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 $\mathbf{W}$ 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

$$I_B=\frac{(\mathbf{s}\mathbf{-}{\overline{\mathbf{s}}}_{\mathbf{w}}{\mathbf{)}}^{\mathrm{\top }}\mathbf{W}\mathbf{(}\mathbf{s}\mathbf{-}{\overline{\mathbf{s}}}_{\mathbf{w}}\mathbf{)}}{\sqrt{\mathbf{(}\mathbf{s}\mathbf{-}{\overline{\mathbf{s}}}_{\mathbf{w}}{\mathbf{)}}^{\mathrm{\top }}\mathbf{D}\mathbf{(}\mathbf{s}\mathbf{-}{\overline{\mathbf{s}}}_{\mathbf{w}}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{-}{\overline{\mathbf{s}}}_{\mathbf{w}}{\mathbf{)}}^{\mathrm{\top }}\mathbf{B}\mathbf{(}\mathbf{s}\mathbf{-}{\overline{\mathbf{s}}}_{\mathbf{w}}\mathbf{)}}},$$

where $\mathbf{D}$ is the diagonal matrix containing the $w_i$,

$${\overline{\mathbf{s}}}_{\mathbf{w}}\mathbf{=}\mathbf{1}\frac{{\mathbf{s}}^{\mathrm{\top }}\mathbf{W}\mathbf{1}}{{\mathbf{1}}^{\mathrm{\top }}\mathbf{W}\mathbf{1}}$$

and

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

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)