Spatially discrete sampling

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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

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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).

Details

To draw a sample of size n from a population of spatial locations X_{i} : i = 1,…,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,…, 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} ≥ δ, increase i by 1 and return to step 2 if i ≤ n, otherwise stop.

  • Step 4. If d_{\min} < δ, draw an integer j at random from 1, 2,…,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,…,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,…,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 ≤ k, otherwise stop.

Value

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

Author(s)

Emanuele Giorgi e.giorgi@lancaster.ac.uk

Peter J. Diggle p.diggle@lancaster.ac.uk

Examples

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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)

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