R: Compute the Ellipsoid Hull or Spanning Ellipsoid of a Point Set

ellipsoidhull {cluster}

R Documentation

Compute the Ellipsoid Hull or Spanning Ellipsoid of a Point Set

Description

Compute the ``ellipsoid hull'' or ``spanning ellipsoid'', i.e. the ellipsoid of minimal volume (`area' in 2D) such that all given points lie just inside or on the boundary of the ellipsoid.

Usage

ellipsoidhull(x, tol=0.01, maxit=5000,
              ret.wt = FALSE, ret.sqdist = FALSE, ret.pr = FALSE)
## S3 method for class 'ellipsoid':
print(x, digits = max(1, getOption("digits") - 2), ...)

Arguments

`x`	the n p-dimensional points asnumeric n x p matrix.
`tol`	convergence tolerance for Titterington's algorithm. Setting this to much smaller values may drastically increase the number of iterations needed, and you may want to increas `maxit` as well.
`maxit`	integer giving the maximal number of iteration steps for the algorithm.
`ret.wt, ret.sqdist, ret.pr`	logicals indicating if additional information should be returned, `ret.wt` specifying the weights, `ret.sqdist` the squared distances and `ret.pr` the final probabilities in the algorithms.
`digits,...`	the usual arguments to `print` methods.

Details

The ``spanning ellipsoid'' algorithm is said to stem from Titterington(1976), in Pison et al(1999) who use it for clusplot.default.
The problem can be seen as a special case of the ``Min.Vol.'' ellipsoid of which a more more flexible and general implementation is cov.mve in the lqs package.

Value

an object of class "ellipsoid", basically a list with several components, comprising at least

`cov`	p x p covariance matrix description the ellipsoid.
`loc`	p-dimensional location of the ellipsoid center.
`d2`	average squared radius.
`wt`	the vector of weights iff `ret.wt` was true.
`sqdist`	the vector of squared distances iff `ret.sqdist` was true.
`prob`	the vector of algorithm probabilities iff `ret.pr` was true.
`it`	number of iterations used.
`tol, maxit`	just the input argument, see above.
`eps`	the achieved tolerance which is the maximal squared radius minus p.
`ierr`	error code as from the algorithm; `0` means ok.
`conv`	logical indicating if the converged. This is defined as `it < maxit && ierr == 0`.

Author(s)

Martin Maechler did the present class implementation; Rousseeuw et al did the underlying code.

References

Pison, G., Struyf, A. and Rousseeuw, P.J. (1999) Displaying a Clustering with CLUSPLOT, Computational Statistics and Data Analysis, 30, 381–392.
A version of this is available as technical report from http://win-www.uia.ac.be/u/statis/abstract/Disclu99.htm

D.N. Titterington. (1976) Algorithms for computing {D}-optimal design on finite design spaces. In Proc. of the 1976 Conf. on Information Science and Systems, 213–216; John Hopkins University.

Examples

x <- rnorm(100)
xy <- unname(cbind(x, rnorm(100) + 2*x + 10))
exy <- ellipsoidhull(xy)
exy # >> calling print.ellipsoid()

plot(xy)
lines(predict(exy))
points(rbind(exy$loc), col = "red", cex = 3, pch = 13)

exy <- ellipsoidhull(xy, tol = 1e-7, ret.wt = TRUE, ret.sq = TRUE)
str(exy) # had small `tol', hence many iterations
(ii <- which(zapsmall(exy $ wt) > 1e-6)) # only about 4 to 6 points
round(exy$wt[ii],3); sum(exy$wt[ii]) # sum to 1

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