Risk-adjusted nearest-neighbour hierarchical clustering

Nearest-neighbour clustering (as nnhclu(), first order) with the threshold distance adjusted for a baseline population at risk, so that incidents dense only because people are dense there are not called a cluster. In nnhclu() the threshold is the mean nearest-neighbour distance under complete spatial randomness plus z standard errors, 0.5 sqrt(A / n) + z sqrt((4 - pi) A / (4 pi)) / n, which is f / sqrt(lambda) with lambda = n / A and f = 0.5 + z sqrt((4 - pi) / (4 pi n)). Here the uniform intensity is replaced by the intensity the n incidents would have if they followed the baseline: lambda_i = n b(p_i) / B, where b is the normal-kernel density (bandwidth h) of the baseline units with sizes bw and B is their total, and two incidents i and j are linked when |p_i - p_j| < f / sqrt((lambda_i + lambda_j) / 2). Taking the mean of the two intensities is this function's reading of "the threshold at the pair's baseline density" (the published description does not say how the two ends combine); a uniform baseline gives back nnhclu()'s threshold. Grouping is nnhclu()'s first order; higher orders are not formed.

Usage

rnnhcl(
  x,
  y = NULL,
  base,
  bw = NULL,
  h,
  p = 0.5,
  minpts = 10,
  k = 1,
  c = 1,
  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.

base

The baseline units: coordinates as for x (zone centroids, or points), in the same system as x (degrees when lonlat = TRUE).

bw

NULL for size 1 each, or the baseline sizes (populations), numbers from 0 with a positive sum.

h

The bandwidth of the baseline's normal kernel, positive, in the unit of the coordinates (metres when lonlat = TRUE).

p

The one-tailed probability that a pair is as close by chance: one number strictly between 0 and 1.

minpts

The fewest points a cluster keeps: a whole number from 2.

k

The divisor offset: 0, 1 or 2; the total weight must exceed it.

c

The multiplier of each cluster's standard deviational ellipse (see sdell()).

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: cl, each point's cluster (a row of clusters) or NA; clusters, a data frame of points, mx, my, wx, wy, a, b and angle as in nnhclu(); thr, each point's own threshold f / sqrt(lambda_i).

References

Johnson, S. C. (1967). Hierarchical clustering schemes. Psychometrika, 32(3), 241-254. doi:10.1007/BF02289588

Clark, P. J. and Evans, F. C. (1954). Distance to nearest neighbor as a measure of spatial relationships in populations. Ecology, 35(4), 445-453. doi:10.2307/1931034

Examples

set.seed(1)
town <- cbind(x = rnorm(400, 0, 1), y = rnorm(400, 0, 1))
inc <- rbind(town[1:60, ], cbind(x = rnorm(15, 3, 0.1),
                                 y = rnorm(15, 3, 0.1)))
r <- rnnhcl(inc, base = town, h = 0.5, minpts = 5)
r$clusters
#>   points       mx      my      wx       wy         a         b    angle
#> 1     15 2.973022 3.00705 2.97886 3.000247 0.1228748 0.1070088 1.886728