Minimal average degree aberration and the state polytope for experimental designs

Berstein, Yael; Maruri-Aguilar, Hugo; Onn, Shmuel; Riccomagno, Eva; Wynn, Henry P.

doi:10.1007/s10463-010-0291-8

Cited by 11 publications

(16 citation statements)

References 24 publications

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“…We denote this generic size as n * and the quotient basis is said to be generic. Indeed, it was previously observed that a generic set of quotient basis exists at certain increments of n where the least aberration is observed [6]. If n is not generic, the quotient basis can be used to identify the polynomial accuracy of the algebraic quadrature but the confounding or aberration effects may have stronger influence on the accuracy of the algebraic quadrature.…”

Section: Univariate Uncertainty Quantificationmentioning

confidence: 99%

“…At these n values, the quotient bases are generic and the design has the least aberration [6]. 4.5.2.…”

Section: 44mentioning

confidence: 99%

“…The main purpose of the paper is to introduce a new theory of quadrature based on the socalled algebraic method in experimental design. This method was introduced originally to study interpolation and aliasing in factorial design [6], but has the advantage of delivering a possible interpolator for any data over any experimental design with an a priori determination of the design's polynomial exactness. We refer to a set of quadrature points as a design to maintain the connection.…”

Section: Introductionmentioning

confidence: 99%

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The Algebraic Method in Quadrature for Uncertainty Quantification

Ko¹,

Wynn²

2016

SIAM/ASA J. Uncertainty Quantification

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Abstract. A general method of quadrature for uncertainty quantification (UQ) is introduced based on the algebraic method in experimental design. This is a method based on the theory of zero dimensional algebraic varieties. It allows quadrature of polynomials or polynomial approximands for quite general sets of quadrature points, here called 'designs'. The method goes some way to explaining when quadrature weights are non-negative and gives exact quadrature for monomials in the quotient ring defined by the algebraic method. The relationship to the classical methods based on zeros of orthogonal polynomials is discussed and numerical comparisons made with methods such as Gaussian quadrature and Smolyak grids. Application to UQ is examined in the context of polynomial chaos expansion and probabilistic collocation method where solution statistics are estimated.

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Section: Univariate Uncertainty Quantificationmentioning

confidence: 99%

“…At these n values, the quotient bases are generic and the design has the least aberration [6]. 4.5.2.…”

Section: 44mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Algebraic Method in Quadrature for Uncertainty Quantification

Ko¹,

Wynn²

2016

SIAM/ASA J. Uncertainty Quantification

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show abstract

“…A study for the class of Latin hypercube designs carried out in Bernstein et al [2010] shows different situations that can occur when computing the algebraic fan. Moreover by Theorem 30 in Pistone et al [2001] for a design whose points are chosen at random (with respect to any Lebesgue absolute continuous measure) the algebraic fan equals the statistical fan with probability one.…”

Section: Motivations For a Numerical Fan Of A Designmentioning

confidence: 99%

Numerical algebraic fan of a design for statistical model building

Rudak¹,

Kuhnt²,

Riccomagno³

2016

STAT SINICA

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“…In [12] the linear aberration for polynomial models is introduced. This is a new concept in experimental designs for quantitative factors which is an alternative to aberration and is strongly related to corner cut models and state polytopes.…”

Section: Advanced Topicsmentioning

confidence: 99%