gell: (U, D) Representation of an Ellipsoid in R^p.

In friendly/gellipsoid: Generalized Ellipsoids

(U, D) Representation of an Ellipsoid in R^p.

Description

gell provides a set of ways to specify a generalized ellipsoid in R^p, using the (U, D) representation to include all special cases, where U is a square orthogonal matrix, and D is diagonal with extended non-negative real numbers, i.e. 0, Inf or a positive real.

Usage

gell(x, ...)

## Default S3 method:
gell(x, center = 0, Sigma, ip, span, A, u, d = 1, epsfac = 2, ...)

## S3 method for class 'gell'
gell(x, ...)

Arguments

x	An object
...	Other arguments
center	A vector specifying the center of the ellipsoid
Sigma	A square, symmetric, non-negative definite dispersion (variance) matrix
ip	A square, symmetric, non-negative definite inner product matrix. See Details.
span	A subspace with a given span. See Details.
A	A matrix giving a linear transformation of the unit sphere.
u	A U matrix
d	Diagonal elements of a D matrix
epsfac	Factor of .Machine$double.eps used to distinguish zero vs. positive singular values

Details

The resulting class of ellipsoids includes degenerate ellipsoids that are flat and/or unbounded. Thus ellipsoids are naturally defined to include lines, hyperplanes, points, cylinders, etc.

gell can currently generate the (U, D) representation from 5 ways of specifying an ellipsoid:

1. From the non-negative definite dispersion (variance) matrix, Sigma: U D^2 U' = Sigma, where some elements of the diagonal matrix D can be 0. This can only generate bounded ellipsoids, possibly flat.

2. From the non-negative definite inner product matrix 'ip': U W^2 U = C where some elements of the diagonal matrix W can be 0. Then set D = W^-1 where 0^-1 = Inf. This can only generate fat (non-empty interior) ellipsoids, possibly unbounded.

3. From a subspace spanned by 'span' Let U_1 be an orthonormal basis of Span('span'), let U_2 be an orthonormal basis of the orthogonal complement, the U = [ U_1 U_2 ] and D = diag( c(Inf,...,Inf, 0,..,0)) where the number of Inf's is equal to the number of columns of U_1.

4. From a transformation of the unit sphere given by A(Unit sphere) where A = UDV', i.e. the SVD.

5. (Generalization of 4): A, d where A is any matrix and d is a vector of factors corresponding to columns of A. These factors can be 0, positive or Inf. In this case U and D are such that U D(Unit sphere) = A diag(d)(Unit sphere). This is the only representation that can be used for all forms of ellipsoids and in which any ellipsoid can be represented.

Value

A (U, D) representation of the ellipsoid, with components

center	center
u	Right singular vectors
d	Singular values

Author(s)

Georges Monette

References

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

See Also

dual, gmult, signature,

Examples

gell(Sigma = diag(3))    # the unit sphere

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

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