Confidence, Prediction, and Tolerance Regions for the Multivariate Normal Distribution

Chew, Victor

doi:10.1080/01621459.1966.10480892

Cited by 112 publications

(66 citation statements)

References 27 publications

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“…we may view Z [i] (ω) as a (noisy) approximation to the Legendre coefficient δ [i] of (11). It is instructive to rewrite (14) in terms of our Karhunen-Loève expansion (now truncated at L terms): from (5), (9), (11), and (14) we obtain…”

Section: Now Letmentioning

confidence: 99%

“…. , I + 1, with y constrained to reside on the elliptical confidence region from [11]. We prefer the rectangular region in anticipation of the Linear Program introduced below; ultimately, however, the elliptical region (or an oriented rectangular region) could be incorporated and would of course sharpen our results.…”

Section: Now Letmentioning

confidence: 99%

“…It is instructive to rewrite (14) in terms of our Karhunen-Loève expansion (now truncated at L terms): from (5), (9), (11), and (14) we obtain…”

Section: Now Letmentioning

confidence: 99%

“…represents a rectangular confidence region which encloses the strict elliptical confidence region [11] -the i th dimension of this rectangular region is derived by maximizing the i th component of y ∈ R I+1 , for i = 1, . .…”

Section: Now Letmentioning

confidence: 99%

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Certified reduced basis model validation: A frequentistic uncertainty framework

Huynh¹,

Knezevic²,

Patera³

2012

Computer Methods in Applied Mechanics and Engineering

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We introduce a frequentistic validation framework for assessment -acceptance or rejection -of the consistency of a proposed parametrized partial differential equation model with respect to (noisy) experimental data from a physical system. Our method builds upon the Hotelling Our approach introduces two new elements: a spectral representation of the misfit which reduces the dimensionality and variance of the underlying multivariate Gaussian but without introduction of the usual regression assumptions; a certified (verified) reduced basis approximation -reduced order model -which greatly accelerates computational performance but without any loss of rigor relative to the full (finite element) discretization. We illustrate our approach with examples from heat transfer and acoustics, both based on synthetic data. We demonstrate that we can efficiently identify possibility regions that characterize parameter uncertainty; furthermore, in the case that the possibility region is empty, we can deduce the presence of "unmodeled physics" such as cracks or heterogeneities.

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Section: Now Letmentioning

confidence: 99%

Section: Now Letmentioning

confidence: 99%

“…It is instructive to rewrite (14) in terms of our Karhunen-Loève expansion (now truncated at L terms): from (5), (9), (11), and (14) we obtain…”

Section: Now Letmentioning

confidence: 99%

Section: Now Letmentioning

confidence: 99%

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Certified reduced basis model validation: A frequentistic uncertainty framework

Huynh¹,

Knezevic²,

Patera³

2012

Computer Methods in Applied Mechanics and Engineering

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“…This is explained under the assumption that the multivariate date follows the normal distribution function (Chew, 1966;Fan and Zhang, 2000;Frank, 1966;Johnson and Whichern, 2002;Sun and Loader, 1994). The (1 − α) confidence regions for the bivariate distribution function are represented with circular or elliptical shapes with respect to values of the correlation coefficient ρ, and the confidence regions for the trivariate distribution function are expressed with spherical or ellipsoid shapes with respect to the types of variance and covariance matrix.…”

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confidence: 99%

Multivariate confidence region using quantile vectors

Hong

Kim

2017

CSAM

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Multivariate confidence regions were defined using a chi-square distribution function under a normal assumption and were represented with ellipse and ellipsoid types of bivariate and trivariate normal distribution functions. In this work, an alternative confidence region using the multivariate quantile vectors is proposed to define the normal distribution as well as any other distributions. These lower and upper bounds could be obtained using quantile vectors, and then the appropriate region between two bounds is referred to as the quantile confidence region. It notes that the upper and lower bounds of the bivariate and trivariate quantile confidence regions are represented as a curve and surface shapes, respectively. The quantile confidence region is obtained for various types of distribution functions that are both symmetric and asymmetric distribution functions. Then, its coverage rate is also calculated and compared. Therefore, we conclude that the quantile confidence region will be useful for the analysis of multivariate data, since it is found to have better coverage rates, even for asymmetric distributions.

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Tolerance Region

Vangel

2005

Encyclopedia of Biostatistics

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In a sample from a multivariate distribution, a tolerance region is a region for which there is a specified high probability (the confidence ) that at least a specified proportion of the distribution (the coverage ) is included. It is thus a generalization of the univariate tolerance interval. The construction using statistically equivalent blocks is described briefly.

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Confidence, Prediction, and Tolerance Regions for the Multivariate Normal Distribution

Cited by 112 publications

References 27 publications

Certified reduced basis model validation: A frequentistic uncertainty framework

Certified reduced basis model validation: A frequentistic uncertainty framework

Multivariate confidence region using quantile vectors

Tolerance Region

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