The Estimation of Probability Densities and Cumulatives by Fourier Series Methods

Kronmal, Richard A.; Tarter, Michael E.

doi:10.2307/2283885

Cited by 176 publications

(99 citation statements)

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“…Of course, there are cases where the first condition is violated for lower-order estimators. Two such cases include orthogonal series estimators (Kronmal and Tarter, 1968;Watson, 1969) and boundary-corrected kernel estimators (Rice, 1984). Note that positive kernel estimators correspond to k = 2.…”

Section: Rates Of Convergencementioning

confidence: 99%

Multivariate Density Estimation and Visualization

Scott

2011

Handbook of Computational Statistics

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Section: Rates Of Convergencementioning

confidence: 99%

Multivariate Density Estimation and Visualization

Scott

2011

Handbook of Computational Statistics

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“…Fourier series were first employed to estimate probability densities in [4]. Recently, [5] and [6] ensured the non-negativity of Fourier series by approximating the square root of the density instead of the density itself.…”

Section: Introductionmentioning

confidence: 99%

Fourier Density Approximation for Belief Propagation in Wireless Sensor Networks

Wang

Obradović

et al. 2009

Lecture Notes in Electrical Engineering

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Abstract-Many distributed inference problems in wireless sensor networks can be represented by probabilistic graphical models, where belief propagation, an iterative message passing algorithm provides a promising solution. In order to make the algorithm efficient and accurate, messages which carry the belief information from one node to the others should be formulated in an appropriate format. This paper presents two belief propagation algorithms where non-linear and non-Gaussian beliefs are approximated by Fourier density approximations, which significantly reduces power consumptions in the belief computation and transmission. We use self-localization in wireless sensor networks as an example to illustrate the performance of this method.

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“…While these measurements and their analysis do not lead to any mathematical ambiguities and their interpretation is not afflicted by ghost phenomena reserved for diffraction data only, orientation density estimation by the application of transform methods and corresponding infinite harmonic series expansion may again result in negative values of the estimated density (Kronmal & Tarter, 1968;Bryan, 1971). Therefore, nonparametric hyperspherical kernel density estimation is suggested here as an alternative that will always yield a non-negative density estimate of the orientation density.…”

Section: Introductionmentioning

confidence: 99%

Towards statistics of crystal orientations in quantitative texture anaylsis

Schaeben¹

1993

J Appl Cryst

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By the application of Rodrigues parameters, crystal orientations are represented as points on the unit semi-hypersphere $4+ c R 4 or equivalently on the projective hyperplane H3c R 4. For the statistical analysis of orientation data, probability models on $4+-H 3 are required. Among different hyperspherical analogs of the normal distribution in Euclidean space, corresponding to various of its characterizations, the Bingham model distribution is" characterized as the hyperspherical analog for statistical purposes. Maximum-likelihood estimates of its parameter matrices and single-sample significance tests of uniformity and rotational symmetry are presented. In terms of probability theory and stochastic processes, the hyperspherical Browniandiffusion distribution is the analog of the normal distribution in Euclidean space. An orientation distribution derived by Savelova lind. Lab. (USSR), (1984), 50, 468-474] and erroneously related to a central-limit-type theorem for rotations by Matthies, Muller & Vinel [Textures Mierostruct. (1988), 10, 77-96], with the Brownian-diffusion distribution on $4+ -H 3, recalls the fact that no simple analog of the central limit theorem in Euclidean space exists for arbitrary spaces and manifolds, e.g. for S p, SP+, SO(p), p > 2. Therefore, interpretations of preferred orientations in terms of mechanisms of texture development related to an imaginary simple analog of the central limit theorem would generally be misleading. Individual crystal-orientation measurements may initially require different nonparametric methods of analysis. As a counterpart of pole-to-orientation density inversion of diffraction data, hyperspherical kernel orientation density estimation is suggested for individual orientation measurements. With the application of non-negative kernels, the estimated orientation density is always non-negative. The critical smoothing parameter of this method is determined on information-theoretical grounds. For patterns of preferred orientation too complex to be sufficiently approximated by the parametric Bingham model distribution, a clustering of individual orientation 0021-8898/93/010112-10506.00 measurements into disjunct classes is suggested that relates similarity of orientations to multimodality of the estimated orientation density; each class of orientations may then be individually analyzed by parametric methods.

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The Estimation of Probability Densities and Cumulatives by Fourier Series Methods

Cited by 176 publications

References 0 publications

Multivariate Density Estimation and Visualization

Multivariate Density Estimation and Visualization

Fourier Density Approximation for Belief Propagation in Wireless Sensor Networks

Towards statistics of crystal orientations in quantitative texture anaylsis

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