Two-level Orthogonal Arrays Research Articles

Dual-Orthogonal Arrays for Order-of-Addition Two-Level Factorial Experiments

A new class of orthogonal arrays called dual-orthogonal arrays is proposed in this paper to design order-of-addition two-level factorial experiments in which both component addition orders and component levels can be varied over treatments. Dual-orthogonal arrays can be viewed as an optimal combination of order-of-addition orthogonal arrays and two-level orthogonal arrays. Based on these two different concepts of orthogonality, when a compound model is used to fit the observed data, both pairwise order effects and component main effects can be estimated with optimal efficiency. Under the assumption of normality, these two kinds of parametric effects can also be inferred independently. A three-drug combination study is first used to show that dual-orthogonal arrays can be practical for real-world studies. Both combinatorial and computational methods are then introduced to construct dual-orthogonal arrays. Additionally, a design catalogue is generated for future work.

Semifoldover plans for three-level orthogonal arrays with quantitative factors

Although foldover designs can de-alias many effects, they involve at least twice the original number of runs. A semifoldover design, one kind of the partial foldover designs, is typically much more efficient since such a design adds only half of the new runs of a foldover design to the initial design. Semifoldover designs for two-level orthogonal arrays have been investigated in recent literatures. With the use of linear-quadratic system, this paper considers semifoldover designs for three-level orthogonal arrays with quantitative factors. We examine when the linear effects can be de-aliased from their aliased two-factor interactions for regular and nonregular designs, and obtain some good properties via semifolding over on partial factors or all factors. Theoretical properties and some examples are provided to illustrate the usefulness of the proposed designs.

Block-regularized repeated learning-testing for estimating generalization error

Repeated learning-testing (RLT) is a popular cross-validation method that is usually adopted in estimation and comparison of algorithm performance in machine learning. However, the variance of the estimator of the generalization error in the RLT method is easily affected by random partitions. Poor data partitioning may cause a large overlap between any two training sets of RLT and enlarge the variance in the RLT estimator of the generalization error. Thus, in this study, we constrain numbers of overlapping samples between any two training sets and construct a novel data partitioning schema in which RLT is called the block-regularized RLT (BRLT). We theoretically prove that the BRLT estimator has a smaller variance than does the RLT estimator. Specifically, the variance of the RLT estimator reaches its minimum when samples in a data set equally occur in all training sets and all the numbers of overlapping samples are equal. Furthermore, we provide easy-to-use construction algorithms of BRLT’s partition set for several training set size and partition count settings by adopting two-level orthogonal arrays. We also illustrate BRLT’s optimal properties with several simulated and real-world examples.

Optimizing two-level orthogonal arrays for simultaneously estimating main effects and pre-specified two-factor interactions

This paper considers the construction of D-optimal two-level orthogonal arrays that allow for the joint estimation of all main effects and a specified set of two-factor interactions. A sharper upper bound on the determinant of the related matrix is derived. To numerically obtain D-optimal and nearly D-optimal orthogonal arrays of large run sizes, an efficient search procedure is proposed based on a discrete optimization algorithm. Results on designs of 20, 24, 28, 36, 44 and 52 runs with three or fewer two-factor interactions are illustrated here to demonstrate the performance of the proposed approach. In addition, two cases with four two-factor interactions are also demonstrated here.

Efficient arrangements of two-level orthogonal arrays in two and four blocks

ABSTRACTWhen orthogonal arrays are used in practical applications, it is often difficult to perform all the designed runs of the experiment under homogeneous conditions. The arrangement of factorial runs into blocks is usually an action taken to overcome such obstacles. However, an arbitrary configuration might lead to spurious analysis results. In this work, the nice properties of two-level orthogonal arrays are taken into consideration and an effective method for arranging experimental runs into two and four blocks of the same size is proposed. This method is based on the so-called J-characteristics of the corresponding array. General theoretical results are given for studying up to four experimental factors in two blocks, as well as for studying up to three experimental factors in four blocks. Finally, we provide best blocking arrangements when the number of the factors of interest is larger, by exploiting the known lists of non-isomorphic orthogonal arrays with two levels and various run sizes.

A comparison of two-level designs to estimate all main effects and two-factor interactions

ABSTRACTWe compare cost-efficient alternatives for the full factorial 24 design, the regular 25-1 fractional factorial design, and the regular 26-1 fractional factorial design that can fit the model consisting of all the main effects as well as all the two-factor interactions. For 4 and 5 factors we examine orthogonal arrays with 12 and 20 runs, respectively. For 6 factors we consider orthogonal arrays with 24 as well as 28 runs. We consult complete catalogs of two-level orthogonal arrays to find the ones that provide the most efficient estimation of all the effects in the model. We compare these arrays with D-optimal designs found using a coordinate exchange algorithm. The D-optimal designs are always preferable to the most efficient orthogonal arrays for fitting the full model in all the factors.

Risk decision-making based on Mahalanobis-Taguchi system and grey cumulative prospect theory for enterprise information investment

For the decision-making problem of enterprise information technology investment risk, based on existing research shortage, this paper proposes a risk decision-making method based on cloud model, Mahalanobis-Taguchi system and grey cumulative prospect theory with the consideration of the fuzziness o f evaluating language and subjective risk attitude of decision makers. The method results in the ``quantitative to qualitative'' transformation of risk evaluation in information technology investment by cloud model and constructs interval numbers decision matrices according to the ``3En'' principle. An m-dimensional interval numbers decision vector is regarded as super cuboids in m-dimensional attributes space, using two-level orthogonal experiment to arrange points uniformly and dispersedly. The numbers of points are assured by testing numbers of two-level orthogonal arrays and these points compose of distribution points set to stand for decision-making project. In order to eliminate the influence of correlation among indices, Mahalanobis distance is used to calculate the distance from each investment program to ideal programs. Since the decision-maker's attitude can affect the decision result when the investment decision-making in practice, the paper defines the grey correlation coefficient and the prospect value function based on Mahalanobis distance, and calculates the comprehensive prospect value of each program and finds the optimal one. Finally, an example application proves that the method is a feasible and effective method of decision-making. Compared with other methods, the result is much more realistic and superior.

Risky Group Decision-Making Method for Distribution Grid Planning

Abstract With rapid speed on electricity using and increasing in renewable energy, more and more research pay attention on distribution grid planning. For the drawbacks of existing research, this paper proposes a new risky group decision-making method for distribution grid planning. Firstly, a mixing index system with qualitative and quantitative indices is built. On the basis of considering the fuzziness of language evaluation, choose cloud model to realize “quantitative to qualitative” transformation and construct interval numbers decision matrices according to the “3En” principle. An m-dimensional interval numbers decision vector is regarded as super cuboids in m-dimensional attributes space, using two-level orthogonal experiment to arrange points uniformly and dispersedly. The numbers of points are assured by testing numbers of two-level orthogonal arrays and these points compose of distribution points set to stand for decision-making project. In order to eliminate the influence of correlation among indices, Mahalanobis distance is used to calculate the distance from each solutions to others which means that dynamic solutions are viewed as the reference. Secondly, due to the decision-maker’s attitude can affect the results, this paper defines the prospect value function based on SNR which is from Mahalanobis-Taguchi system and attains the comprehensive prospect value of each program as well as the order. At last, the validity and reliability of this method is illustrated by examples which prove the method is more valuable and superiority than the other.

Construction of minimum generalized aberration two-level orthogonal arrays

In this paper we explore the problem of constructing two-level Minimum Generalized Aberration (MGA) orthogonal arrays with strength $t$, $n$ runs and $q>t$ columns, using a method that employs the $J$-characteristics of a two-level design. General results for the construction of MGA orthogonal arrays with $t+1$, $t+2$ and $t+3$ columns are given, while all MGA designs with strength $t\ge 2$, $n \equiv$ 0 mod 4 runs and $q\le 6$ are constructed. Results are also given for two-level orthogonal arrays with $q=7$ factors, but with strength greater than two. Projection properties of the MGA designs that have been identified, are also discussed.

D-optimal two-level orthogonal arrays for estimating main effects and some specified two-factor interactions

Theoretical results are derived for finding D-optimal two-level orthogonal arrays for estimating main effects and some specified two-factor interactions. The upper bounds of the determinant of the related matrix for D-optimality are obtained. For run sizes 12 and 20, D-optimal orthogonal arrays are found for all requirement sets with three or less two-factor interactions, and all the D-optimal orthogonal arrays except one can be constructed by sequentially collecting columns. Examples are also given for run sizes 36 and 52 to discuss the guidelines to construct D-optimal orthogonal arrays.

Two-Level Designs of Strength 3 and up to 48 Runs

SummaryThe paper will help practitioners to select strength 3 designs that are useful for screening both main effects and two-factor interactions. We calculated word length patterns, correlations of four-factor interaction contrast vectors with the intercept and ranks of the two-factor interaction matrices for all non-equivalent two-level orthogonal arrays of strength 3 and run sizes up to 48. On the basis of these characteristics, there are a limited number of designs that can be recommended for practical use.

Improved WLP and GWP lower bounds based on exact integer programming

By using exact integer programming (IP) (integer programming in infinite precision) bounds on the word-length patterns (WLPs) and generalized word-length patterns (GWPs) for fractional factorial designs are improved. In the literature, bounds on WLPs are formulated as linear programming (LP) problems. Although the solutions to such problems must be integral, the optimization is performed without the integrality constraints. Two examples of this approach are bounds on the number of words of length four for resolution IV regular designs, and a lower bound for the GWP of two-level orthogonal arrays. We reformulate these optimization problems as IP problems with additional valid constraints in the literature and improve the bounds in many cases. We compare the improved bound to the enumeration results in the literature to find many cases for which our bounds are achieved. By using the constraints in our integer programs we prove that f ( 16 λ , 2 , 4 ) ⩽ 9 if λ is odd where f ( 2 t λ , 2 , t ) is the maximum n for which an OA ( N , n , 2 , t ) exists. We also present a theorem for constructing GMA OA ( N , N / 2 - u , 2 , 3 ) for u = 1 , … , 5 .

A construction of [formula omitted] by nonlinear functions and some classification for [formula omitted

In the case of s = 2 , Seiden and Zemach [1966. On orthogonal arrays. Ann. Math. Statist. 37, 1355–1370] classified all OA ( 2 t λ , t + 1 , 2 , t ) for any λ . Fujii et al. [1989. Classification of two symbol orthogonal arrays of strength t, t + 3 constraints and index 4, II. SUT J. Math. 25, 161–177] classified OA ( 2 t + 1 , t + 2 , 2 , t ) and OA ( 2 t + 2 , t + 3 , 2 , t ) . Recently, Stufken and Tang [2006. Complete enumeration of two-level orthogonal arrays of strength d with d + 2 constraints. Preprint] completely enumerated OA ( 2 t λ , t + 2 , 2 , t ) . Moreover, in the case of s = 3 , Hedayat et al. [1997a. On the maximum number of factors and the enumeration of 3-symbol orthogonal arrays of strength 3 and index 2. J. Statist. Plann. Inference 58, 43–63; 1997b. On the construction and existence of orthogonal arrays with three levels and indexes 1 and 2. Ann. Statist. 25, 2044–2053] showed that all OA ( 3 t , t + 1 , 3 , t ) are isomorphic and linear. Hedayat et al. [1997a. On the maximum number of factors and the enumeration of 3-symbol orthogonal arrays of strength 3 and index 2. J. Statist. Plann. Inference 58, 43–63] classified all OA ( 2 · 3 3 , 5 , 3 , 3 ) . In this paper, we treat the case of s ⩾ 4 . Firstly, we show a construction theorem of OA ( s t , t + 1 , s , t ) by utilizing nonlinear functions over GF ( s ) , where s = p l , p a prime, and l a multiple of p. Secondly, we classify OA ( 4 t , t + 1 , 4 , t ) for t = 2 , 3 and 4 by utilizing their functional representations.

Complete enumeration of two-Level orthogonal arrays of strength d with d + 2 constraints

Enumerating nonisomorphic orthogonal arrays is an important, yet very difficult, problem. Although orthogonal arrays with a specified set of parameters have been enumerated in a number of cases, general results are extremely rare. In this paper, we provide a complete solution to enumerating nonisomorphic two-level orthogonal arrays of strength d with d+2 constraints for any d and any run size n=λ2d. Our results not only give the number of nonisomorphic orthogonal arrays for given d and n, but also provide a systematic way of explicitly constructing these arrays. Our approach to the problem is to make use of the recently developed theory of J-characteristics for fractional factorial designs. Besides the general theoretical results, the paper presents some results from applications of the theory to orthogonal arrays of strength two, three and four.

Optimal Additions to and Deletions from Two-Level Orthogonal Arrays

SummaryConsider the problem of selecting a two-level factorial design. It is well known that two-level orthogonal arrays of strength 4 or more with e extra runs have various optimality properties including generalized Cheng (type 1) optimality when e=1, restricted Cheng (type 1) optimality when e=2 and E-optimality when 3⩽e⩽5. More general Schur optimality results are derived for more general values of e within the more restricted class of augmented two-level orthogonal arrays. Similar results are derived for the class of orthogonal arrays with deletions. Examples are used to illustrate the results and in many cases the designs are confirmed to be optimal across all two-level designs.

Non-isomorphic orthogonal arrays obtained by juxtaposition

In this paper, we obtain a large class of non-isomorphic two-level orthogonal arrays using juxtaposition of certain orthogonal arrays that arise from Hadamard matrices.

Estimation of Error Variance in the Analysis of Experiments Using Two-Level Orthogonal Arrays

We consider estimation of an unknown error variance based on the experiments using two-level orthogonal arrays under squared error loss. We address two shrinkage type estimators obtained by pooling sums of squares for factorial effects when they are smaller than the natural estimator or the unbiased one. We discuss whether these two shrinkage estimators uniformly improve upon the natural one or the unbiased one, respectively. It depends on the combination of the number of columns to which the error term is assigned and that of all factorial effects. Through numerical evaluation, we examine the conditions for uniform improvement. Performance of the improved estimators compared with the unbiased estimator is also examined by taking into account both the bias and MSE reduction.

Another Look at Projection Properties of Hadamard Matrices

Suppose a large number of factors (q) is examined in an experimental situation. It is often anticipated that only a few (k) of these will be important. Usually, it is not known which of the q factors will be the important ones, that is, it is not known which k columns of the experimental design will be of further interest. Screening designs are useful for such situations. It is of practical interest for a given k to know all the classes of inequivalent projections of the design into the k dimensions that have certain statistical properties, since it helps experimenters in selecting a screening design with favorable properties. In this paper we study all the classes of inequivalent projections of certain two-level orthogonal arrays that arise from Hadamard matrices, using well known statistical criteria, such as generalized resolution and generalized minimum aberration. We also pay attention to each projection's distinct runs. Results are given for orthogonal arrays with n = 16, 20 and 24 runs, when they are projected into small dimensions. Useful remarks on design selection are made based on the frequency of appearance of each projection design in every Hadamard matrix.

Blocked Nonregular Two-Level Factorial Designs

This article discusses the optimal blocking criteria for nonregular two-level designs. We extend the optimal blocking criteria of Cheng and Wu to nonregular designs by adapting the G- and G2-minimum aberration criteria discussed by Tang and Deng. To define word-length pattern for nonregular designs, we extend the notion of “word” to nonregular designs through a polynomial representation of factorial designs. We define treatment resolution and block resolution for evaluating the degrees of aliasing and confounding. We propose four new criteria, which we use to search for optimal blocking schemes of 12-run, 16-run, and 20-run two-level orthogonal arrays.

The Idle Column Method: Design Construction, Properties and Comparisons

Taguchi's idle column method for constructing arrays is investigated and generalized. A general construction method is proposed via a necessary and sufficient condition on two-level orthogonal arrays to which the method is applicable. A method is developed for identifying the idle column and pairs of columns in the arrays. Mixed two- and three-level idle column arrays with 64 or fewer runs are tabulated. Some guidelines are offered on when the idle column arrays should be used and when they should be replaced by orthogonal main-effect plans or D-optimal designs.