ellipsoidhull: 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)
"print"(x, digits = max(1, getOption("digits") - 2), ...)

Arguments

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.

Value

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

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 MASS package.

References

Pison, G., Struyf, A. and Rousseeuw, P.J. (1999) Displaying a Clustering with CLUSPLOT, Computational Statistics and Data Analysis, 30, 381--392.

D.M. 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

Run this code

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

plot(xy, main = "ellipsoidhull(<Gauss data>) -- 'spanning points'")
lines(predict(exy), col="blue")
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  "spanning ellipsoid" points
round(exy$wt[ii],3); sum(exy$wt[ii]) # weights summing to 1
points(xy[ii,], pch = 21, cex = 2,
       col="blue", bg = adjustcolor("blue",0.25))

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