Computing the distribution of quadratic forms: Further comparisons between the Liu–Tang–Zhang approximation and exact methods

Duchesne, Pierre; Micheaux, Pierre Lafaye de

doi:10.1016/j.csda.2009.11.025

Cited by 189 publications

(144 citation statements)

References 13 publications

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“…CMB observations, for example, are constraining flat ΛCDM to a precision where posteriors are well approximated by Gaussian distributions. Whenever updating or comparing CMB con-120.6 (25). The contours cover 68% and 95% of the overall posterior volume and the dotted lines show the means.…”

Section: Discussionmentioning

confidence: 99%

Quantifying concordance in cosmology

Seehars

Grandis

Amara³

et al. 2016

Phys. Rev. D

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Quantifying the concordance between different cosmological experiments is important for testing the validity of theoretical models and systematics in the observations. In earlier work, we thus proposed the Surprise, a concordance measure derived from the relative entropy between posterior distributions. We revisit the properties of the Surprise and describe how it provides a general, versatile, and robust measure for the agreement between datasets. We also compare it to other measures of concordance that have been proposed for cosmology. As an application, we extend our earlier analysis and use the Surprise to quantify the agreement between WMAP 9, Planck 13 and Planck 15 constraints on the ΛCDM model. Using a principle component analysis in parameter space, we find that the large Surprise between WMAP 9 and Planck 13 (S = 17.6 bits, implying a deviation from consistency at 99.8% confidence) is due to a shift along a direction that is dominated by the amplitude of the power spectrum. The Planck 15 constraints deviate from the Planck 13 results (S = 56.3 bits), primarily due to a shift in the same direction. The Surprise between WMAP and Planck consequently disappears when moving to Planck 15 (S = −5.1 bits). This means that, unlike Planck 13, Planck 15 is not in tension with WMAP 9. These results illustrate the advantages of the relative entropy and the Surprise for quantifying the disagreement between cosmological experiments and more generally as an information metric for cosmology.

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Section: Discussionmentioning

confidence: 99%

Quantifying concordance in cosmology

Seehars

Grandis

Amara³

et al. 2016

Phys. Rev. D

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“…For a similar, more general approach, see [10]. For alternative approaches, see [11]. Unlike these alternatives, the approach of the present section has an upper bound on its required number of floating-point operations that depends only on the number n of bins and on the precision ε of computations, not on the values of σ 1 , σ 2 , .…”

Section: The Sum Of the Squares Of Independent Centered Gaussian Randmentioning

confidence: 99%

Computing the confidence levels for a root-mean-square test of goodness-of-fit

Perkins

Tygert

Ward

2011

Applied Mathematics and Computation

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The classic χ 2 statistic for testing goodness-of-fit has long been a cornerstone of modern statistical practice. The statistic consists of a sum in which each summand involves division by the probability associated with the corresponding bin in the distribution being tested for goodness-of-fit. Typically this division should precipitate rebinning to uniformize the probabilities associated with the bins, in order to make the test reasonably powerful. With the now widespread availability of computers, there is no longer any need for this. The present paper provides efficient black-box algorithms for calculating the asymptotic confidence levels of a variant on the classic χ 2 test which omits the problematic division. In many circumstances, it is also feasible to compute the exact confidence levels via Monte Carlo simulation.

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“…(9) and the scope of this paper. Duchesne and De Micheaux (2010) have compared the properties of several approximation approaches to problem (11) with "exact" methods that can bound the approximation error and make it arbitrarily small. All methods are implemented in C language and interfaced to R in the package CompQuadForm.…”

Section: Conclusion and Next Stepsmentioning

confidence: 99%

Simulating mixtures of multivariate data with fixed cluster overlap in FSDA library

Riani

Cerioli

Perrotta

et al. 2015

Adv Data Anal Classif

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We extend the capabilities of MixSim, a framework which is useful for evaluating the performance of clustering algorithms, on the basis of measures of agreement between data partitioning and flexible generation methods for data, outliers and noise. The peculiarity of the method is that data are simulated from normal mixture distributions on the basis of pre-specified synthesis statistics on an overlap measure, defined as a sum of pairwise misclassification probabilities. We provide new tools which enable us to control additional overlapping statistics and departures from homogeneity and sphericity among groups, together with new outlier contamination schemes. The output of this extension is a more flexible framework for generation of data to better address modern robust clustering scenarios in presence of possible contamination. WeThe FSDA project and the related research are partly supported by the University of Parma through the MIUR PRIN "MISURA-Multivariate models for risk assessment" project and by the Joint Research Centre of the European Commission, through its work program 2014-2020. Electronic supplementary materialThe online version of this article (doi:10.1007/s11634-015-0223-9) contains supplementary material, which is available to authorized users. also study the properties and the implications that this new way of simulating clustering data entails in terms of coverage of space, goodness of fit to theoretical distributions, and degree of convergence to nominal values. We demonstrate the new features using our MATLAB implementation that we have integrated in the Flexible Statistics for Data Analysis (FSDA) toolbox for MATLAB. With MixSim, FSDA now integrates in the same environment state of the art robust clustering algorithms and principled routines for their evaluation and calibration. A spin off of our work is a general complex routine, translated from C language to MATLAB, to compute the distribution function of a linear combinations of non central χ 2 random variables which is at the core of MixSim and has its own interest for many test statistics.

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Computing the distribution of quadratic forms: Further comparisons between the Liu–Tang–Zhang approximation and exact methods

Cited by 189 publications

References 13 publications

Quantifying concordance in cosmology

Quantifying concordance in cosmology

Computing the confidence levels for a root-mean-square test of goodness-of-fit

Simulating mixtures of multivariate data with fixed cluster overlap in FSDA library

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