L1median {robustX}R Documentation

Compute the Multivariate L1-Median

Description

Compute the multivariate L1-median m, i.e., the minimizer of

sum(i = 1:n; || x[i] - m ||),

where || u || = sqrt(sum(j = 1:p; u[j]^2)).

As a convex problem, there's always a global minimizer, computable not by a closed formula but rather an iterative search. As the (partial) first derivatives of the objective function is undefined the data points, the minimization is not entirely trivial.

Usage

L1median(X, m.init = apply(X, 2, median), weights = NULL,
        method = c("nlm", "HoCrJo", "VardiZhang", optimMethods, nlminbMethods),
        pscale = apply(abs(centr(X, m.init)), 2, mean, trim = 0.40),
        tol = 1e-08, maxit = 200, trace = FALSE,
        zero.tol = 1e-15, ...)

Arguments

X numeric matrix of dimension n x p, say.
m.init starting value for m; typically and by default the coordinatewise median.
weights optional numeric vector of non-negative weights; currently only implemented for method "VardiZhang".
method character string specifying the computational method, i.e., the algorithm to be used (can be abbreviated).
pscale numeric p-vector of positive numbers, the coordinate-wise scale (typical size of delta(m[j])), where m is the problem's solution.
tol positive number specifying the (relative) convergence tolerance.
maxit positive integer specifying the maximal number of iterations (before the iterations are stopped prematurely if necessary).
trace an integer specifying the tracing level of the iterations; 0 does no tracing
zero.tol for method "VardiZhang", a small positive number specifying the tolerance for determining that the iteration is ‘exactly’ at a data point (which is singularity).
... optional arguments to nlm() or the control (list) arguments of optim(), or nlminb(), respectively.

Details

Currently, we have to refer to the “References” below.

Value

currently the result depends strongly on the method used.
FIXME. This will change considerably.

Author(s)

Martin Maechler. Method "HoCrJo" is mostly based on Kristel Joossens' function, implementing Hossjer and Croux (1995).

References

Hossjer and Croux, C. (1995). Generalizing Univariate Signed Rank Statistics for Testing and Estimating a Multivariate Location Parameter. Non-parametric Statistics 4, 293–308.

Vardi, Y. and Zhang, C.-H. (2000). The multivariate L_1-median and associated data depth. Proc. National Academy of Science 97(4), 1423–1426.

See Also

median, covMcd

Examples

data(stackloss)
L1median(stackloss)
L1median(stackloss, method = "HoCrJo")

[Package robustX version 1.1-2 Index]