Nonparametric regression on the hyper-sphere with uniform design

Monnier, Jean-Baptiste

doi:10.1007/s11749-011-0233-7

Cited by 10 publications

(10 citation statements)

References 22 publications

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“…Typically, the threshold is chosen such that τ j,k = κ (ln n/n), where κ depends explicitly on two parameters, namely, the radius R of the Besov ball on which the function f is defined and its supremum M ; see, e.g., Baldi et al [1]. An alternative and partially data-driven choice for κ is proposed by Monnier [38], i.e., here…”

Section: Comparison With Other Resultsmentioning

confidence: 99%

“…In the nonparametric setting, needlets have found various applications on directional statistics. Baldi et al [1] established minimax rates of convergence for the L p -risk of nonlinear needlet density estimators within the hard local thresholding paradigm, while analogous results concerning regression function estimation were established by Monnier [38]. The block thresholding framework was investigated in Durastanti [9].…”

Section: An Overview Of the Literaturementioning

confidence: 98%

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Adaptive global thresholding on the sphere

Durastanti

2016

Journal of Multivariate Analysis

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This work is concerned with the study of the adaptivity properties of nonparametric regression estimators over the d-dimensional sphere within the global thresholding framework. The estimators are constructed by means of a form of spherical wavelets, the so-called needlets, which enjoy strong concentration properties in both harmonic and real domains. The author establishes the convergence rates of the L p -risks of these estimators, focussing on their minimax properties and proving their optimality over a scale of nonparametric regularity function spaces, namely, the Besov spaces.

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Section: Comparison With Other Resultsmentioning

confidence: 99%

Section: An Overview Of the Literaturementioning

confidence: 98%

Adaptive global thresholding on the sphere

Durastanti

2016

Journal of Multivariate Analysis

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“…Di Marzio et al (2019b) built on their construction in Di Marzio et al (2014) to perform local polynomial logistic regression with a spherical predictor. Monnier (2011) proposed needlet-based regression for a uniformly distributed predictor on S d and Gaussian noise, whilst Lin (2019) weakened those assumptions and introduced regularisation on the needlet coefficients.…”

Section: Linear Responsementioning

confidence: 99%

Recent advances in directional statistics

Pewsey

García‐Portugués

2021

TEST

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Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere, and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper, we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (Wiley 1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, space situational awareness, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. AnWe are most grateful to three anonymous referees and, in alphabetical order, to

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“…A frequently used technique is that of series estimators based on spherical harmonics [see Abrial et al (2008) for example], which -roughly speaking -generalise estimators of a regression function on the line based on Fourier series to data on the sphere. Alternative series estimators have been proposed by Narcowich et al (2006), Baldi et al (2009) and Monnier (2011) who suggest to use spherical wavelets (needlets) in situations where better localisation properties are required. Most authors consider the 2-dimensional sphere S 3 in R 3 as they are interested in the development of statistical methodology for concrete applications such as earth and planetary sciences.…”

Section: Introductionmentioning

confidence: 99%

Optimal designs for regression with spherical data

Dette¹,

Konstantinou²,

Schorning³

et al. 2019

Electron. J. Statist.

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In this paper optimal designs for regression problems with spherical predictors of arbitrary dimension are considered. Our work is motivated by applications in material sciences, where crystallographic textures such as the missorientation distribution or the grain boundary distribution (depending on a four dimensional spherical predictor) are represented by series of hyperspherical harmonics, which are estimated from experimental or simulated data. For this type of estimation problems we explicitly determine optimal designs with respect to Kiefers Φ p -criteria and a class of orthogonally invariant information criteria recently introduced in the literature. In particular, we show that the uniform distribution on the m-dimensional sphere is optimal and construct discrete and implementable designs with the same information matrices as the continuous optimal designs. Finally, we illustrate the advantages of the new designs for series estimation by hyperspherical harmonics, which are symmetric with respect to the first and second crystallographic point group.

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Nonparametric regression on the hyper-sphere with uniform design

Abstract: Nonparametric regression, Uniform design, Minimax rate, Needlets, Needlet-shrinkage, Stochastic thresholding, 62G08, 62G05, 62C20,

Cited by 10 publications

References 22 publications

Adaptive global thresholding on the sphere

Adaptive global thresholding on the sphere

Recent advances in directional statistics

Optimal designs for regression with spherical data

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