Rational formulae for the production of a spherically symmetric probability distribution

Cook, J.M.

doi:10.1090/s0025-5718-1957-0690630-7

Cited by 41 publications

(31 citation statements)

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“…Several methods for generating random points uniformly distributed on the boundary surface of a d-sphere have been proposed by Cook [3], Marsaglia [16] and Tashiro [24]. The methods by Cook [3] and Marsaglia [16] are limited to d = 4 dimensions.…”

Section: Resultsmentioning

confidence: 99%

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On a combination method of VDR and patchwork for generating uniform random points on a unit sphere

Yang

Pang

Hou

et al. 2005

Journal of Multivariate Analysis

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In this paper, we use a combination of VDR theory and patchwork method to derive an efficient algorithm for generating uniform random points on a unit d-sphere. We first propose an algorithm to generate random vector with uniform distribution on a unit 2-sphere on the plane. Then we use VDR theory to reduce random vector X d with uniform distribution on a unit d-sphere into X d = (X d−2 , 1 − X d−2 2 (X d−1 , X d )), such that the random vector (X d−1 , X d ) is uniformly distributed on a unit 2-sphere and X d−2 has conditional uniform distribution on a (d − 2)-sphere of2 . Finally, we arrive by induction at an algorithm for generating uniform random points on a unit d-sphere.

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Section: Resultsmentioning

confidence: 99%

“…The methods by Cook [3] and Marsaglia [16] are limited to d = 4 dimensions. Tashiro's method is an improved version of the method by Marsaglia [16] and can be extended to any finite dimension.…”

Section: Resultsmentioning

confidence: 99%

On a combination method of VDR and patchwork for generating uniform random points on a unit sphere

Yang

Pang

Hou

et al. 2005

Journal of Multivariate Analysis

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“…In this sense, trying give answer to the proposed question we consider different values of s and search how the ranked institutions react under small changes in the values of s. As one can observe from Appendix 2, the ranked positions achieved by the French universities almost remain unchanged no matter the perturbations introduced in the weights. Only the expected ranked positions in the range [5 th -quantile, 95 thquantile] for a level of perturbation about 30% of w j present small differences for some universities. For the German universities, the ranked positions remain unchanged.…”

Section: Inserm) Industry and Others This Simple Indicator Correspomentioning

confidence: 95%

Improving quality assessment of composite indicators in university rankings: a case study of French and German universities of excellence

Benito

Romera

2011

Scientometrics

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Composite indicators play an essential role for benchmarking higher education institutions. One of the main sources of uncertainty building composite indicators and, undoubtedly, the most debated problem in building composite indicators is the weighting schemes (assigning weights to the simple indicators or subindicators) together with the aggregation schemes (final composite indicator formula). Except the ideal situation where weights are provided by the theory, there clearly is a need for improving quality assessment of the final rank linked with a fixed vector of weights. We propose to use simulation techniques to generate random perturbations around any initial vector of weights to obtain robust and reliable ranks allowing to rank universities in a range bracket. The proposed methodology is general enough to be applied no matter the weighting scheme used for the composite indicator. The immediate benefit achieved is a reduction of the uncertainty associated with the assessment of a specific rank which is not representative of the real performance of the university, and an improvement of the quality assessment of composite indicators used to rank. AbstractComposite indicators play an essential role for benchmarking higher education institutions. One of the main sources of uncertainty building composite indicators and, undoubtedly, the most debated problem in building composite indicators is the weighting schemes (assigning weights to the simple indicators or subindicators) together with the aggregation schemes (final composite indicator formula). Except the ideal situation where weights are provided by the theory, there clearly is a need for improving quality assessment of the final rank linked with a fixed vector of weights. We propose to use simulation techniques to generate random perturbations around any initial vector of weights to obtain robust and reliable ranks allowing to rank universities in a range bracket. The proposed methodology is general enough to be applied no matter the weighting scheme used for the composite indicator. The immediate benefit achieved is a reduction of the uncertainty associated with the assessment of a specific rank which is not representative of the real performance of the university, and an improvement of the quality assessment of composite indicators used to rank. To illustrate the proposed methodology we rank the French and the German universities involved in their respective 2008 Excellence Initiatives.

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“…Therefore the area of Dt is, to within o(5), Remarks, (a) Matrices A of the form required in the theorem may be constructed by taking a basis for R* containing the vector (1/2, • ■ •, 1/2), and orthonormalizing to generate the remaining three rows of A. (b) The use of points uniformly distributed on a sphere is numerically practical ; for methods of generating pseudo-random points with that distribution see, e.g., [6] and [7].…”

Section: =1mentioning

confidence: 99%

“…Let Ki, ■ • ■, KM be the subcubes into which He is partitioned, enumerated somehow. Then, since Q is applied to the different K's independently, M (6) °-\Qm(Î))= T,°\QKÁf)).…”

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confidence: 99%

Stochastic Quadrature Formulas

Haber¹

1969

Mathematics of Computation

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Abstract. A class of formulas for the numerical evaluation of multiple integrals is described, which combines features of the Monte-Carlo and the classical methods. For certain classes of functions-defined by smoothness conditions-these formulas provide the fastest possible rate of convergence to the integral. Asymptotic error estimates are derived, and a method is described for obtaining good a posteriori error bounds when using these formulas. Equal-coefficients formulas of this class, of degrees up to 3, are constructed.

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Rational formulae for the production of a spherically symmetric probability distribution

Cited by 41 publications

References 0 publications

On a combination method of VDR and patchwork for generating uniform random points on a unit sphere

On a combination method of VDR and patchwork for generating uniform random points on a unit sphere

Improving quality assessment of composite indicators in university rankings: a case study of French and German universities of excellence

Stochastic Quadrature Formulas

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