Mean centre and dispersion of points

The (weighted) mean centre of a point pattern, the standard deviations of the two coordinates, the standard distance deviation (the root mean squared distance from the mean centre), and the geometric and harmonic means of the coordinates (Bachi 1957; Neft 1962). The mean centre is the point that minimises the sum of squared distances to the points. The dispersions divide by the total weight less k: k = 1 makes the squared values unbiased for the variances; k = 2 is the convention of some crime-mapping texts; k = 0 divides by the total weight. That the mean centre minimises the weighted sum of squared distances is proved in Lean: see proof("cgstat").

Usage

cgstat(x, y = NULL, w = NULL, k = 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.

w

NULL for equal weights, or one non-negative weight per point (counts at a location, say), with a positive sum.

k

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

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 named numeric vector: w (the total weight), mx, my (the mean centre, in degrees when lonlat = TRUE), sx, sy (standard deviations of the coordinates), sdd (the standard distance deviation), gx, gy (geometric means) and hx, hy (harmonic means). The last four are NaN for a coordinate with a non-positive value under positive weight, and for lonlat = TRUE (they depend on the origin of the coordinates). Lengths are in the unit of the coordinates, or metres.

References

Bachi, R. (1957). Statistical analysis of geographical series. Bulletin de l'Institut International de Statistique, 36(2), 229-240.

Neft, D. S. (1966). Statistical Analysis for Areal Distributions. Regional Science Research Institute, Philadelphia.

Examples

set.seed(1)
p <- cbind(x = rnorm(50, 100, 10), y = rnorm(50, 200, 5))
cgstat(p)
#>          w         mx         my         sx         sy        sdd         gx 
#>  50.000000 101.004483 200.586632   8.313939   4.844140   9.622228 100.653418 
#>         gy         hx         hy 
#> 200.529543 100.284986 200.472685 
cgstat(p[, 1], p[, 2], w = rep(1:2, 25))
#>          w         mx         my         sx         sy        sdd         gx 
#>  75.000000 100.685776 200.457192   8.553285   4.753294   9.785321 100.307901 
#>         gy         hx         hy 
#> 200.401917  99.908680 200.346969 
cgstat(p, k = 0)  # dispersions over n, not n - 1
#>          w         mx         my         sx         sy        sdd         gx 
#>  50.000000 101.004483 200.586632   8.230380   4.795453   9.525520 100.653418 
#>         gy         hx         hy 
#> 200.529543 100.284986 200.472685