bw_cv_polysph: Cross-validation bandwidth selection for polyspherical kernel density estimator

Description

Likelihood Cross-Validation (LCV) and Least Squares Cross-Validation (LSCV) bandwidth selection for the polyspherical kernel density estimator.

Usage

bw_cv_polysph(X, d, kernel = 1, kernel_type = 1, k = 10,
  intrinsic = FALSE, type = c("LCV", "LSCV")[1], M = 10000, bw0 = NULL,
  na.rm = FALSE, h_min = 0, upscale = FALSE, deriv = 0,
  imp_mc = TRUE, seed_mc = NULL, exact_vmf = FALSE, common_h = FALSE,
  spline = FALSE, opt = c("optim", "nlm")[1], ncores = 1, ...)

Value

A list with entries bw (optimal bandwidth) and opt, the latter containing the output of nlm, optim, or optimParallel.

Arguments

X: a matrix of size c(n, sum(d) + r) with the sample.
d: vector of size r with dimensions.
kernel: kernel employed: 1 for von Mises--Fisher (default); 2 for Epanechnikov; 3 for softplus.
kernel_type: type of kernel employed: 1 for product kernel (default); 2 for spherically symmetric kernel.
k: softplus kernel parameter. Defaults to 10.0.
intrinsic: use the intrinsic distance, instead of the extrinsic-chordal distance, in the kernel? Defaults to FALSE.
type: cross-validation type, either "LCV" (default) or "LSCV".
M: Monte Carlo samples to use for approximating the integral in the LSCV loss.
bw0: initial bandwidth vector for minimizing the CV loss. If NULL, it is computed internally by magnifying the bw_rot_polysph bandwidths by 50%. Can be also a matrix of initial bandwidth vectors.
na.rm: remove NAs in the objective function? Defaults to FALSE.
h_min: minimum h enforced (componentwise). Defaults to 0.
upscale: rescale the resulting bandwidths to work for derivative estimation? Defaults to FALSE.
deriv: derivative order to perform the upscaling. Defaults to 0.
imp_mc: use importance sampling in the Monte Carlo approximation of the integral in the LSCV loss? It is more accurate but also more time consuming. Defaults to TRUE.
seed_mc: seed for the Monte Carlo simulations used to estimate the integral in the LSCV loss. Defaults to NULL (no seed is fixed for different bandwidths).
exact_vmf: use the closed-form for the LSCV loss with the von Mises--Fisher kernel? Defaults to FALSE.
common_h: use the same bandwidth for all dimensions? Defaults to FALSE.
spline: use a faster spline approximation to compute Bessel functions? Defaults to FALSE.
opt: optimizer to use; either "optim" (default) or "nlm".
ncores: number of cores used during the optimization. Defaults to 1.
...: further arguments passed to optim or nlm (if ncores = 1) or optimParallel (if ncores > 1).

Details

If bw0 is a matrix, then the optimization is started at that row of bandwidths that is most promising for the optimization, i.e., the bandwidths that minimized the CV loss.

Examples

Run this code

n <- 20
d <- 1:2
kappa <- rep(10, 2)
X <- r_vmf_polysph(n = n, d = d, mu = r_unif_polysph(n = 1, d = d),
                   kappa = kappa)
bw_cv_polysph(X = X, d = d, type = "LCV")$bw
bw_cv_polysph(X = X, d = d, type = "LSCV", exact_vmf = TRUE)$bw

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