wpikInv {WaveSampling}

Stratification matrix from inverse inclusion probabilities

Description

The stratification matrix is calculated from the inverse inclusion probabilities. It is a direct implementation of the spatial weights specified in Tillé et al., (2018).

Usage

wpikInv(X, pik, tore = FALSE, shift = FALSE, toreBound = -1)

Arguments

X

matrix representing the spatial coordinates.

pik

vector of the inclusion probabilities. The length should be equal to N.

tore

an optional logical value, if we are considering the distance on a tore. Default is FALSE.

shift

an optional logical value, if you would use a shift perturbation. See Details for more informations. Default is FALSE.

toreBound

a numeric value that specify the size of the grid. Default is -1.

Details

Entries of the stratification matrix indicates how the units are close from each others. Hence a large value w_{kl} means that the unit k is close to the unit l. This function considers that if unit k were selected in the sample drawn from the population then k would represent 1/\pi_k units in the population and, as a consequence, it would be natural to consider that k has n_k = (1/\pi_k - 1) neighbours in the population. The n_k neighbours are the nearest neighbours of k according to distances. The weights are so calculated as follows :

• w_{kl} = 1 if unit l \in N_{\lfloor n_k \rfloor }

• w_{kl} = n_k - \lfloor n_k \rfloor if unit l is the \lceil n_k \rceil nearest neighbour of k.

• w_{kl} = 0 otherwise.

\lfloor n_k \rfloor and \lceil n_k \rceil are the inferior and the superior integers of n_k.

The option shift add a small normally distributed perturbation rnorm(0,0.01) to the coordinates of the centroid of the stratum considered. This could be useful if there are many unit that have the same distances. Indeed, if two units have the same distance and are the last unit before that the bound is reached, then the weights of the both units is updated. If a shift perturbation is used then all the distances are differents and only one unit weight is update such that the bound is reached.

The shift perturbation is generated at the beginning of the procedure such that each stratum is shifted by the same perturbation.

Value

A sparse matrix representing the spatial weights.

Author(s)

Raphaël Jauslin [e-mail]

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, distUnitk, wave.

Examples

N <- 25
n <- 5
X <- as.matrix(cbind(runif(N),runif(N)))
pik <- rep(n/N,N)
W <- wpikInv(X,pik)