gellipsoid: Generalized Ellipsoids

The gellipsoid package extends the class of geometric ellipsoids to “generalized ellipsoids”, which allow degenerate ellipsoids that are flat and/or unbounded. Thus, ellipsoids can be naturally defined to include lines, hyperplanes, points, cylinders, etc. The methods can be used to represent generalized ellipsoids in a -dimensional space , with plots in up to 3D.

The goal is to be able to think about, visualize, and compute a linear transformation of an ellipsoid with central matrix or its inverse which apply equally to unbounded and/or degenerate ellipsoids. This permits exploration of a variety to statistical issues that can be visualized using ellipsoids as discussed by Friendly, Fox & Monette (2013), Elliptical Insights: Understanding Statistical Methods Through Elliptical Geometry doi:10.1214/12-STS402.

The implementation uses a representation, based on the singular value decomposition (SVD) of an ellipsoid-generating matrix, , where is square orthogonal and is diagonal.

For the usual, “proper” ellipsoids, is positive-definite so all elements of are positive. In generalized ellipsoids, is extended to non-negative real numbers, i.e.  its elements can be 0, Inf or a positive real.

Definitions

A proper ellipsoid in can be defined by where is a non-negative definite central matrix, In applications, is typically a variance-covariance matrix A proper ellipsoid is bounded, with a non-empty interior. We call these fat ellipsoids.

A degenerate flat ellipsoid corresponds to one where the central matrix is singular or when there are one or more zero singular values in . In 3D, a generalized ellipsoid that is flat in one dimension () collapses to an ellipse; one that is flat in two dimensions () collapses to a line, and one that is flat in three dimensions collapses to a point.

An unbounded ellipsoid is one that has infinite extent in one or more directions, and is characterized by infinite singular values in . For example, in 3D, an unbounded ellipsoid with one infinite singular value is an infinite cylinder of elliptical cross-section.

Principal functions

• gell() Constructs a generalized ellipsoid using the representation. The inputs can be specified in a variety of ways:

  • a non-negative definite variance matrix;
  • an inner-product matrix
  • a subspace with a given span
  • a matrix giving a linear transformation of the unit sphere

• dual() calculates the dual or inverse of a generalized ellipsoid

• gmult() calculates a linear transformation of a generalized ellipsoid

• signature() calculates the signature of a generalized ellipsoid, a vector of length 3 giving the number of positive, zero and infinite singular values in the (U, D) representation.

• ell3d() Plots generalized ellipsoids in 3D using the rgl package

Installation

You can install the development version of gellipsoid from GitHub with:

# install.packages("devtools")
devtools::install_github("friendly/gellipsoid")

Example

Properties of generalized ellipsoids

The following examples illustrate gell objects and their properties. Each of these may be plotted in 3D using ell3d(). These objects can be specified in a variety of ways, but for these examples the span is simplest.

A unit sphere in has a central matrix of the identity matrix.

library(gellipsoid)
(zsph <- gell(Sigma = diag(3)))  # a unit sphere in R^3
#> $center
#> [1] 0 0 0
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    0    0    1
#> [2,]    0    1    0
#> [3,]    1    0    0
#> 
#> $d
#> [1] 1 1 1
#> 
#> attr(,"class")
#> [1] "gell"
signature(zsph)
#>  pos zero  inf 
#>    3    0    0
isBounded(zsph)
#> [1] TRUE
isFlat(zsph)
#> [1] FALSE

A plane in is flat in one dimension.

(zplane <- gell(span = diag(3)[, 1:2]))  # a plane
#> $center
#> [1] 0 0 0
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
#> 
#> $d
#> [1] Inf Inf   0
#> 
#> attr(,"class")
#> [1] "gell"
signature(zplane)
#>  pos zero  inf 
#>    0    1    2
isBounded(zplane)
#> [1] FALSE
isFlat(zplane)
#> [1] TRUE

dual(zplane)  # line orthogonal to that plane
#> $center
#> [1] 0 0 0
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    0    0    1
#> [2,]    0    1    0
#> [3,]    1    0    0
#> 
#> $d
#> [1] Inf   0   0
#> 
#> attr(,"class")
#> [1] "gell"
signature(dual(zplane))
#>  pos zero  inf 
#>    0    2    1

A hyperplane. Note that the gell object with a center contains more information than the geometric plane.

(zhplane <- gell(center = c(0, 0, 2), 
                 span = diag(3)[, 1:2]))  # a hyperplane
#> $center
#> [1] 0 0 2
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
#> 
#> $d
#> [1] Inf Inf   0
#> 
#> attr(,"class")
#> [1] "gell"
signature(zhplane)
#>  pos zero  inf 
#>    0    1    2

dual(zhplane)  # orthogonal line through same center
#> $center
#> [1] 0 0 2
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    0    0    1
#> [2,]    0    1    0
#> [3,]    1    0    0
#> 
#> $d
#> [1] Inf   0   0
#> 
#> attr(,"class")
#> [1] "gell"

A point:

zorigin <- gell(span = cbind(c(0, 0, 0)))
signature(zorigin)
#>  pos zero  inf 
#>    0    3    0

# what is the dual (inverse) of a point?
dual(zorigin)
#> $center
#> [1] 0 0 0
#> 
#> $u
#>      [,1] [,2] [,3]
#> [1,]    0    0    1
#> [2,]    0    1    0
#> [3,]    1    0    0
#> 
#> $d
#> [1] Inf Inf Inf
#> 
#> attr(,"class")
#> [1] "gell"

signature(dual(zorigin))
#>  pos zero  inf 
#>    0    0    3

Drawing generalized ellipsoids

The following figure shows views of two generalized ellipsoids. (blue) determines a proper, fat ellipsoid; it’s inverse also generates a proper ellipsoid. (red) determines an improper, flat ellipsoid, whose inverse is an unbounded cylinder of elliptical cross-section. is the projection of onto the plane where . The scale of these ellipsoids is defined by the gray unit sphere.

This figure illustrates the orthogonality of each and its dual, .

References

Friendly, M., Monette, G. and Fox, J. (2013). Elliptical Insights: Understanding Statistical Methods through Elliptical Geometry. Online paper

Friendly, M. (2013). Supplementary materials for “Elliptical Insights …”, https://www.datavis.ca/papers/ellipses/