Generation of qualitative process with spatial structure

The purpose of the function dgp.spq is to generate a random dataset with the dimensions and spatial structure decided by the user. This function may be useful in pure simulation experiments or with the aim of showing specific properties and characteristics of a spatial qualitative dataset.

Usage

dgp.spq(listw = listw, p = p,  rho = rho, control = list())

Arguments

listw: A listw object of the class nb, knn, listw o matrix created for example by nb2listw from spatialreg package; if nb2listw not given, set to the same spatial weights as the listw argument. It can also be a spatial weighting matrix of order (NxN) instead of a listw object. Default = NULL.
p: a vector with the percentage of elements of each categories. The lengths must be the number of categories. The sum of the elements of vector must be 1.
rho: the level of spatial dependence (values between -1 y 1)
control: List of additional control arguments. See control argument section.

Value

a factor of length N (listw is a matrix of order (NxN)) with levels the first capital letters: "A", "B", ....

The description of the DGP is available in Páez et al. 2010 (pag 291) and in details section.

Details

In order to obtain categorical random variables with controlled degrees of spatial dependence, we have designed a two- stage data generating process:

Firstly, we simulate autocorrelated data using the following model: $$Y = (I - \rho W)^{-1} \epsilon$$ where $\epsilon = N(0,1)$ I is the $N \times N$ identity matrix, $\rho$ is a parameter of spatial dependence, and W is a connectivity matrix that determines the set of spatial relationships among points.

In the second step of the data generation process, the continuous spatially autocorrelated variable Y is used to define a discrete spatial process as follows. Let $b_{ij}$ be defined by:
$$p(Y \leq b_{ij})= {i \over j} \ \ \ with \ \ \ i<j$$ Let $A =\{a_1,a_2,...,a_k\}$ and define the discrete spatial process as: $$X_s=a_1 \ \ \ if \ \ \ Y_s \leq b_{1k}$$ $$X_s=a_i \ \ \ if \ \ \ b_{i-1k} < Y_s \leq b_{ik}$$ $$X_s=a_k \ \ \ if \ \ \ Y_s > b_{k-1k}$$

Control arguments

seedinit: seed to generate the data sets

References

Ruiz M, López FA, A Páez. (2010). Testing for spatial association of qualitative data using symbolic dynamics. Journal of Geographical Systems. 12 (3) 281-309

Author

Fernando López	fernando.lopez@upct.es
Román Mínguez	roman.minguez@uclm.es
Antonio Páez	paezha@gmail.com
Manuel Ruiz	manuel.ruiz@upct.es

Examples

#
N <- 100
cx <- runif(N)
cy <- runif(N)
coor <- cbind(cx,cy)
p <- c(1/6,3/6,2/6)
rho = 0.5
listw <- spdep::nb2listw(spdep::knn2nb(spdep::knearneigh(cbind(cx,cy), k = 4)))
xf <- dgp.spq(list = listw, p = p, rho = rho)

data(provinces_spain)
listw <- spdep::poly2nb(provinces_spain, queen = FALSE)
#> although coordinates are longitude/latitude, st_intersects assumes that they
#> are planar
#> Warning: some observations have no neighbours;
#> if this seems unexpected, try increasing the snap argument.
#> Warning: neighbour object has 4 sub-graphs;
#> if this sub-graph count seems unexpected, try increasing the snap argument.
p <- c(1/6,3/6,2/6)
rho = 0.9
xf <- dgp.spq(p = p, listw = listw, rho = rho)
provinces_spain$xf <- xf
plot(provinces_spain["xf"])