continuous.sample {PrevMap}

Spatially continuous sampling

Description

Draws a sample of spatial locations within a spatially continuous polygonal sampling region.

Usage

continuous.sample(poly, n, delta, k = 0, rho = NULL)

Arguments

poly

boundary of a polygon.

n

number of events.

delta

minimum permissible distance between any two events in preliminary sample.

k

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)

rho

maximum distance between close pairs of locations in final sample.

Details

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.

Value

A matrix of dimension n by 2 containing event locations.

Author(s)

Emanuele Giorgi [e-mail]

Peter J. Diggle [e-mail]