Draws a sample of spatial locations within a spatially continuous polygonal sampling region.
continuous.sample(poly, n, delta, k = 0, rho = NULL)boundary of a polygon.
number of events.
minimum permissible distance between any two events in preliminary sample.
number of locations in preliminary sample to be replaced by near neighbours of other preliminary sample locations in final sample (must be between 0 and n/2)
maximum distance between close pairs of locations in final sample.
A matrix of dimension n by 2 containing event locations.
To draw a sample of size n from a spatially continuous region \(A\), with the property that the distance between any two sampled locations is at least delta, the following algorithm is used.
Step 1. Set \(i = 1\) and generate a point \(x_{1}\) uniformly distributed on \(A\).
Step 2. Increase \(i\) by 1, generate a point \(x_{i}\) uniformly distributed on \(A\) and calculate the minimum, \(d_{\min}\), of the distances from \(x_{i}\) to all \(x_{j}: j < i \).
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 \(d_{\min} < \delta\), return to step 2 without increasing \(i\).
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. Choose a value rho such that a close pair of points will be a pair of points separated by a distance of at most rho.
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. Replace \(x_{i_{1}}\) by \(x_{i_{2}} + u\) , where \(u\) is uniformly distributed on the disc with centre \(x_{i_{2}}\) and radius rho, increase \(i\) by 1 and return to step 5 if \(i \le k\), otherwise stop.