# Product Within Distance (Spatially Balanced Sampling Design) Selects spatially balanced samples through the use of the Product Within Distance design (PWD). To have constant inclusion probabilities $\pi_{i}=n/N$, where $n$ is sample size and $N$ is population size, the distance matrix has to be standardized with function `stprod`. ```r pwd(dis, n, beta = 10, nrepl = 1L, niter = 10L) ``` ## Arguments - `dis`: A distance matrix NxN that specifies how far all the pairs of units in the population are. - `n`: Sample size. - `beta`: Parameter $\beta$ for the algorithm. The higher $\beta$ is, the more the sample is going to be spread (default = 10). - `nrepl`: Number of samples to draw (default = 1). - `niter`: Maximum number of iterations for the algorithm. More iterations are better but require more time. Usually 10 is very efficient (default = 10). ## Returns Returns a list with the following components: * `s`, a matrix `nrepl` x `n`, which contains the `nrepl` selected samples, each of them stored in a row. In particular, the i-th row contains all labels of units selected in the i-th sample. * `iterations`, number of iterations run by the algorithm. ## Examples ```r # Example 1 # Draw 1 sample of dimension 15 without constant inclusion probabilities dis <- as.matrix(dist(cbind(lucas_abruzzo$x, lucas_abruzzo$y))) # distance matrix s <- pwd(dis = dis, n = 15)$s # drawn sample # Example 2 # Draw 1 sample of dimension 15 with constant inclusion probabilities # equal to n/N, with N = population size dis <- as.matrix(dist(cbind(lucas_abruzzo$x, lucas_abruzzo$y))) # distance matrix con <- rep(0, nrow(dis)) # vector of constraints stand_dist <- stprod(mat = dis, con = con) # standardized matrix s <- pwd(dis = stand_dist$mat, n = 15)$s # drawn sample # Example 3 # Draw 2 samples of dimension 15 with constant inclusion probabilities # equal to n/N, with N = population size, and an increased level of spread, beta = 20 dis <- as.matrix(dist(cbind(lucas_abruzzo$x, lucas_abruzzo$y))) # distance matrix con <- rep(0, nrow(dis)) # vector of constraints stand_dist <- stprod(mat = dis, con = con) # standardized matrix s <- pwd(dis = stand_dist$mat, n = 15, beta = 20, nrepl = 2)$s # drawn samples ``` ## References Benedetti R, Piersimoni F (2017). A spatially balanced design with probability function proportional to the within sample distance. **Biometrical Journal**, 59 (5), 1067-1084. tools:::Rd_expr_doi("10.1002/bimj.201600194")

pwd function