Envelopes of Ripley's L function and a test of complete spatial randomness

L(r) - r for the data, its pointwise minimum, maximum and mean over nsim patterns of as many uniform points in the rectangle, and a global Monte Carlo test on the maximum absolute deviation T = max_r |L(r) - r|, whose p-value (1 + #{T_sim >= T}) / (nsim + 1) has exact size (Ripley 1977; Diggle 2013). The pointwise envelopes are not a test of the whole curve: each radius alone falls outside them with probability 2 / (nsim + 1) under randomness.

Usage

ripenv(x, y = NULL, win, r, corr = "iso", nsim = 99, seed = NULL)

Arguments

x, y: The points in planar coordinates (project longitude and latitude with eqproj() first): x and y numeric vectors of equal length, or x a matrix or data frame with columns named x and y (an unnamed two-column matrix is read as x then y). At least two points, all inside win; no missing values.
win: The study rectangle c(xmin, xmax, ymin, ymax), in the unit of the coordinates. The intensity is the number of points over its area.
r: The radii: non-negative numbers in strictly increasing order, in the unit of the coordinates.
corr: The edge correction: "iso", "border", "modif" or "none".
nsim: The number of simulated patterns: one whole number from 1.
seed: NULL to draw the simulations' seed from R's generator (so set.seed() fixes them), or one whole number from 0.

Value

A list. env, a data frame with one row per radius: r, obs (L(r) - r of the data), lo, hi (the pointwise envelope) and mean. test, a named vector: t (the maximum absolute deviation) and p (its Monte Carlo p-value).

Details

Euclidean distance only; ripk() takes metric.

References

Diggle, P. J. (2013). Statistical Analysis of Spatial and Spatio-Temporal Point Patterns (3rd ed.), chapter 6. CRC Press.

Ripley, B. D. (1977). Modelling spatial patterns. Journal of the Royal Statistical Society, Series B, 39(2), 172-212. doi:10.1111/j.2517-6161.1977.tb01615.x

Examples

set.seed(1)
ripenv(runif(100), runif(100), win = c(0, 1, 0, 1),
       r = seq(0.01, 0.25, 0.02), nsim = 39)$test
#>          t          p 
#> 0.01033936 0.15000000