sphunif (version 1.4.3)

F_from_f: Distribution and quantile functions from angular function

Description

Numerical computation of the distribution function \(F\) and the quantile function \(F^{-1}\) for an angular function \(f\) in a tangent-normal decomposition. \(F^{-1}(x)\) results from the inversion of $$F(x) = \int_{-1}^x \omega_{p - 1}c_f f(z) (1 - z^2)^{(p - 3) / 2} \,\mathrm{d}z$$ for \(x\in [-1, 1]\), where \(c_f\) is a normalizing constant and \(\omega_{p - 1}\) is the surface area of \(S^{p - 2}\).

Usage

F_from_f(f, p, Gauss = TRUE, N = 320, K = 1000, tol = 1e-06, ...)
F_inv_from_f(f, p, Gauss = TRUE, N = 320, K = 1000, tol = 1e-06, ...)

Value

A splinefun object ready to evaluate \(F\) or \(F^{-1}\), as specified.

Arguments

f

angular function defined on \([-1, 1]\). Must be vectorized.

p

integer giving the dimension of the ambient space \(R^p\) that contains \(S^{p-1}\).

Gauss

use a Gauss--Legendre quadrature rule to integrate \(f\) with N nodes? Otherwise, rely on integrate Defaults to TRUE.

N

number of points used in the Gauss--Legendre quadrature. Defaults to 320.

K

number of equispaced points on \([-1, 1]\) used for evaluating \(F^{-1}\) and then interpolating. Defaults to 1e3.

tol

tolerance passed to uniroot for the inversion of \(F\). Also, passed to integrate's rel.tol and abs.tol if Gauss = FALSE. Defaults to 1e-6.

...

further parameters passed to f.

Details

The normalizing constant \(c_f\) is such that \(F(1) = 1\). It does not need to be part of f as it is computed internally.

Interpolation is performed by a monotone cubic spline. Gauss = TRUE yields more accurate results, at expenses of a heavier computation.

If f yields negative values, these are silently truncated to zero.

Examples

f <- function(x) rep(1, length(x))
plot(F_from_f(f = f, p = 4, Gauss = TRUE), ylab = "F(x)", xlim = c(-1, 1))
plot(F_from_f(f = f, p = 4, Gauss = FALSE), col = 2, add = TRUE,
     xlim = c(-1, 1))
curve(p_proj_unif(x = x, p = 4), col = 3, add = TRUE, n = 300)
plot(F_inv_from_f(f = f, p = 4, Gauss = TRUE), ylab = "F^{-1}(x)")
plot(F_inv_from_f(f = f, p = 4, Gauss = FALSE), col = 2, add = TRUE)
curve(q_proj_unif(u = x, p = 4), col = 3, add = TRUE, n = 300)