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.
gell(x, ...)# S3 method for default
gell(x, center = 0, Sigma, ip, span, A, u, d = 1, epsfac = 2, ...)
# S3 method for gell
gell(x, ...)
A (U, D) representation of the ellipsoid, with components
center
Right singular vectors
Singular values
An object
Other arguments
A vector specifying the center of the ellipsoid
A square, symmetric, non-negative definite dispersion (variance) matrix
A square, symmetric, non-negative definite inner product matrix. See Details.
A subspace with a given span. See Details.
A matrix giving a linear transformation of the unit sphere.
A U matrix
Diagonal elements of a D matrix
Factor of .Machine$double.eps used to distinguish zero
vs. positive singular values
Georges Monette
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:
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.
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.
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.
From a transformation of the unit sphere given by A(Unit sphere) where A = UDV', i.e. the SVD.
(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.
Friendly, M., Monette, G. and Fox, J. (2013). Elliptical Insights: Understanding Statistical Methods through Elliptical Geometry. Statistical Science, 28(1), 1-39.
dual, gmult, signature,
gell(Sigma = diag(3)) # the unit sphere
(zplane <- gell(span = diag(3)[,1:2])) # a plane
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