Goodness‐of‐fit and Discordancy Tests for Samples From the Watson Distribution on the Sphere

Best, D. J.; Fisher, N. I.

doi:10.1111/j.1467-842x.1986.tb00580.x

Cited by 27 publications

(20 citation statements)

References 12 publications

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“…In this study, we focused on fitting girdle type data to the more generalized Watson girdle distribution. We performed both graphical and formal goodness of fit (GOF) tests [38] on the data fitted for the Watson girdle model to determine the adequacy of their representation. The Watson girdle distribution can be characterized by three parameters, κ, α, and β.…”

Section: Methodsmentioning

confidence: 99%

Spherical statistics for characterizing the spatial distribution of deep brain stimulation effects on neuronal activity

Xiao

Johnson

2015

Journal of Neuroscience Methods

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Background Computational models of deep brain stimulation (DBS) have played a key role in understanding its physiological mechanisms. By estimating a volume of tissue directly modulated by DBS, one can relate the neuronal pathways within those volumes to the therapeutic efficacy of a particular DBS setting. New Method A spherical statistical framework is described to quantify and determine salient features of such morphologies using visualization techniques, empirical shape analysis, and formal hypothesis testing. This framework is shown using a 3D model of thalamocortical neurons surrounding a radially-segmented DBS array. Results We show that neuronal population volumes modulated by various DBS electrode configurations can be characterized by parametric distribution models, such as Kent and Watson girdle models. Distribution parameters were found to change with stimulus settings, including amplitude and radial distance from the DBS array. Increasing stimulation amplitude through a single electrode resulted in more diffuse neuronal activation and increased rotational symmetry about the mean direction of the activated population. When stimulation amplitude was held constant, the activated neuronal population distribution was more concentrated with distance from the DBS array and was also more rotationally asymmetric. We also show how data representation (e.g. stimulus-entrained cell body vs. axon node) can significantly alter model distribution shape. Comparison to Existing Methods This statistical framework provides a quantitative method to analyze the spatial morphologies of DBS-induced effects on neuronal activity. Conclusions The application of spherical statistics to assess spatial distributions of neuronal activity has potential usefulness for numerous other recording, labeling, and stimulation modalities.

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Section: Methodsmentioning

confidence: 99%

Spherical statistics for characterizing the spatial distribution of deep brain stimulation effects on neuronal activity

Xiao

Johnson

2015

Journal of Neuroscience Methods

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“…If µ 1 has a unique median m 1 then µ 0 is automatically the uniform distribution on S 0 . The nested spheres in (2) are reminiscent of the principal nested spheres of [11] but, whereas principal nested spheres may be small spheres and are chosen to give closest fit to the data, the spheres in (2) are great spheres and are chosen to be orthogonal to m p−1 , . .…”

Section: Spheresmentioning

confidence: 99%

Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds

Jupp

Kume

2020

Journal of Multivariate Analysis

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The standard method of transforming a continuous distribution on the line to the uniform distribution on [0, 1] is the probability integral transform. Analogous transforms exist on compact Riemannian manifolds, X , in that for each distribution with continuous positive density on X , there is a continuous mapping of X to itself that transforms the distribution into the uniform distribution. In general, this mapping is far from unique. This paper introduces the construction of a version of such a probability integral that (under mild conditions) is canonical. The construction is extended to shape spaces, simply-connected spaces of non-positive curvature, and simplices.The probability integral transform is used to derive tests of goodness of fit from tests of uniformity. Illustrative examples of these tests of goodness of fit are given involving (i) Fisher distributions on S 2 , (ii) isotropic Mardia-Dryden distributions on the shape space Σ 5 2 . Their behaviour is investigated by simulation.

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“…Large-sample asymptotic properties. An appropriate setting for largesample asymptotic results is that in which the mapping t given by (2.2) is allowed to depend on the sample size n. Thus, there is a sequence t (1) , t (2) , . .…”

Section: 2mentioning

confidence: 99%

“…The corresponding goodness-of-fit statistic is the weighted Sobolev statistic (3.1) with t replaced by t (n) . If t (1) , t (2) , . .…”

Section: 2mentioning

confidence: 99%

Sobolev tests of goodness of fit of distributions on compact Riemannian manifolds

Jupp¹

2005

Ann. Statist.

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Goodness‐of‐fit and Discordancy Tests for Samples From the Watson Distribution on the Sphere

Cited by 27 publications

References 12 publications

Spherical statistics for characterizing the spatial distribution of deep brain stimulation effects on neuronal activity

Spherical statistics for characterizing the spatial distribution of deep brain stimulation effects on neuronal activity

Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds

Sobolev tests of goodness of fit of distributions on compact Riemannian manifolds

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