Generalized resolution and minimum aberration criteria for plackett-burman and other nonregular factorial designs
01 Jan 1999-
TL;DR: In this paper, a generalized resolution criterion is defined and used for assessing non-regular fractional factorials, notably Plackett-Burman designs, which is intended to capture projection properties, complementing that of Webb (1964) whose concept of resolution concerns the estimability of lower order fractional fractional factors under the assumption that higher order effects are negligible.
Abstract: Resolution has been the most widely used criterion for comparing regular fractional factorials since it was introduced in 1961 by Box and Hunter. In this pa- per, we examine how a generalized resolution criterion can be defined and used for assessing nonregular fractional factorials, notably Plackett-Burman designs. Our generalization is intended to capture projection properties, complementing that of Webb (1964) whose concept of resolution concerns the estimability of lower order ef- fects under the assumption that higher order effects are negligible. Our generalized resolution provides a fruitful criterion for ranking different designs while Webb's resolution is mainly useful as a classification rule. An additional advantage of our approach is that the idea leads to a natural generalization of minimum aberration. Examples are given to illustrate the usefulness of the new criteria.
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17 Jan 2017
TL;DR: In this article, Wu et al. proposed three new types of frequency tables for assessing the quality of level-balanced factorial designs, which are coding invariant and behave more favorably when used for mixed-level arrays.
Abstract: Quality assessment of factorial designs, particularly mixed-level factorial designs, is a nontrivial task. Existing methods for orthogonal arrays include generalized minimum aberration, a modification thereof that was proposed by Wu and Zhang for mixed two- and four-level arrays, and minimum projection aberration. For supersaturated designs, E(s2) or χ2-based criteria are widely used. Based on recent insights by Gromping and Xu regarding the interpretation of the projected aR values used in minimum projection aberration, this article proposes three new types of frequency tables for assessing the quality of level-balanced factorial designs. These are coding invariant, which is particularly important for designs with qualitative factors. The proposed tables are used in the same way as those used in minimum projection aberration and behave more favorably when used for mixed-level arrays. Furthermore, they are much more manageable than the above-mentioned approach by Wu and Zhang. The article justifie...
4 citations
Cites background from "Generalized resolution and minimum ..."
...Grömping and Xu (2014) defined the generalized resolution (GR) as a generalized version of the definition of Deng and Tang (1999). In terms of average R2 values, their definition can be written as GR = R + 1 − √ aveR(2)worst;, i....
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...Grömping and Xu (2014) defined the generalized resolution (GR) as a generalized version of the definition of Deng and Tang (1999)....
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Patent•
10 Jul 2019TL;DR: In this paper, a computing device obtains a metric N indicating a quantity of a plurality of test cases for an output design of an experiment Each element of a test case of the output design is a test condition for testing one of factors for the experiment.
Abstract: A computing device obtains a metric N indicating a quantity of a plurality of test cases for an output design of an experiment Each element of a test case of the output design is a test condition for testing one of factors for the experiment. The computing device obtains input indicating a quantity p of an indicated plurality of factors for the output design. The computing device determines whether there are stored instructions for generating an initial screening design for the experiment. The computing device responsive to determining that there are stored instructions, selects, using the stored instructions, the initial screening design for the experiment. The computing device determines whether to modify the initial screening design based on modification criteria comprising a secondary criterion, the metric N, and/or the quantity p. The computing device outputs an indication of the updated screening design for the output design of the experiment.
4 citations
Patent•
22 Nov 2019TL;DR: In this paper, a computing device receives data comprising inputs representing a respective option for each of factors in each of test cases, and receives a request requesting an evaluation of the data for generating a model (e.g. a machine learning algorithm) to predict responses based on the factors.
Abstract: A computing device receives data comprising inputs representing a respective option for each of factors in each of test cases. The data comprises a response of the system for each of the test cases. The computing device receives a request requesting an evaluation of the data for generating a model (e.g. a machine learning algorithm) to predict responses based on the factors. The computing device obtains different group identifiers for each of groups for distributing the test cases for the system (e.g., groups of a K-fold cross-validation). The computing device for each of validation(s): generates a data set comprising a respective data element for each of the test cases of the plurality of test cases; and controls assignment of a group identifier of the different group identifiers to each of the respective data elements. The computing device outputs an indication of one or more generated data sets for the validation(s).
4 citations
01 Jan 2010
TL;DR: In this paper, the authors give an expository review of applications of computational algebraic statistics to design and analysis of fractional factorial experiments based on their recent works, which greatly enlarges the scope of factorial designs.
Abstract: We give an expository review of applications of computational algebraic statistics to design and analysis of fractional factorial experiments based on our recent works. For the purpose of design, the techniques of Gröbner bases and indicator functions allow us to treat fractional factorial designs without distinction between regular designs and nonregular designs. For the purpose of analysis of data from fractional factorial designs, the techniques of Markov bases allow us to handle discrete observations. Thus the approach of computational algebraic statistics greatly enlarges the scope of fractional factorial designs.
4 citations
Cites result from "Generalized resolution and minimum ..."
...We also compared our criterion to a generalized minimum aberration criterion by [12] and affinely full dimensionality....
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03 Jan 2012
TL;DR: In this paper, the authors present several criteria for choosing fractional factorial designs, including the popular criterion of minimum aberration, and the issue of optimal blocking of fractional Factorial designs was discussed in Section 13.8.3.4 of HK2.
Abstract: Fractional factorial designs have a long history of successful use in scientific investigations and industrial experiments. This important subject was treated in Chapter 13 of Hinkelmann and Kempthorne (2005), Design and Analysis of Experiments, Volume 2, hereafter referred to as HK2. Several criteria for choosing fractional factorial designs, including the popular criterion of minimum aberration, were briefly presented in Section 13.3.4 of HK2, and the issue of optimal blocking of fractional factorial designs was discussed in Section 13.8.3. These criteria were proposed for choosing designs with better capability of estimating lower-order effects, albeit with different interpretations of such capability. The differences in the interpretations sometimes lead to inconsistences or even contradictions among the different criteria, and one should not expect any criterion to work in all circumstances. In this chapter, we give a
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References
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3,546 citations
TL;DR: In this paper, the problem of choosing a design such that the polynomial f(ξ) = f (ξ1, ξ2, · · ·, ξ k ) fitted by the method of least squares most closely represents the true function over some region of interest R in the ξ space, no restrictions being introduced that the experimental points should necessarily lie inside R, is considered.
Abstract: The general problem is considered of choosing a design such that (a) the polynomial f(ξ) = f(ξ1, ξ2, · · ·, ξ k ) in the k continuous variables ξ' = (ξ1, ξ2, · · ·, ξ k ) fitted by the method of least squares most closely represents the true function g(ξ1, ξ2, · · ·, ξ k ) over some “region of interest” R in the ξ space, no restrictions being introduced that the experimental points should necessarily lie inside R; and (b) subject to satisfaction of (a), there is a high chance that inadequacy of f(ξ) to represent g(ξ) will be detected. When the observations are subject to error, discrepancies between the fitted polynomial and the true function occur: i. due to sampling error (called here “variance error”), and ii. due to the inadequacy of the polynomial f(ξ) exactly to represent g(ξ) (called here “bias error”). To meet requirement (a) the design is selected so as to minimize J, the expected mean square error averaged over the region R. J contains two components, one associated entirely with varian...
697 citations
"Generalized resolution and minimum ..." refers result in this paper
...Finally, we note that our argument for minimizing biases is similar to that in Box and Draper (1959)....
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471 citations
"Generalized resolution and minimum ..." refers methods in this paper
...For a detailed discussion on the concept of resolution for regular factorials, we refer to Box and Hunter (1961)....
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TL;DR: The 2 k-p Fractional Factorial Designs Part I. as discussed by the authors is a collection of fractional fractional factorial designs with a focus on the construction of the construction.
Abstract: (2000). The 2 k—p Fractional Factorial Designs Part I. Technometrics: Vol. 42, No. 1, pp. 28-47.
449 citations
TL;DR: In this article, the concept of aberration is proposed as a way of selecting the best designs from those with maximum resolution, and algorithms are presented for constructing these minimum aberration designs.
Abstract: For studying k variables in N runs, all 2 k–p designs of maximum resolution are not equally good. In this paper the concept of aberration is proposed as a way of selecting the best designs from those with maximum resolution. Algorithms are presented for constructing these minimum aberration designs.
420 citations
"Generalized resolution and minimum ..." refers methods in this paper
...For results on minimum aberration designs, we refer to Fries and Hunter (1980), Franklin (1984), Chen and Wu (1991), Chen (1992), Tang and Wu (1996), Chen and Hedayat (1996) and Cheng, Steinberg and Sun (1999)....
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