Space-time K function of Diggle, Chetwynd, Haggkvist and Morris (1995)

For events \((x_i, t_i)\) in a rectangle A and interval the interval from \(T_0\) to \(T_1\) (length T), with \(w_{ij}\) Ripley's isotropic factor and \(v_{ij} = 2\) when the interval \(t_i \pm u\), \(u = |t_i - t_j|\), reaches either time limit (else 1): \(K_s(s) = |A| / (n(n-1)) \sum w_{ij} I(d_{ij} \le s)\), \(K_t(t) = T / (n(n-1)) \sum v_{ij} I(u_{ij} \le t)\) and \(K_{st}(s, t) = |A| T / (n(n-1)) \sum w_{ij} v_{ij} I(d_{ij} \le s) I(u_{ij} \le t)\). Under no space-time interaction \(K_{st} = K_s K_t\); \(D = K_{st} - K_s K_t\) and \(D_0 = D / (K_s K_t)\) are returned. These are the estimators of splancs::stkhat.

Usage

SpaceTimeK(points, times, window, tlimits, s, t)

Arguments

points

Event locations (n x 2 matrix) inside window.

times

Event times inside tlimits.

window

Rectangle c(xmin, xmax, ymin, ymax).

tlimits

c(T0, T1).

s

Spatial distances (increasing).

t

Time lags (increasing).

Value

List with ks, kt, kst (matrix, rows s), D, D0, s, t.

References

Diggle, P. J., Chetwynd, A. G., Haggkvist, R. and Morris, S. E. (1995). Second-order analysis of space-time clustering. Statistical Methods in Medical Research 4, 124-136.

Examples

P <- rbind(c(0.1, 0.2), c(0.3, 0.25), c(0.8, 0.9), c(0.5, 0.5))
SpaceTimeK(P, c(1, 1.5, 4, 2), c(0, 1, 0, 1), c(0, 5), 0.3, 1)$ks
#> [1] 0.2262165