On a projection-based class of uniformity tests on the hypersphere

Abstract

We propose a projection-based class of uniformity tests on the hypersphere using statistics that integrate, along all possible directions, the weighted quadratic discrepancy between the empirical cumulative distribution function of the projected data and the projected uniform distribution. Simple expressions for several test statistics are obtained for the circle and sphere, and relatively tractable forms for higher dimensions. Despite its different origin, the proposed class is shown to be related with the well-studied Sobolev class of uniformity tests. Our new class proves itself advantageous by allowing to derive new tests for hyperspherical data that neatly extend the circular tests by Watson, Ajne, and Rothman, and by introducing the first instance of an Anderson-Darling-like test for such data. The asymptotic distributions and the local optimality against certain alternatives of the new tests are obtained. A simulation study evaluates the theoretical findings and evidences that, for certain scenarios, the new tests are competitive against previous proposals. The new tests are employed in three astronomical applications.

The shape of aroma: measuring and modeling citrus oil gland distribution

“…We tested whether the gland point cloud follows a uniform distribution, where every unit area of the skin has the same probability of containing oil glands; or if the underlying distribution is rotationally symmetric, where the oil glands pattern is symmetrical around a fixed direction. Uniformity was tested with Projected Anderson-Darling (PAD) test (García-Portugués et al, 2020) with the R package sphunif (García-Portugués and Verdebout, 2021). The rotational symmetry was tested with a scatter-location hybrid test with an unspecified direction of symmetry (García-Portugués et al, 2020) with the R package rotasym (García-Portugués et al, 2021).…”

“…Uniformity was tested with Projected Anderson-Darling (PAD) test (García-Portugués et al, 2020) with the R package sphunif (García-Portugués and Verdebout, 2021). The rotational symmetry was tested with a scatter-location hybrid test with an unspecified direction of symmetry (García-Portugués et al, 2020) with the R package rotasym (García-Portugués et al, 2021). Additionally, we visually examined the distribution of oil glands for most fruits and compared them to simulated uniform distributions by projecting them to 2D via Lambert azimuthal equal-area projections (Mardia and Jupp, 1999, Ch.…”

On the power of Sobolev tests for isotropy under local rotationally symmetric alternatives

“…Tests in the high-dimensional context were considered in Cai and Jiang (2012), Cai et al (2013), and Cutting et al (2017). Projection-based tests were proposed in Cuesta-Albertos et al (2009) and García-Portugués et al (2020a). Uniformity tests were used in García-Portugués et al (2020b) to construct tests for rotational symmetry.…”

Data-driven stabilizations of goodness-of-fit tests

“…While a sizable number of tests of uniformity on S p−1 exist (see a review in García-Portugués and Verdebout (2018)), perhaps the two most known omnibus tests are that of Kuiper (1960) and Watson (1961) on S 1 : their statistics, V n and U 2 n , can be regarded as the rotation-invariant versions of the Kolmogorov-Smirnov and Cramér-von Mises tests of uniformity, respectively. Moving beyond S 1 has proven a challenging task for ecdf-based tests up to relatively recent years, with Cuesta-Albertos et al ( 2009) using a Kolmogorov-Smirnov test on random projections data and García-Portugués et al (2020) proposing a class of projected-ecdf statistics that extends Watson (1961) test to S p−1 (see Section 3.1). As in the classical setting, tests of uniformity on S p−1 allow for testing the goodnessof-fit of more general distributions: in S 1 , this is a straightforward application of the probability integral transform in the angles space [−π, π); the case S p−1 , p ≥ 3, is remarkably more complex and has been recently put forward in Jupp and Kume (2020).…”

“…are provided inGarcía-Portugués et al (2020), while the sphunif R package (García-Portugués and Verdebout, 2021) provides implementations for P CvM n,p , P AD n,p , and N n,p , for all p ≥ 2.…”