PKBD

class QuadratiK.spherical_clustering.PKBD

Class for estimating density and generating samples of Poisson-kernel based distribution (PKBD). Details on PKBD can be found in User Guide.

Methods

PKBD.dpkb(x, mu, rho[, logdens])	Computes the density value for a given point x from the Poisson kernel-based distribution with mean direction vector mu and concentration parameter rho.
PKBD.rpkb(n, mu, rho[, method, random_state])	Function for generating a random sample from PKBD.

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PKBD.dpkb(x: ndarray | DataFrame, mu: ndarray, rho: float, logdens: bool = False) → ndarray

Computes the density value for a given point x from the Poisson kernel-based distribution with mean direction vector mu and concentration parameter rho.

Parameters

xnumpy.ndarray, pandas.DataFrame

A matrix with a number of columns >= 2.

mufloat

Location parameter with the same length as the rows of x. Normalized to length one.

rhofloat

Concentration parameter. \(\rho \in (0,1]\).

logdensbool, optional

If True, densities d are given as \(\log(d)\). Defaults to False.

Returns

densitynumpy.ndarray

An array with the evaluated density values.

PKBD.rpkb(n: int, mu: ndarray | list[float], rho: float, method: str = 'rejvmf', random_state: int | None = None) → ndarray

Function for generating a random sample from PKBD. The number of observation generated is determined by n.

Parameters

nint

Sample size.

munp.ndarray, list

Location parameter with the same length as the quantiles.

rhofloat

Concentration parameter. \(\rho \in (0,1]\).

methodstr, optional

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);

Defaults to ‘rejvmf’.

random_stateint, None, optional.

Seed for random number generation. Defaults to None.

Returns

samplesnumpy.ndarray

Generated observations from a poisson kernel-based density. This function returns a matrix of generated observations.

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

from QuadratiK.spherical_clustering import PKBD
pkbd_data = PKBD().rpkb(10,[0.5,0],0.5, "rejvmf", random_state= 42)
dens_val  = PKBD().dpkb(pkbd_data, [0.5,0.5],0.5)
print(dens_val)

[0.46827108 0.05479605 0.21163936 0.06195099 0.39567698 0.40473724
 0.26561508 0.36791766 0.09324676 0.46847274]