Geometric Median

Compute the “geometric median” of points in n-dimensional space, that is the point with the least sum of (Euclidean) distances to all these points.

geo_median(P, tol = 1e-07, maxiter = 200)

Arguments

P

matrix of points, x_i-coordinates in the ith column.

tol

relative tolerance.

maxiter

maximum number of iterations.

Details

The task is solved applying an iterative process, known as Weiszfeld's algorithm. The solution is unique whenever the points are not collinear.

If the dimension is 1 (one column), the median will be returned.

Value

Returns a list with components p the coordinates of the solution point, d the sum of distances to all the sample points, reltol the relative tolerance of the iterative process, and niter the number of iterations.

References

See Wikipedia's entry on “Geometric median”.

Note

This is also known as the “1-median problem” and can be generalized to the “k-median problem” for k cluster centers; see kcca in the `flexclust' package.

See also

L1linreg

Examples

# Generate 100 points on the unit sphere in the 10-dim. space
set.seed(1001)
P <- rands(n=100, N=9)
( sol <- geo_median(P) )
#> $p
#>  [1] -0.009481361 -0.007643410 -0.001252910  0.006437703 -0.019982885
#>  [6] -0.045337987  0.036249563  0.003232175  0.035040592  0.046713023
#> 
#> $d
#> [1] 99.6638
#> 
#> $reltol
#> [1] 3.069063e-08
#> 
#> $niter
#> [1] 10
#> 
# $p
#  [1] -0.009481361 -0.007643410 -0.001252910  0.006437703 -0.019982885 -0.045337987
#  [7]  0.036249563  0.003232175  0.035040592  0.046713023
# $d
# [1] 99.6638
# $reltol
# [1] 3.069063e-08
# $niter
# [1] 10