pk.test: Poisson kernel-based quadratic distance test of Uniformity on...

In QuadratiK: Collection of Methods Constructed using Kernel-Based Quadratic Distances

Poisson kernel-based quadratic distance test of Uniformity on the sphere

Description

This function performs the kernel-based quadratic distance goodness-of-fit tests for Uniformity for spherical data x using the Poisson kernel with concentration parameter rho.
The Poisson kernel-based test for uniformity exhibits excellent results especially in the case of multimodal distributions, as shown in the example of the Uniformity test on the Sphere vignette.

Usage

pk.test(x, rho, B = 300, Quantile = 0.95)

## S4 method for signature 'ANY'
pk.test(x, rho, B = 300, Quantile = 0.95)

## S4 method for signature 'pk.test'
show(object)

Arguments

x	A numeric d-dim matrix of data points on the Sphere S^(d-1).
rho	Concentration parameter of the Poisson kernel function.
B	Number of Monte Carlo iterations for critical value estimation of Un (default: 300).
Quantile	The quantile to use for critical value estimation, 0.95 is the default value.
object	Object of class pk.test

Details

Let x_1, x_2, ..., x_n be a random sample with empirical distribution function \hat F. We test the null hypothesis of uniformity on the d-dimensional sphere, i.e. H_0:F=G, where G is the uniform distribution on the d-dimensional sphere \mathcal{S}^{d-1}. We compute the U-statistic estimate of the sample KBQD (Kernel-Based Quadratic Distance)

U_{n}=\frac{1}{n(n-1)}\sum_{i=2}^{n}\sum_{j=1}^{i-1}K_{cen} (\mathbf{x}_{i}, \mathbf{x}_{j}),

then the first test statistic is given as

T_{n}=\frac{U_{n}}{\sqrt{Var(U_{n})}},

with

Var(U_{n})= \frac{2}{n(n-1)} \left[\frac{1+\rho^{2}}{(1-\rho^{2})^{d-1}}-1\right],

and the V-statistic estimate of the KBQD

V_{n} = \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}K_{cen} (\mathbf{x}_{i}, \mathbf{x}_{j}),

where K_{cen} denotes the Poisson kernel K_\rho centered with respect to the uniform distribution on the d-dimensional sphere, that is

K_{cen}(\mathbf{u}, \mathbf{v}) = K_\rho(\mathbf{u}, \mathbf{v}) -1

and

K_\rho(\mathbf{u}, \mathbf{v}) = \frac{1-\rho^{2}}{\left(1+\rho^{2}- 2\rho (\mathbf{u}\cdot \mathbf{v})\right)^{d/2}},

for every \mathbf{u}, \mathbf{v} \in \mathcal{S}^{d-1} \times \mathcal{S}^{d-1}.

The asymptotic distribution of the V-statistic is an infinite combination of weighted independent chi-squared random variables with one degree of freedom. The cutoff value is obtained using the Satterthwaite approximation c \cdot \chi_{DOF}^2, where

c=\frac{(1+\rho^{2})- (1-\rho^{2})^{d-1}}{(1+\rho)^{d}-(1-\rho^{2})^{d-1}}

and

DOF(K_{cen} )=\left(\frac{1+\rho}{1-\rho} \right)^{d-1}\left\{ \frac{\left(1+\rho-(1-\rho)^{d-1} \right )^{2}} {1+\rho^{2}-(1-\rho^{2})^{d-1}}\right \}.

. For the U-statistic the cutoff is determined empirically:

• Generate data from a Uniform distribution on the d-dimensional sphere;

• Compute the test statistics for B Monte Carlo(MC) replications;

• Compute the 95th quantile of the empirical distribution of the test statistic.

Value

An S4 object of class pk.test containing the results of the Poisson kernel-based tests. The object contains the following slots:

• method: Description of the test performed.

• x Data matrix.

• Un The value of the U-statistic.

• CV_Un The empirical critical value for Un.

• H0_Vn A logical value indicating whether or not the null hypothesis is rejected according to Un.

• Vn The value of the V-statistic Vn.

• CV_Vn The critical value for Vn computed following the asymptotic distribution.

• H0_Vn A logical value indicating whether or not the null hypothesis is rejected according to Vn.

• rho The value of concentration parameter used for the Poisson kernel function.

• B Number of replications for the critical value of the U-statistic Un.

Note

A U-statistic is a type of statistic that is used to estimate a population parameter. It is based on the idea of averaging over all possible distinct combinations of a fixed size from a sample. A V-statistic considers all possible tuples of a certain size, not just distinct combinations and can be used in contexts where unbiasedness is not required.

References

Ding, Y., Markatou, M. and Saraceno, G. (2023). “Poisson Kernel-Based Tests for Uniformity on the d-Dimensional Sphere.” Statistica Sinica. doi:10.5705/ss.202022.0347

See Also

pk.test

Examples

# create a pk.test object
x_sp <- sample_hypersphere(3, n_points=100)
unif_test <- pk.test(x_sp,rho=0.8)
unif_test

QuadratiK documentation built on Oct. 29, 2024, 5:08 p.m.