• Corpus ID: 88523748

An overview of uniformity tests on the hypersphere

@article{GarciaPortugues2018AnOO,
  title={An overview of uniformity tests on the hypersphere},
  author={Eduardo Garc'ia-Portugu'es and Thomas Verdebout},
  journal={arXiv: Methodology},
  year={2018}
}
When modeling directional data, that is, unit-norm multivariate vectors, a first natural question is to ask whether the directions are uniformly distributed or, on the contrary, whether there exist modes of variation significantly different from uniformity. We review in this article a reasonably exhaustive collection of uniformity tests for assessing uniformity in the hypersphere. Specifically, we review the classical circular-specific tests, the large class of Sobolev tests with its many… 
View PDF on arXiv
12 Citations
Background Citations
5
Methods Citations
4
Results Citations
1

12 Citations

Citation Type

Citation Type

A Cramér–von Mises Test of Uniformity on the Hypersphere
Testing uniformity of a sample supported on the hypersphere is one of the first steps when analysing multivariate data for which only the directions (and not the magnitudes) are of interest. In this…
On Sobolev tests of uniformity on the circle with an extension to the sphere
Circular and spherical data arise in many applications, especially in Biology, Earth Sciences and Astronomy. In dealing with such data one of the preliminary steps before any further inference, is to…
On the power of axial tests of uniformity on spheres
Testing uniformity on the $p$-dimensional unit sphere is arguably the most fundamental problem in directional statistics. In this paper, we consider this problem in the framework of axial data, that…
On the power of Sobolev tests for isotropy under local rotationally symmetric alternatives
We consider one of the most classical problems in multivariate statistics, namely the problem of testing isotropy, or equivalently, the problem of testing uniformity on the unit hypersphere Sp−1 of…
Testing uniformity on high-dimensional spheres: The non-null behaviour of the Bingham test
TLDR
A lower bound on the minimax separation rate is derived and it is established that the Bingham test is minimax rate-optimal in the class of Watson distributions.
A Stein Goodness-of-fit Test for Directional Distributions
TLDR
This study proposes non-parametric goodness-of-fit testing procedures for general directional distributions based on kernel Stein discrepancy based on Stein's operator on spheres, which is derived by using Stokes' theorem.
Recent advances in directional statistics
TLDR
This paper provides a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics, and considers developments for the exploratory analysis of directional data.
The Statistics of Circular Optimal Transport
Empirical optimal transport (OT) plans and distances provide effective tools to compare and statistically match probability measures defined on a given ground space. Fundamental to this are…
Harmonic analysis and distribution-free inference for spherical distributions
Reliable Covariance Estimation
TLDR
This work develops a consistent and easily applicable goodness-of-fit test against all alternatives to ellipticity when the scatter matrix is unknown and reformulated as an asymptotic uniformity test for certain sequences of exchangeable vectors on the unit sphere.
...
...

References

SHOWING 1-10 OF 61 REFERENCES
Comparison of tests of uniformity defined on the hypersphere
Power of Tests of Uniformity Defined on the Hypersphere
Abstract In multivariate data with n individuals described by p variables it is usual to assume that the variables are fixed and the individuals are randomly selected; in the present study we assume…
On projection-based tests for directional and compositional data
A new class of nonparametric tests, based on random projections, is proposed. They can be used for several null hypotheses of practical interest, including uniformity for spherical (directional) and…
Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives
We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in non-null issues. We show that rotationally symmetric alternatives lead to two Local…
Supersmooth testing on the sphere over analytic classes
We consider the nonparametric goodness-of-fit test of the uniform density on the sphere when we have observations whose density is the convolution of an error density and the true underlying density.…
A new Sobolev test for uniformity on the circle
A new invariant test for uniformity on the circle is presented. The motivation of this paper was to find a test consistent against all possible alternatives that would be powerful against both…
A decomposition for invariant tests of uniformity on the sphere
  • J. Pycke
  • Mathematics, Computer Science
  • 2007
TLDR
It is shown that this type of U-statistic, whose kernel is invariant by isometries, can be separated into a set of statistics whose limiting random variables are independent, and an interpretation of Watson type identities between quadratic functionals of Gaussian processes is proposed.
Modifications of the Rayleigh and Bingham Tests for Uniformity of Directions
The general machinery of Cordeiro and Ferrari (1991, Biometrika78, 573?582) and Chandra and Mukerjee (1991, J. Multivariate Anal.36, 103?112) provides modifications of score test statistics which…
A Data Driven Smooth Test for Circular Uniformity
TLDR
The proposed omnibus test for uniformity on the circle is based upon the idea of data driven smooth tests as presented in Ledwina and it seems to outperform other known omnibus tests when testing against multimodal alternatives.
Tests for randomness of directions against equatorial and bimodal alternatives
SUMMARY Tests of randomness of directions in three-dimensional space or equivalently tests of uniform distribution of points on the unit sphere are treated. One test is against alternatives which…
...
...