Compute the Ellipsoid Hull or Spanning Ellipsoid of a Point Set
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.
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), ...)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 |
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, |
digits,... |
the usual arguments to |
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.
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. Further, d2 = t^2, where
t is “the value of a t-statistic on the ellipse
boundary” (from |
wt |
the vector of weights iff |
sqdist |
the vector of squared distances iff |
prob |
the vector of algorithm probabilities iff |
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; |
conv |
logical indicating if the converged. This is defined as
|
Martin Maechler did the present class implementation; Rousseeuw et al did the underlying original code.
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.
predict.ellipsoid which is also the
predict method for ellipsoid objects.
volume.ellipsoid for an example of ‘manual’
ellipsoid object construction;
further ellipse from package ellipse
and ellipsePoints from package sfsmisc.
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))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.