In friendly/gellipsoid: Generalized Ellipsoids

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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 $d$-dimensional space $\mathbf{R}^d$, 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 $\mathbf{C}$ or its inverse $\mathbf{C}^{-1}$ 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 .

The implementation uses a $(\mathbf{U}, \mathbf{D})$ representation, based on the singular value decomposition (SVD) of an ellipsoid-generating matrix, $\mathbf{A} = \mathbf{U} \mathbf{D} \mathbf{V}^{T}$, where $\mathbf{U}$ is square orthogonal and $\mathbf{D}$ is diagonal.

For the usual, "proper" ellipsoids, $\mathbf{A}$ is positive-definite so all elements of $\mathbf{D}$ are positive. In generalized ellipsoids, $\mathbf{D}$ is extended to non-negative real numbers, i.e. its elements can be 0, Inf or a positive real.

Definitions

A proper ellipsoid in $\mathbf{R}^d$ can be defined by $\mathbf{E} := {x \; : \; x^T \mathbf{C} x \le 1 }$ where $\mathbf{C}$ is a non-negative definite central matrix, In applications, $\mathbf{C}$ 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 $\mathbf{C}$ is singular or when there are one or more zero singular values in $\mathbf{D}$. In 3D, a generalized ellipsoid that is flat in one dimension ($\mathbf{D} = \mathrm{diag} {X, X, 0}$) collapses to an ellipse; one that is flat in two dimensions ($\mathbf{D} = \mathrm{diag} {X, 0, 0}$) 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 $\mathbf{D}$. 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 $(\mathbf{U}, \mathbf{D})$ 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 gellipsoid package from CRAN as follows:

install.packages("gellipsoid")

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

# install.packages("remotes")
remotes::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 $R^3$ has a central matrix of the identity matrix.

library(gellipsoid)
(zsph <- gell(Sigma = diag(3)))  # a unit sphere in R^3
signature(zsph)
isBounded(zsph)
isFlat(zsph)

A plane in $R^3$ is flat in one dimension.

(zplane <- gell(span = diag(3)[, 1:2]))  # a plane
signature(zplane)
isBounded(zplane)
isFlat(zplane)

dual(zplane)  # line orthogonal to that plane
signature(dual(zplane))

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
signature(zhplane)

dual(zhplane)  # orthogonal line through same center

A point:

zorigin <- gell(span = cbind(c(0, 0, 0)))
signature(zorigin)

# what is the dual (inverse) of a point?
dual(zorigin)

signature(dual(zorigin))

Drawing generalized ellipsoids

The following figure shows views of two generalized ellipsoids. $C_1$ (blue) determines a proper, fat ellipsoid; it's inverse $C_1^{-1}$ also generates a proper ellipsoid. $C_2$ (red) determines an improper, flat ellipsoid, whose inverse $C_2^{-1}$ is an unbounded cylinder of elliptical cross-section. $C_2$ is the projection of $C_1$ onto the plane where $z = 0$. The scale of these ellipsoids is defined by the gray unit sphere.

knitr::include_graphics("man/figures/gell3d-1.png")

This figure illustrates the orthogonality of each $C$ and its dual, $C^{-1}$.

knitr::include_graphics("man/figures/gell3d-4.png")

References

Friendly, M., Monette, G. and Fox, J. (2013). Elliptical Insights: Understanding Statistical Methods through Elliptical Geometry. Statistical Science, 28(1), 1–39. Online paper; DOI

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

friendly/gellipsoid documentation built on Nov. 15, 2024, 1:16 p.m.