A new Sobolev test for uniformity on the circle

Hermans, M.; Rasson, Jean‐Paul

doi:10.1093/biomet/72.3.698

Cited by 28 publications

(18 citation statements)

References 5 publications

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“…The investigation that has been undertaken (Stephens 1969; Bergin 1991; Bogdan et al 2002; Pycke 2010) has highlighted potential for low power (as discussed above). For this reason, alternative tests have been proposed that were designed specifically to have better power against multimodal alternatives: the two most developed that we could find in the literature were due to Hermans and Rasson (1985) and Bogdan et al (2002); see Pycke (2010) for theoretical underpinning of general classes to tests for circular uniformity. Both have been subject to very little published evaluation, and neither is available in any software package that we know of.…”

Section: Introductionmentioning

confidence: 99%

Circular data in biology: advice for effectively implementing statistical procedures

Landler

Ruxton

Malkemper

2018

Behav Ecol Sociobiol

134

120

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Circular data are common in biological studies. The most fundamental question that can be asked of a sample of circular data is whether it suggests that the underlying population is uniformly distributed around the circle, or whether it is concentrated around at least one preferred direction (e.g. a migratory goal or activity phase). We compared the statistical power of five commonly used tests (the Rayleigh test, the V-test, Watson’s test, Kuiper’s test and Rao’s spacing test) across a range of different unimodal scenarios. The V-test showed higher power for symmetrical distributions, Rao’s spacing performed worst for all explored unimodal distributions tested and the remaining three tests showed very similar performance. However, the V-test only applies if the hypothesis is restricted to one (pre-specified) direction of interest. In all other unimodal cases, we recommend using the Rayleigh test. Much less explored is the multimodal case with data concentrated around several directions. We performed power simulations for a variety of multimodal situations, testing the performance of the widely used Rayleigh, Rao’s, Watson, and Kuiper’s tests as well as the more recent Bogdan and Hermans-Rasson tests. Our analyses of alternative statistical methods show that the commonly used tests lack statistical power in many of multimodal cases. Transformation of the raw data (e.g. doubling the angles) can overcome some of the issues, but only in the case of perfect f-fold symmetry. However, the Hermans-Rasson method, which is not yet implemented in any software package, outcompetes the alternative tests (often by substantial margins) in most of the multimodal situations explored. We recommend the wider uptake of the powerful but hitherto neglected Hermans-Rasson method. In summary, we provide guidance for biologists helping them to make decisions when testing circular data for single or multiple departures from uniformity.Electronic supplementary materialThe online version of this article (10.1007/s00265-018-2538-y) contains supplementary material, which is available to authorized users.

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Section: Introductionmentioning

confidence: 99%

Circular data in biology: advice for effectively implementing statistical procedures

Landler

Ruxton

Malkemper

2018

Behav Ecol Sociobiol

134

120

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“…This should not be seen as a failing or weakness of the Rayleigh test, since it was not originally designed to detect deviations other than unimodal von Mises ones. In [8], we highlighted a test due to Hermans and Rasson [5] that showed considerably more power than the Rayleigh test to detect some types of multimodal deviations, however we did not explore its performance for unimodal distributions. We argued that the power gain in multimodal scenarios should encourage more widespread uptake of the Hermans–Rasson test.…”

Section: Introductionmentioning

confidence: 99%

The Hermans–Rasson test as a powerful alternative to the Rayleigh test for circular statistics in biology

2019

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Background Circular data are gathered in diverse fields of science where measured traits are cyclical in nature: such as compass directions or times of day. The most common statistical question asked of a sample of circular data is whether the data seems to be drawn from a uniform distribution or one that is concentrated around one or more preferred directions. The overwhelmingly most-popular test of the null hypothesis of uniformity is the Rayleigh test, even though this test is known to have very low power in some circumstances. Here we present simulation studies evaluating the performance of tests developed as alternatives to the Rayleigh test. Results The results of our simulations demonstrate that a single test, the Hermans and Rasson test is almost as powerful as the Rayleigh test in unimodal situations (when the Rayleigh test does well) but substantially outperforms the Rayleigh test in multimodal situations. Conclusion We recommend researchers switch to routine use of the new Hermans and Rasson test. We also demonstrate that all available tests have low power to detect departures from uniformity involving more than two concentrated regions: we recommend that where researchers suspect such complex departures that they collect substantially-sized samples and apply another recent test due to Pycke that was designed specifically for such complex cases. We provide clear textual descriptions of how to implement each of these recommended tests and encode them in R functions that we provide. Electronic supplementary material The online version of this article (10.1186/s12898-019-0246-8) contains supplementary material, which is available to authorized users.

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“…Giné (1975a) carried out the program for the circle, sphere, and the projective plane, recovering many known examples in the literature and introducing new tests of uniformity. Several authors have studied and derived Sobolev tests for different examples including circular and directional data, tests of symmetry, and unitary eigenvalues; see Prentice (1978); Wellner (1979); Spurr (1983, 1985); Hermans and Rasson (1985); Baringhaus (1991); Sengupta and Pal (2001); Coram and Diaconis (2003).…”

Section: Spectral Tests On General Spacesmentioning

confidence: 99%

New tests of uniformity on the compact classical groups as diagnostics for weak-$^{*}$ mixing of Markov chains

Sepehri¹

2019

Bernoulli

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This paper introduces two new families of non-parametric tests of goodness-of-fit on the compact classical groups. One of them is a family of tests for the eigenvalue distribution induced by the uniform distribution, which is consistent against all fixed alternatives. The other is a family of tests for the uniform distribution on the entire group, which is again consistent against all fixed alternatives. The construction of these tests heavily employs facts and techniques from the representation theory of compact groups. In particular, new Cauchy identities are derived and proved for the characters of compact classical groups, in order to accomodate the computation of the test statistic. We find the asymptotic distribution under the null and general alternatives.The tests are proved to be asymptotically admissible. Local power is derived and the global properties of the power function against local alternatives are explored.The new tests are validated on two random walks for which the mixing-time is studied in the literature. The new tests, and several others, are applied to the Markov chain sampler proposed by Jones, Osipov and Rokhlin (2011), providing strong evidence supporting the claim that the sampler mixes quickly.MSC 2010 subject classifications: 62G10, 60B15, 62M15, 20C15.

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A new Sobolev test for uniformity on the circle

Cited by 28 publications

References 5 publications

Circular data in biology: advice for effectively implementing statistical procedures

Circular data in biology: advice for effectively implementing statistical procedures

The Hermans–Rasson test as a powerful alternative to the Rayleigh test for circular statistics in biology

New tests of uniformity on the compact classical groups as diagnostics for weak-$^{*}$ mixing of Markov chains

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