WaveSampling (version 0.1.4)

IB: Spreading measure based on Moran's \(I\) index

Description

This function implements the spreading measure based on Moran's \(I\) index.

Usage

IB(W, s)

Value

A numeric value that represents the spatial balance. It could be any real value between -1 (spread) and 1 (clustered).

Arguments

W

a stratification matrix inheriting from sparseMatrix that represents the spatial weights. See wpik.

s

a vector of size \(N\) with elements equal to 0 or 1. The value 1 indicates that the unit is selected while the value 0 is for non-chosen units.

Details

This index is developped by Tillé et al. (2018) and measure the spreading of a sample drawn from a population. It uses a corrected version of the traditional Moran's \(I\) index. Each row of the matrix \(\bf W\) should represents a stratum. Each stratum is defined by a particular unit and its neighbouring units. See wpik. The spatial balance measure is equal to

$$I_B =\frac{( \bf s- \bar{s}_w)^\top W ( s- \bar{s}_w)}{\bf \sqrt{( s- \bar{s}_w)^\top D ( s- \bar{s}_w) ( s- \bar{s}_w)^\top B ( s- \bar{s}_w)}},$$

where \(\bf D\) is the diagonal matrix containing the \(w_i\),

$$ \bf \bar{s}_w = 1 \frac{ s^\top W 1}{ 1^\top W 1}$$

and

$$ \bf B = W^\top D^{-1} W - \frac{ W^\top 1 1^\top W}{1^\top W 1}.$$

To specify the spatial weights uses the argument W.

References

Tillé, Y., Dickson, M.M., Espa, G., and Guiliani, D. (2018). Measuring the spatial balance of a sample: A new measure based on Moran's I index. Spatial Statistics, 23, 182-192.

See Also

wpik

Examples

  N <- 36
  n <- 12
  x <- seq(1,sqrt(N),1)
  X <- expand.grid(x,x)
  pik <- rep(n/N,N)
  W <- wpik(as.matrix(X),pik,bound = 1,tore = TRUE,shift = FALSE,toreBound = sqrt(N))
  W <- W - diag(diag(W))
  s <- wave(as.matrix(X),pik,tore = TRUE,shift = TRUE,comment = TRUE)
  IB(W,s)