Knox test of near repeats

Asks whether incidents close together in space also happen close together in time more often than chance gives: the near-repeat pattern of burglary and shootings (Knox 1964; Townsley, Homel and Chaseling 2003). Every pair of incidents within the largest distance is put in a band of distance, (ds[k - 1], ds[k]], and a band of time, (dt[l - 1], dt[l]] (the first bands start at 0 and include it), and each cell's count is compared with its expectation when the times are exchangeable among the locations, which is exact: E = S_k T_l / P, with S_k the pairs in distance band k, T_l the pairs in time band l and P = n (n - 1) / 2. Each cell has a one-sided permutation p-value, the times shuffled among the locations nsim times from seed, and the Poisson tail P(X >= obs) with the expectation as its mean, the classical Knox approximation for a rare cell. Cells are tested separately: a table of many cells shows some small p-values by chance alone. A near-repeat pattern also arises when one offender commits a series; the test speaks to timing, not to its cause.

Usage

knxnr(x, y = NULL, t, ds, dt, nsim = 999, seed = NULL, lonlat = FALSE)

Arguments

x, y

The points: x and y numeric vectors of equal length; or x a matrix or data frame with columns named x and y (or X and Y, as sf::st_coordinates() names them, or lon and lat with lonlat = TRUE) and y NULL; or x an sf or sfc object of POINT geometries, whose coordinate reference system must agree with lonlat. An unnamed two-column matrix is read as x then y; other columns are never taken by position. Vectors given separately (x, y, and w or t where a function takes them) are matched by name when every one is named and no name repeats, as elsewhere in the package; otherwise by position. No missing or infinite values.

t

The times: numbers (days, say), or Date or POSIXct (taken in days), one per incident.

ds

The upper edges of the distance bands: positive and increasing, in the unit of the coordinates (metres when lonlat = TRUE).

dt

The upper edges of the time bands: positive and increasing, in the unit of t.

nsim

The number of permutations: a whole number from 0 (none).

seed

NULL to draw the simulations' seed from R's generator (so set.seed() fixes them), or one whole number from 0.

lonlat

FALSE (the default) when x and y are planar coordinates in one length unit; TRUE when they are longitude and latitude in degrees, which are then projected to metres by eqproj() about their own centre, and centres are returned in degrees. It is never guessed.

Value

A list of class knxnr: tab, a data frame with one row per cell, distance bands varying fastest: dlo, dhi, tlo, thi, obs (pairs), exp, ratio (obs / exp, NaN when nothing is expected), p ((1 + #{obs* >= obs}) / (nsim + 1), NaN without permutations) and ppois; n (incidents), pairs (pairs within the largest distance) and nsim.

References

Knox, E. G. (1964). The detection of space-time interactions. Journal of the Royal Statistical Society, Series C, 13(1), 25-30. doi:10.2307/2985220

Townsley, M., Homel, R. and Chaseling, J. (2003). Infectious burglaries: a test of the near repeat hypothesis. British Journal of Criminology, 43(3), 615-633. doi:10.1093/bjc/43.3.615

Examples

set.seed(1)
p <- cbind(x = runif(150, 0, 2000), y = runif(150, 0, 2000))
k <- knxnr(p, t = runif(150, 0, 365), ds = c(200, 400), dt = c(14, 28),
           nsim = 99, seed = 2)
k
#> knxnr: 150 incidents, 1322 close pairs, 99 permutations
#>  dlo dhi tlo thi obs      exp     ratio    p      ppois
#>    0 200   0  14  31 25.41369 1.2198149 0.12 0.15624997
#>  200 400   0  14  94 76.08752 1.2354195 0.02 0.02588842
#>    0 200  14  28  29 23.99195 1.2087390 0.16 0.17702659
#>  200 400  14  28  56 71.83087 0.7796091 0.99 0.97655462