Synthetic point-referenced spatial data

Dataset of size 500 with spatial coordinates sampled uniformly from the unit square, two covariates, a spatial effect with a checkerboard pattern and one response for each of the Gaussian, Poisson, binomial and binary families, plus a Gaussian and a Poisson response with spatially varying coefficients (a varying intercept with a ring pattern and a varying slope of x1). All the responses share the covariates and the spatial effects, so that every spatial model in the package can be fitted to the same data.

Usage

data(simSpatial)

Format

a data.frame object with 500 rows and 13 columns

s1, s2: 2-D coordinates in the unit square.
x1, x2: covariates sampled from the standard normal distribution.
y_gauss: Gaussian response.
y_pois: Poisson count response.
y_binom: binomial count response, out of n_trials trials.
n_trials: number of trials of the binomial response (5 to 20).
y_bin: binary response.
y_svc: Gaussian response with the spatially varying intercept z2 and the spatially varying slope 5 + z of x1.
y_svc_pois: Poisson count response with the spatially varying intercept z2 and the spatially varying slope -0.5 + z of x1.
z_true: true spatial effect (checkerboard) that generated the data; the spatially varying part of the slope of x1 in y_svc and y_svc_pois.
z2_true: true spatially varying intercept (rings) of y_svc and y_svc_pois.

Details

The spatial effect is the checkerboard $$ z(s) = 1.5 \sin(2 \pi s_1) \sin(2 \pi s_2), $$ and the spatially varying intercept is the rings $$ z_2(s) = \cos(4 \pi \lVert s - s_0 \rVert) - c, \quad s_0 = (0.5, 0.5), $$ where the constant $c \approx 0.129$ makes the mean of $z_2$ over the unit square zero (the mean of $z$ is zero by symmetry, and the two surfaces are orthogonal on the unit square). Both are deterministic surfaces, not draws from a Gaussian process. The responses are generated as $$ \begin{aligned} y_{\mathrm{gauss}}(s) &\sim N(2 + 5 x_1(s) - x_2(s) + z(s), 0.5^2),\\ y_{\mathrm{pois}}(s) &\sim \mathrm{Poisson}(\exp\{2 - 0.5 x_1(s) + 0.3 x_2(s) + z(s)\}),\\ y_{\mathrm{binom}}(s) &\sim \mathrm{Binomial}(m(s), \mathrm{ilogit}\{0.5 - 0.5 x_1(s) + 0.5 x_2(s) + z(s)\}),\\ y_{\mathrm{bin}}(s) &\sim \mathrm{Bernoulli}(\mathrm{ilogit}\{0.5 - 0.5 x_1(s) + 0.5 x_2(s) + z(s)\}),\\ y_{\mathrm{svc}}(s) &\sim N(2 + z_2(s) + \{5 + z(s)\} x_1(s) - x_2(s), 0.5^2),\\ y_{\mathrm{svc\_pois}}(s) &\sim \mathrm{Poisson}(\exp\{2 + z_2(s) + \{-0.5 + z(s)\} x_1(s) + 0.3 x_2(s)\}), \end{aligned} $$ where $m(s)$ is the number of trials. This data can be generated with the code given in the example.

Author

Soumyakanti Pan span18@ucla.edu

Examples

set.seed(1729)
n <- 500
s1 <- runif(n)
s2 <- runif(n)
x1 <- rnorm(n)
x2 <- rnorm(n)
z <- 1.5 * sin(2 * pi * s1) * sin(2 * pi * s2)
ring <- function(s1, s2) cos(4 * pi * sqrt((s1 - 0.5)^2 + (s2 - 0.5)^2))
g <- (seq_len(1000) - 0.5) / 1000
z2 <- ring(s1, s2) - mean(outer(g, g, ring))
n_trials <- sample(5:20, n, replace = TRUE)
dat <- data.frame(
  s1 = s1, s2 = s2, x1 = x1, x2 = x2,
  y_gauss = rnorm(n, 2 + 5 * x1 - x2 + z, sd = 0.5),
  y_pois = rpois(n, exp(2 - 0.5 * x1 + 0.3 * x2 + z)),
  y_binom = rbinom(n, n_trials, plogis(0.5 - 0.5 * x1 + 0.5 * x2 + z)),
  n_trials = n_trials,
  y_bin = rbinom(n, 1, plogis(0.5 - 0.5 * x1 + 0.5 * x2 + z)),
  y_svc = rnorm(n, 2 + z2 + (5 + z) * x1 - x2, sd = 0.5),
  y_svc_pois = rpois(n, exp(2 + z2 + (-0.5 + z) * x1 + 0.3 * x2)),
  z_true = z,
  z2_true = z2
)
all.equal(dat, simSpatial)
#> [1] TRUE