Harmonic analysis and distribution-free inference for spherical distributions

@article{Jammalamadaka2019HarmonicAA,
  title={Harmonic analysis and distribution-free inference for spherical distributions},
  author={Sreenivasa Rao Jammalamadaka and Gy{\"o}rgy H. Terdik},
  journal={J. Multivar. Anal.},
  year={2019},
  volume={171},
  pages={436-451}
}
Abstract Fourier analysis, and representation of circular distributions in terms of their Fourier coefficients, is quite commonly discussed and used for model-free inference such as testing uniformity and symmetry, in dealing with 2-dimensional directions. However, a similar discussion for spherical distributions, which are used to model 3-dimensional directional data, is not readily available in the literature in terms of their harmonics. This paper, in what we believe is the first such… Expand
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References

SHOWING 1-10 OF 60 REFERENCES
On Optimal Tests for Rotational Symmetry Against New Classes of Hyperspherical Distributions
Abstract Motivated by the central role played by rotationally symmetric distributions in directional statistics, we consider the problem of testing rotational symmetry on the hypersphere. We adopt aExpand
Probably Approximately Symmetric: Fast Rigid Symmetry Detection With Global Guarantees
TLDR
This work presents a fast algorithm for global rigid symmetry detection with approximation guarantees, and proves that the density of the sampling depends on the total variation of the shape, allowing for formal bounds on the algorithm's complexity and approximation quality. Expand
Directional Statistics, I
In many natural and physical sciences the measurements are directions—either in two- or three-dimensions. This chapter briefly introduces this novel area of statistics, and provides a good startingExpand
Dispersion on a sphere
  • R. Fisher
  • Mathematics
  • Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1953
Any topological framework requires the development of a theory of errors of characteristic and appropriate mathematical form. The paper develops a form of theory which appears to be appropriate toExpand
An overview of uniformity tests on the hypersphere
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 existExpand
Spherical Deconvolution
This paper proposes nonparametric deconvolution density estimation overS2. Here we would think of theS2elements of interest being corrupted by randomSO(3) elements (rotations). The resulting densityExpand
Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions
Supposing a given collection Y, YN of i.i.d. random points on a Riemannian manifold, we discuss how to estimate the underlying distribution from a differential geometric viewpoint. The mainExpand
A Reflective Symmetry Descriptor for 3D Models
TLDR
A new reflective symmetry descriptor that represents a measure of reflective symmetry for an arbitrary 3D model for all planes through the model’s center of mass (even if they are not planes of symmetry), and is insensitive to noise and stable under point sampling. Expand
Representations of SO(3) and angular polyspectra
TLDR
The findings of the present paper constitute a basis upon which one can build formal procedures for the statistical analysis and the probabilistic modelization of the Cosmic Microwave Background radiation, which is currently a crucial topic of investigation in cosmology. Expand
Introduction to Fourier Analysis on Euclidean Spaces.
The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the actionExpand
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