W.estimate {CARBayesST}

Estimate an appropriate neighbourhood matrix for a set of spatial data using a baseline neighbourhood matrix and a graph based optimisation algorithm.

Description

Estimate an appropriate neighbourhood matrix (W.est) for a given set of spatial data (spdata) from a baseline neighbourhood matrix (W) using the graph-based optimisation algorithm proposed by Lee, Meeks and Pettersson (2021). The baseline neighbourhood matrix W should be binary and based on a simple geographical specification such as the border sharing rule. The estimated neighbourhood matrix is constructed by removing neighbour relations (i.e. setting w_ij = w_ji = 0) if they are not appropriate for the data. Note, new edges not in the initial W matrix cannot be added when creating W.est.

Usage

W.estimate(W, spdata, add=FALSE, remove=TRUE, remove_first=FALSE)

Arguments

W

A binary K * K neighbourhood matrix, where K is the number of spatial units. A typical specification would have the jkth element equalling one if areas (j, k) are spatially close (e.g. share a common border) and zero otherwise. Each row must contain at least one non-zero entry.

spdata

A K * 1 vector of spatial data that you wish to optimise the neighbourhood matrix for. The kth element of this vector must correspond to the area represented by the kth row of the W matrix.

add

Allow the optimiser to add edges back in to the graph if they have previously been removed. Defaults to FALSE.

remove

Allow the optimiser to remove edges from the graph if they have previously been added in. Defaults to TRUE.

remove_first

If only one of add or remove are TRUE, then this option has no effect. If both add and remove are TRUE, then this option determines whether the optimiser first tries to add or remove edges (before alternating between the two phases). Defaults to FALSE, meaning that the optimiser tries to add edges first.

Value

W.est

An optimised K by K neighbourhood matrix.

Author(s)

William Pettersson ([e-mail]) and Kitty Meeks

References

Lee, D and Meeks, K (2020). Improved inference for areal unit count data using graph-based optimisation. arXiv:2010.10893.

Examples

####################################################
#### Run the optmiser on simulated data on a lattice
####################################################
#### Load other libraries required
library(MASS)

#### Set up a square lattice region
x.easting <- 1:10
x.northing <- 1:10
Grid <- expand.grid(x.easting, x.northing)
K <- nrow(Grid)

#### Split the area into two groups between which there will be a boundary.
groups <-rep(1, K) 
groups[Grid$Var1>5] <- 2

#### set up distance and neighbourhood (W, based on sharing a common border) matrices
distance <- as.matrix(dist(Grid))
W <-array(0, c(K,K))
W[distance==1] <-1 	

#### Generate the response data
phi <- mvrnorm(n=1, mu=groups, Sigma=0.05 * exp(-0.1 * distance))
lp <- 4 + phi
Y <- rpois(n=K, lambda =exp(lp))


# Compute the spdata object on the linear predictor scale
spdata <- log(Y) - mean(log(Y))

#### Run W matrix optmiser
W.est <- W.estimate(W, spdata)