \bf 1\bf 1^\top \bf W}{\bf1^\top \bf W \bf 1}.\]
The Moran’s \(I\) index is implemented in the function IB(). It is possible to specify your own spatial weights with the argument W. There is no natural way of defining \(\bf W\), here we propose to consider for each unit only the neighbour such that the sum of the inclusion probabilities of the stratum sum up to 1. It is implemented in the function wpik(). Another way of estimating the spatial weights is developed by Tillé et al. (2018) and use the inverse of the inclusion probabilities \(1/\pi_i\) to estimate the neighbours of the unit \(i\). It is implemented in the function wpikInv(). As explain by Tillé et al. (2018) \(w_{ii}\) is supposed to be equal to 0 for all \(i \in U\). By construction the function wpik does not return the diagonal equal to zero. So if we want to calculate the Moran’s I index with wpik, we need to subtract the diagonal of the returned matrix.
W <- wpik(X,pik)
W <- W - diag(diag(W))
IB(W,s)
#> [1] -0.3786126
W1 <- wpikInv(X,pik)
IB(W1,s)
#> [1] -0.3872553
