dpkb {QuadratiK}

The Poisson kernel-based Distribution (PKBD)

Description

Density function and random number generation from the Poisson kernel-based Distribution with mean direction vector mu and concentration parameter rho.

Usage

dpkb(x, mu, rho, logdens = FALSE)

rpkb(
  n,
  mu,
  rho,
  method = "rejvmf",
  tol.eps = .Machine$double.eps^0.25,
  max.iter = 1000
)

Arguments

x

Matrix (or data.frame) with number of columns >=2.

mu

location vector parameter with length indicating the dimension of generated points.

rho

is the concentration parameter, with 0 <= rho < 1.

logdens

Logical; if 'TRUE', densities d are given as log(d).

n

number of observations.

method

string that indicates the method used for sampling observations. The available methods are

• 'rejvmf' acceptance-rejection algorithm using von Mises-Fisher envelops (Algorithm in Table 2 of Golzy and Markatou 2020);

• 'rejacg' using angular central Gaussian envelops (Algorithm in Table 1 of Sablica et al. 2023);

• 'rejpsaw' using projected Saw distributions (Algorithm in Table 2 of Sablica et al. 2023).

tol.eps

the desired accuracy of convergence tolerance (for 'rejacg' method).

max.iter

the maximum number of iterations (for 'rejacg' method).

Details

If the chosen method is 'rejacg', the function uniroot, from the stat package, is used to estimate the beta parameter. In this case, the complete results are provided as output.

Value

dpkb gives the density value. rpkb generates random observations from the PKBD.

The number of observations generated is determined by n for rpkb. This function returns a list with the matrix of generated observations x, the number of tries numTries and the number of acceptances numAccepted.

References

Golzy, M., Markatou, M. (2020) Poisson Kernel-Based Clustering on the Sphere: Convergence Properties, Identifiability, and a Method of Sampling, Journal of Computational and Graphical Statistics, 29:4, 758-770, DOI: 10.1080/10618600.2020.1740713.

Sablica L., Hornik K., Leydold J. (2023) "Efficient sampling from the PKBD distribution", Electronic Journal of Statistics, 17(2), 2180-2209.

Examples

# Generate some data from pkbd density
pkbd_dat <- rpkb(10, c(0.5,0), 0.5)

# Calculate the PKBD density values
dens_val <- dpkb(pkbd_dat$x, c(0.5,0.5),0.5)