PrevMap (version 1.5.4)

discrete.sample: Spatially discrete sampling

Description

Draws a sub-sample from a set of units spatially located irregularly over some defined geographical region by imposing a minimum distance between any two sampled units.

Usage

discrete.sample(xy.all, n, delta, k = 0)

Arguments

xy.all

set of locations from which the sample will be drawn.

n

size of required sample.

delta

minimum distance between any two locations in preliminary sample.

k

number of locations in preliminary sample to be replaced by nearest neighbours of other preliminary sample locations in final sample (must be between 0 and n/2).

Value

A matrix of dimension n by 2 containing the final sampled locations.

Details

To draw a sample of size n from a population of spatial locations \(X_{i} : i = 1,\ldots,N\), with the property that the distance between any two sampled locations is at least delta, the function implements the following algorithm.

• Step 1. Draw an initial sample of size n completely at random and call this \(x_{i} : i = 1,\dots, n\).

• Step 2. Set \(i = 1\) and calculate the minimum, \(d_{\min}\), of the distances from \(x_{i}\) to all other \(x_{j}\) in the initial sample.

• Step 3. If \(d_{\min} \ge \delta\), increase \(i\) by 1 and return to step 2 if \(i \le n\), otherwise stop.

• Step 4. If \(d_{\min} < \delta\), draw an integer \(j\) at random from \(1, 2,\ldots,N\), set \(x_{i} = X_{j}\) and return to step 3.

Samples generated in this way will exhibit a more regular spatial arrangement than would a random sample of the same size. The degree of regularity achievable will be influenced by the spatial arrangement of the population \(X_{i} : i = 1,\ldots,N\), the specified value of delta and the sample size n. For any given population, if n and/or delta are too large, a sample of the required size with the distance between any two sampled locations at least delta will not be achievable; the suggested solution is then to run the algorithm with a smaller value of delta.

Sampling close pairs of points. For some purposes, it is desirable that a spatial sampling scheme include pairs of closely spaced points. In this case, the above algorithm requires the following additional steps to be taken. Let k be the required number of close pairs.

• Step 5. Set \(j = 1\) and draw a random sample of size 2 from the integers \(1, 2,\ldots,n\), say \((i_{1}, i_{2} )\).

• Step 6. Find the integer \(r\) such that the distances from \(x_{i_{1}}\) to \(X_{r}\) is the minimum of all \(N-1\) distances from \(x_{i_{1}}\) to the \(X_{j}\).

• Step 7. Replace \(x_{i_{2}}\) by \(X_{r}\), increase \(i\) by 1 and return to step 5 if \(i \le k\), otherwise stop.

Examples

# NOT RUN {
x<-0.015+0.03*(1:33)
xall<-rep(x,33)
yall<-c(t(matrix(xall,33,33)))
xy<-cbind(xall,yall)+matrix(-0.0075+0.015*runif(33*33*2),33*33,2)
par(pty="s",mfrow=c(1,2))
plot(xy[,1],xy[,2],pch=19,cex=0.25,xlab="Easting",ylab="Northing",
   cex.lab=1,cex.axis=1,cex.main=1)

set.seed(15892)
# Generate spatially random sample
xy.sample<-xy[sample(1:dim(xy)[1],50,replace=FALSE),]
points(xy.sample[,1],xy.sample[,2],pch=19,col="red")
points(xy[,1],xy[,2],pch=19,cex=0.25)
plot(xy[,1],xy[,2],pch=19,cex=0.25,xlab="Easting",ylab="Northing",
   cex.lab=1,cex.axis=1,cex.main=1)

set.seed(15892)
# Generate spatially regular sample
xy.sample<-discrete.sample(xy,50,0.08)
points(xy.sample[,1],xy.sample[,2],pch=19,col="red")
points(xy[,1],xy[,2],pch=19,cex=0.25)

# }