The Poisson kernel-based Distribution (PKBD)

The Poisson kernel-based densities are based on the normalized Poisson kernel and are defined on the $(d-1)$-dimensional unit sphere. Given a vector $\mathbf{\mu} \in \mathcal{S}^{d-1}$, where $\mathcal{S}^{d-1}= \{x \in \mathbb{R}^d : ||x|| = 1\}$, and a parameter $\rho$ such that $0 < \rho < 1$, the probability density function of a $d$-variate Poisson kernel-based density is defined by: $$f(\mathbf{x}|\rho, \mathbf{\mu}) = \frac{1-\rho^2}{\omega_d ||\mathbf{x} - \rho \mathbf{\mu}||^d},$$ where $\mu$ is a vector orienting the center of the distribution, $\rho$ is a parameter to control the concentration of the distribution around the vector $\mu$ and it is related to the variance of the distribution. Recall that, for $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$, $||x|| = \sqrt{x_1^2 + \ldots + x_d^2}$. Furthermore, $\omega_d = 2\pi^{d/2} [\Gamma(d/2)]^{-1}$ is the surface area of the unit sphere in $\mathbb{R}^d$ (see Golzy and Markatou, 2020). When $\rho \to 0$, the Poisson kernel-based density tends to the uniform density on the sphere. Connections of the PKBDs to other distributions are discussed in detail in Golzy and Markatou (2020). Here we note that when $d=2$, PKBDs reduce to the wrapped Cauchy distribution. Additionally, with precise choice of the parameters $\rho$ and $\mu$ the two-dimensional PKBD becomes a two-dimensional projected normal distribution. However, the connection with the $d$-dimensional projected normal distributions does not carry beyond $d=2$. Golzy and Markatou (2020) proposed an acceptance-rejection method for simulating data from a PKBD using von Mises-Fisher envelops (rejvmf method). Furthermore Sablica, Hornik and Leydold (2023) proposed new ways for simulating from the PKBD, using angular central Gaussian envelops (rejacg) or using the projected Saw distributions (rejpsaw).

Usage

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

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

Arguments

x

$n \times d$-matrix (or data.frame) of $n$ data point on the sphere $\mathcal{S}^{d-1}$, with $d \ge 2$.

mu

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

rho

Concentration parameter, with $0 \le$ rho $< 1$.

logdens

Logical; if 'TRUE', densities are returned in logarithmic scale.

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).

Value

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

Details

This function dpkb() computes the density value for a given point x from the Poisson kernel-based distribution with mean direction vector mu and concentration parameter rho.

The number of observations generated is determined by n for rpkb(). This function returns the $(n \times d)$-matrix of generated $n$ observations on $\mathcal{S}^{(d-1)}$.

A limitation of the rejvmf is that the method does not ensure the computational feasibility of the sampler for $\rho$ approaching 1.

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.

Note

If the required packages (movMF for rejvmf method, and Tinflex for rejpsaw) are not installed, the function will display a message asking the user to install the missing package(s).

References

Golzy, M. and 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. and 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, c(0.5,0.5),0.5)