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Cross-type K function and the segregation test

Source: R/crossk.R
crossk.Rd

Whether incidents of one type (i) have incidents of another (j) near them more than independence gives: the cross-type K function K_ij(r) = A / (n_i n_j) sum e_ab 1(d_ab <= r) over pairs of a point a of type i and a point b of type j in the rectangle win of area A, with Ripley's (1977) isotropic correction e_ab (the length of the circle about a through b over its length inside win), and L = sqrt(K / pi) (Lotwick and Silverman 1982). Under independence K = pi r^2. With i equal to j it is Ripley's K of that type. These are the values of spatstat.explore::Kcross() with correction = "isotropic".

Usage

crossk(x, y = NULL, marks, i, j, win, r, sigma = NULL, 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. At least two points; no missing values.

marks

The type of each point: labels with no missing values.

i, j

The two types, by label or number (of the sorted labels, or a factor's levels).

win

The study rectangle c(xmin, xmax, ymin, ymax), holding every point.

r

The distances: increasing, from 0.

sigma

NULL for no segregation test, or the kernel's standard deviation, positive.

nsim

The number of relabellings: a whole number from 0.

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 of class crossk: k, a data frame of r, k, l and theo (pi r^2); seg, NULL or a named vector of t, p (NaN without relabellings) and nsim; i and j, the types' labels.

Details

With sigma, the segregation test of Diggle, Zheng and Durr (2005) across all the types: at each point the share of each type among the other points, weighted by a normal kernel of standard deviation sigma, against the type's overall share, T = sum_points sum_types (p_m - pbar_m)^2, with a p-value from nsim random relabellings of the types ((1 + #{T* >= T}) / (nsim + 1)), the statistic of spatstat.explore::segregation.test().

References

Ripley, B. D. (1977). Modelling spatial patterns. Journal of the Royal Statistical Society, Series B, 39(2), 172-192. doi:10.1111/j.2517-6161.1977.tb01615.x

Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several types of points. Journal of the Royal Statistical Society, Series B, 44(3), 406-413. doi:10.1111/j.2517-6161.1982.tb01221.x

Diggle, P. J., Zheng, P. and Durr, P. (2005). Nonparametric estimation of spatial segregation in a multivariate point process: bovine tuberculosis in Cornwall, UK. Journal of the Royal Statistical Society, Series C, 54(3), 645-658. doi:10.1111/j.1467-9876.2005.05373.x

Examples

set.seed(1)
p <- cbind(x = runif(120), y = runif(120))
m <- ifelse(p[, 1] < 0.5, "burglary", "robbery")
crossk(p, marks = m, i = "burglary", j = "robbery",
       win = c(0, 1, 0, 1), r = c(0.05, 0.1, 0.2), sigma = 0.1,
       nsim = 19, seed = 1)
#> crossk: K of robbery about burglary
#>     r           k           l        theo
#>  0.05 0.000278474 0.009414936 0.007853982
#>  0.10 0.001113896 0.018829872 0.031415927
#>  0.20 0.023148107 0.085838634 0.125663706
#> segregation T 47.42, p 0.05 (19 relabellings)