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 )
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. |
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.
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
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)