# ellipsoidhull

##### 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.

##### Usage

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

##### Arguments

- x
the \(n\) \(p\)-dimensional points asnumeric \(n\times p\) 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`

theand**sq**uared**dist**ances`ret.pr`

the final**pr**obabilities 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 `MASS`

package.

##### Value

an object of class `"ellipsoid"`

, basically a `list`

with several components, comprising at least

\(p\times p\) *covariance* matrix description
the ellipsoid.

\(p\)-dimensional location of the ellipsoid center.

average squared radius. Further, \(d2 = t^2\), where
\(t\) is “the value of a t-statistic on the ellipse
boundary” (from `ellipse`

in the
ellipse package), and hence, more usefully,
`d2 = qchisq(alpha, df = p)`

, where `alpha`

is the
confidence level for p-variate normally distributed data with
location and covariance `loc`

and `cov`

to lie inside the
ellipsoid.

the vector of weights iff `ret.wt`

was true.

the vector of squared distances iff `ret.sqdist`

was true.

the vector of algorithm probabilities iff `ret.pr`

was true.

number of iterations used.

just the input argument, see above.

the achieved tolerance which is the maximal squared radius minus \(p\).

error code as from the algorithm; `0`

means *ok*.

logical indicating if the converged. This is defined as
`it < maxit && ierr == 0`

.

##### 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.

##### See Also

`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.

`chull`

for the convex hull,
`clusplot`

which makes use of this; `cov.mve`

.

##### Examples

```
# NOT RUN {
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))
# }
```

*Documentation reproduced from package cluster, version 2.1.0, License: GPL (>= 2)*