F_from_f function

Distribution and quantile functions from angular function

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 [REMOVE_ME]$F(x) = \int_{-1}^x \omega_{p - 1}c_f f(z) (1 - z^2)^{(p - 3) / 2}\,\mathrm{d}zF(x) = \int_{-1}^x \omega_{p - 1}c_f f(z)(1 - z^2)^{(p - 3) / 2} dz [REMOVE_ME_2]$

for $x\in [-1, 1]$, where $c_f$ is a normalizing constant and $\omega_{p - 1}$ is the surface area of $S^{p - 2}$.

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, ...)

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.

Returns

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

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

for $x\in [-1, 1]$, where $c_f$ is a normalizing constant and $\omega_{p - 1}$ is the surface area of $S^{p - 2}$.

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)