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

This function performs the kernel-based quadratic distance goodness-of-fit tests for Uniformity for multivariate spherical data x on $\mathcal{S}^{d-1}$ 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 ANY
pk.test(x, rho, B = 300, Quantile = 0.95)

# S4 method for pk.test
show(object)

Arguments

x: A numeric $(n \times d)$-matrix of $n$ data points on the Sphere $\mathcal{S}^(d-1)$ as rows.
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

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.

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 $(d-1)$-dimensional sphere, i.e. $H_0:F=G$, where $G$ is the uniform distribution on the $(d-1)$-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 $(d-1)$-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.

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

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
#> 
#>  Poisson Kernel-based quadratic distance test of 
#>                         Uniformity on the Sphere 
#> Selected consentration parameter rho:  0.8 
#> 
#> U-statistic:
#> 
#> H0 is rejected:  FALSE 
#> Statistic Un:  0.4735996 
#> Critical value:  1.543174 
#> 
#> V-statistic:
#> 
#> H0 is rejected:  FALSE 
#> Statistic Vn:  46.27503 
#> Critical value:  52.23077 
#>