kde_polysph: Polyspherical kernel density estimator

In polykde: Polyspherical Kernel Density Estimation

Polyspherical kernel density estimator

Description

Computes the kernel density estimator for data on the polysphere \mathbb{S}^{d_1} \times \cdots \times \mathbb{S}^{d_r}. Given a sample \boldsymbol{X}_1,\ldots,\boldsymbol{X}_n, this estimator is

\hat{f}(\boldsymbol{x};\boldsymbol{h})=\sum_{i=1}^n L_{\boldsymbol{h}}(\boldsymbol{x},\boldsymbol{X}_i)

for a kernel L and a vector of bandwidths \boldsymbol{h}.

Usage

kde_polysph(
  x,
  X,
  d,
  h,
  weights = as.numeric(c()),
  log = FALSE,
  wrt_unif = FALSE,
  normalized = TRUE,
  intrinsic = FALSE,
  norm_x = FALSE,
  norm_X = FALSE,
  kernel = 1L,
  kernel_type = 1L,
  k = 10,
  loo = FALSE
)

Arguments

x	a matrix of size c(nx, sum(d) + r) with the evaluation points.
X	a matrix of size c(n, sum(d) + r) with the sample.
d	vector of size r with dimensions.
h	vector of size r with bandwidths.
weights	weights for each observation. If provided, a vector of size n with the weights for multiplying each kernel. If not provided, set internally to rep(1 / n, n), which gives the standard estimator.
log	compute the logarithm of the density? Defaults to FALSE.
wrt_unif	flag to return a density with respect to the uniform measure. If FALSE (default), the density is with respect to the Lebesgue measure.
normalized	flag to compute the normalizing constant of the kernel and include it in the kernel density estimator. Defaults to TRUE.
intrinsic	use the intrinsic distance, instead of the extrinsic-chordal distance, in the kernel? Defaults to FALSE.
norm_x, norm_X	ensure a normalization of the data? Defaults to FALSE.
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.
loo	leave-one-out: skip the i-th observation when evaluating at \boldsymbol{X}_i? Requires x = X and n \ge 2. Defaults to FALSE.

Value

A column matrix of size c(nx, 1) with the evaluation of the kernel density estimator.

References

García-Portugués, E. and Meilán-Vila, A. (2026). Kernel density estimation with polyspherical data and its applications. Journal of the American Statistical Association, 121(553):427–439. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.2025.2521898")}

See Also

log_cv_kde_polysph, grad_hess_kde_polysph, r_kde_polysph, bw_rot_polysph, bw_cv_polysph, bw_mrot_polysph.

Examples

# Simple check on S^1 x S^2
n <- 1e3
d <- c(1, 2)
mu <- c(0, 1, 0, 0, 1)
kappa <- c(5, 5)
h <- c(0.2, 0.2)
X <- r_vmf_polysph(n = n, d = d, mu = mu, kappa = kappa)
kde_polysph(x = rbind(mu), X = X, d = d, h = h)
d_vmf_polysph(x = rbind(mu), d = d, mu = mu, kappa = kappa)

polykde documentation built on Nov. 13, 2026, 1:08 a.m.