Centre of minimum distance

The point c that minimises the weighted sum of distances sum(w * |p - c|) to the points: the Weber point or spatial median, which, unlike the median centre, does not depend on the axes. It is found by Weiszfeld's iteration with the modification of Vardi and Zhang (2000), which keeps every step defined and the sum non-increasing when an iterate lands on a data point. The optimality gap certifies the answer: it is 0 exactly at a minimiser. The minimiser is unique unless all the points lie on one line with the weight split evenly.

Usage

webpt(
  x,
  y = NULL,
  w = NULL,
  start = NULL,
  tol = 1e-10,
  maxit = 1000,
  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.

start

NULL to start at the mean centre, or c(x, y) (in degrees when lonlat = TRUE).

tol

Stop when a step is at most tol times the root mean squared distance to the mean centre: one positive number.

maxit

The largest number of steps: 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 named numeric vector: x, y (the centre, in degrees when lonlat = TRUE), obj (the weighted sum of distances), gap (the optimality gap, 0 at a minimiser), iter (steps taken) and code: 2 at an exact minimiser, 1 when the step fell below tol, 0 when maxit was reached.

References

Weiszfeld, E. and Plastria, F. (2009). On the point for which the sum of the distances to n given points is minimum. Annals of Operations Research, 167, 7-41 (translation of the 1937 paper). doi:10.1007/s10479-008-0352-z

Vardi, Y. and Zhang, C.-H. (2000). The multivariate L1-median and associated data depth. Proceedings of the National Academy of Sciences, 97(4), 1423-1426. doi:10.1073/pnas.97.4.1423

Examples

webpt(c(0, 1, 0, 10), c(0, 0, 1, 10))
#>            x            y          obj          gap         iter         code 
#> 5.000000e-01 5.000000e-01 1.555635e+01 4.086973e-10 2.200000e+01 1.000000e+00