These functions create rejection samplers, and uniform manifold samplers based on them, using user-provided parameterization and Jacobian functions.
make_rejection_sampler(
parameterization,
jacobian,
min_params,
max_params,
max_jacobian
)Arguments
- parameterization
A function that takes parameter vector arguments and returns a matrix of coordinates.
- jacobian
A function that takes parameter vector arguments and returns a vector of Jacobian determinants.
- min_params, max_params
(Optionally named) vectors of minimum and maximum values of the parameters, used for uniform sampling.
- max_jacobian
An (ideally sharp) upper bound on the Jacobian determinant.
Details
The rejection sampling technique of Diaconis, Holmes, and
Shahshahani (2013) uses a parameterized embedding from a parameter space to
a coordinate space and relies on a formula for its jacobian determinant.
The parameterization must be a function that takes vector arguments of
equal length and returns a coordinate matrix of the same number of rows.
The jacobian must be a function that takes the same arguments and returns
a scalar value. The parameters must range from their respective minima
min_params to their respective maxima max_params. max_jacobian must
be provided, though it may be larger than the theoretical maximum of the
jacobian determinant.
References
P Diaconis, S Holmes, and M Shahshahani (2013) Sampling from a Manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton, 102--125. doi:10.1214/12-IMSCOLL1006
Examples
set.seed(47569L)
# parameterization and Jacobian for Klein bottle tube embedding
klein_parameterization <- function(theta, phi) {
cbind(
w = (1 + .5 * cos(theta)) * cos(phi),
x = (1 + .5 * cos(theta)) * sin(phi),
y = .5 * sin(theta) * cos(phi/2),
z = .5 * sin(theta) * sin(phi/2)
)
}
klein_jacobian <- function(theta, phi) {
unname(.5 * sqrt((1 + .5 * cos(theta)) ^ 2 + (.5 * .5 * sin(theta)) ^ 2))
}
# custom sampler based on these functions
klein_sampler <- make_rejection_sampler(
klein_parameterization,
klein_jacobian,
max_params = c(theta = 2*pi, phi = 2*pi),
max_jacobian = klein_jacobian(cbind(theta = 0))
)
# compare custom sampler to `sample_klein_tube()`
pairs(klein_sampler(n = 360), asp = 1, pch = 19, cex = .5)
pairs(sample_klein_tube(n = 360, ar = 2), asp = 1, pch = 19, cex = .5)
