Centered L2-discrepancy Research Articles

Design Efficiency of the Asymmetric Minimum Projection Uniform Designs

Highly efficient designs and uniform designs are widely applied in many fields because of their good properties. The purpose of this paper is to study the issue of design efficiency for asymmetric minimum projection uniform designs. Based on the centered L2 discrepancy, the uniformity of the designs with mixed levels is defined, which is used to measure the projection uniformity of the designs. The analytical relationship between the uniformity pattern and the design efficiency is established for mixed-level orthogonal arrays with a strength of two. Moreover, a tight lower bound of the uniformity pattern is presented. The research is relevant in the field of experimental design by providing a theoretical basis for constructing the minimum number of projection uniform designs with a high design efficiency under a certain condition. These conclusions are verified by some numerical examples, which illustrate the theoretical results obtained in this paper.

Convergence of Uniformity Criteria and the Application in Numerical Integration

Quasi-Monte Carlo (QMC) methods have been successfully used for the estimation of numerical integrations arising in many applications. In most QMC methods, low-discrepancy sequences have been used, such as digital nets and lattice rules. In this paper, we derive the convergence rates of order of some improved discrepancies, such as centered L2-discrepancy, wrap-around L2-discrepancy, and mixture discrepancy, and propose a randomized QMC method based on a uniform design constructed by the mixture discrepancy and Baker’s transformation. Moreover, the numerical results show that the proposed method has better approximation than the Monte Carlo method and many other QMC methods, especially when the number of dimensions is less than 10.

Modification of the Maximin and ϕp (Phi) Criteria to Achieve Statistically Uniform Distribution of Sampling Points

This article proposes a sampling technique that delivers robust designs, that is, point sets selected from a design domain in the shape of a unit hypercube. The designs are guaranteed to provide a statistically uniform point distribution, meaning that every location has the same probability of being selected. Moreover, the designs are sample uniform, meaning that each individual design has its points spread evenly throughout the domain. The sample uniformity (often measured via a discrepancy criterion) is achieved using distance-based criteria ( or Maximin), that is, criteria normally used in space-filling designs. We show that the standard intersite metrics employed in distance-based criteria (Maximin and (phi)) do not deliver statistically uniform designs. Similarly, designs optimized via centered L2 discrepancy or support points are also not statistically uniform. When these designs (after optimization based on intersite distances) are used for Monte Carlo type of integration, their statistical nonuniformity is a serious problem as it may lead to a systematic bias. This article proposes using a periodic metric to guarantee the statistical uniformity of the family of distance-based designs. The presented designs used as benchmarks in the article are only taken from the class of Latin hypercube designs, which forces univariate projections to be uniform and improves accuracy in Monte Carlo integration of some functions. Supplementary materials for this article are available online.

An efficient approach for high-dimensional structural reliability analysis

Abstract This paper presents a new method to address the challenge of high-dimensional reliability analysis based on a small number of samples. The method is established based on the maximum entropy method (MEM) with the low-order fractional moments as constraints. A coordinate transformation is first implemented since a positive random variable is required for fractional operations. Then, an estimator-corrector scheme is employed to obtain the probability density function (PDF) of the performance function. This scheme first promptly provides an estimated fractional orders and Lagrange multipliers by solving a linear system of equations as initial values, and then searches the more accurate solutions around the initial values to recover the PDF. Besides, the local optimum can be avoided by using the estimator-corrector scheme. The centered L2 (CL2) discrepancy oriented sequential Latin-hyper cube simulation is proposed to evaluate the low-order fractional moments involved in MEM, which is of critical importance to the efficiency and accuracy for high-dimensional reliability analysis. Typical numerical examples are investigated to validate the proposed method. The results show that the proposed method is of efficacy for high-dimensional reliability problems, even with very low failure probabilities.

Sliced Latin hypercube designs with both branching and nested factors

One special kind of sliced Latin hypercube designs (SLHDs) for computer experiments with branching and nested factors is proposed here, where not only the whole design is an SLHD, but all its slices are also SLHDs. In addition, the SLHD in the first layer has a flexible number of slices, and the slice numbers of the SLHDs in the second layer can be flexible (either the same or different). The construction method is easy to implement, and the resulting designs are orthogonal under some mild conditions. Based on the centered L2-discrepancy, uniform SLHDs with branching and nested factors are further constructed.

Lower bound of average centered -discrepancy for -type designs

ABSTRACTUniform designs are widely used in various scientific investigations and industrial applications. By considering all possible level permutation of the factors, a connection between average centered L2-discrepancy and generalized wordlength pattern for asymmetrical fractional factorial designs is derived. Moreover, we present new lower bounds to the average centered L2-discrepancy for symmetrical and asymmetrical U-type designs. For illustration of the theoretical results, the lower bounds for symmetrical and asymmetrical U-type designs are tabulated, and numerical results indicate that our lower bounds behave well and can be recommended for use in practice.

Uniform minimum moment aberration designs

This paper discusses the issue of constructing uniform minimum moment aberration designs under discrepancies criteria. By considering all possible level permutations of factors, we establish a linear relationship between power moments and average discrepancy defined by a reproducing kernel for an asymmetrical or symmetrical design. We prove that minimum moment aberration designs often have low average discrepancies. Moreover, the average centered L2-discrepancy is expressed as a linear combination of power moments for a given design. An efficient method for constructing uniform minimum moment aberration designs is proposed. Some asymmetrical uniform minimum moment aberration designs obtained by our method have low centered L2-discrepancy and can be recommended for use in practice.

A study of uniformity pattern for extended designs

ABSTRACTThe objective of this paper is to investigate the issue of projection discrepancy for extended U-type or nearly U-type extended designs along the line of Fang and Qin (2005) based on the centered L2-discrepancy proposed in Hickernell (1998). Extended designs are obtained through augmenting optimally few runs (or points) to an optimal U-type design. Lower bounds to projection discrepancy with reference to the centered L2-discrepancy of extended designs have been obtained. Some illustrative examples are also provided.

A two-step methodology to apply low-discrepancy sequences in reliability assessment of structural dynamic systems

This study introduces various low-discrepancy sequences and then develops a new methodology for reliability assessment for structural dynamic systems. In this methodology, a two-step algorithm is first proposed, in which the most uniformly scattered point set among the low-discrepancy sequences is selected according to the centered L2-discrepancy (CL2 discrepancy) and then rearranged to minimize the generalized F-discrepancy (GF discrepancy). After that, the developed point set is incorporated into the maximum entropy method to capture the fractional moments for deriving the extreme value distribution for reliability assessment of structural dynamic systems. Numerical examples are investigated, where the results are compared with those obtained from Monte Carlo simulations, demonstrating the accuracy and efficiency of the proposed methodology.

Fold Over Plan for Uniform Design

This article discusses the fold over problem for uniform design in terms of the centered L2 discrepancy. Both full fold over plan and partial fold over plan are studied. Illustrative examples are given.

A note on optimal foldover four-level factorials

The foldover is a quick and useful technique in construction of fractional factorial designs, which typically releases aliased factors or interactions. The issue of employing the uniformity criterion measured by the centered L2-discrepancy to assess the optimal foldover plans was studied for four-level design. A new analytical expression and a new lower bound of the centered L2-discrepancy for fourlevel combined design under a general foldover plan are respectively obtained. A necessary condition for the existence of an optimal foldover plan meeting this lower bound was described. An algorithm for searching the optimal four-level foldover plans is also developed. Illustrative examples are provided, where numerical studies lend further support to our theoretical results. These results may help to provide some powerful and efficient algorithms for searching the optimal four-level foldover plans.

Lower bound of centered [formula omitted]-discrepancy for mixed two and three levels [formula omitted]-type designs

In this paper, we investigate a new analytical expression of the centered L2-discrepancy measure of uniformity for mixed two and three-level U-type designs in depth. Based on this new formulation, we present a new lower bound to the centered L2-discrepancy for U-type designs with mixed two and three-level, which can be used as a benchmark in searching uniform U-type designs. We also describe a necessary condition for the existence of uniform designs meeting this lower bound. For illustration of the usage of our theoretical results, a catalog of lower bounds for U-type designs in U(n;2m1×3m2) is tabulated, where 0≤m1≤10, 0≤m2<25 and 6≤n<60.

Constructing uniform designs under mixture discrepancy

The discrepancies have played an important role in quasi-Monte Carlo methods and uniform design. Zhou et al. (2013) proposed a new type of discrepancy, mixture discrepancy (MD), and showed that MD may be a better uniformity measure than wrap-around L2-discrepancy and centered L2-discrepancy. In this paper, some constructing methods for uniform designs under MD are shown. The relationship between MD and the generalized wordlength pattern for multi-level design is given, then the level permutation technique is shown as a useful tool to search uniform designs. Moreover, it is shown that MD can be represented by quadratic form and the global optimum solution of experimental design under MD is also given. Furthermore, by such quadratic form, the relationship between a design and its complementary design is shown, which can be used to construct uniform design with large size.

Resolvable orthogonal array-based uniform sliced Latin hypercube designs

Sliced Latin hypercube designs, introduced by Qian (2012), are widely used for computer experiments with qualitative and quantitative factors, multiple experiments, cross-validation and stochastic optimization. In this paper, we propose a new class of sliced Latin hypercube design, called the resolvable orthogonal array-based uniform sliced Latin hypercube design. Such designs are constructed via both symmetric and asymmetric resolvable orthogonal arrays, and measured by the centered L2 discrepancy criterion. When the construction is based on a resolvable orthogonal array with strength w+1, the resulting design not only possesses stratification in any w-dimensional projection for each slice, but also achieves stratification in any (w+1)-dimensional projection for the whole design. Furthermore, the uniformity of the resulting design is also highly improved with respect to the centered L2 discrepancy criterion.

New lower bound for centered [formula omitted]-discrepancy of four-level [formula omitted]-type designs

A new lower bound of the centered L2-discrepancy for four-level U-type designs is obtained. Our new lower bound is sharper and valid for a lot of designs more than other existing lower bound, which is a useful complement to the lower bounds of discrepancies.

Sampling efficiency in Monte Carlo based uncertainty propagation strategies: Application in seawater intrusion simulations

The implementation of Monte Carlo simulations (MCSs) for the propagation of uncertainty in real-world seawater intrusion (SWI) numerical models often becomes computationally prohibitive due to the large number of deterministic solves needed to achieve an acceptable level of accuracy. Previous studies have mostly relied on parallelization and grid computing to decrease the computational time of MCSs. However, another approach which has received less attention in the literature is to decrease the number of deterministic simulations by using more efficient sampling strategies. Sampling efficiency is a measure of the optimality of a sampling strategy. A more efficient sampling strategy requires fewer simulations and less computational time to reach a certain level of accuracy. The efficiency of a sampling strategy is highly related to its space-filling characteristics.This paper illustrates that the use of optimized Latin hypercube sampling (OLHS) strategies instead of the widely employed simple random sampling (SRS) and Latin hypercube sampling (LHS) strategies, can significantly improve sampling efficiency and hence decrease the simulation time of MCSs. Nine OLHS strategies are evaluated including: improved Latin hypercube sampling (IHS); optimum Latin hypercube (OLH) sampling; genetic optimum Latin hypercube (GOLH) sampling; three sampling strategies based on the enhanced stochastic evolutionary (ESE) algorithm namely φp-ESE which employs the φp space-filling criterion, CLD-ESE which utilizes the centered L2-discrepancy (CLD) space-filling criterion, and SLD-ESE which uses the star L2-discrepancy (SLD) space-filling criterion; and three sampling strategies based on the simulated annealing (SA) algorithm namely φp-SA which employs the φp criterion, CLD-SA which uses the CLD criterion, and SLD-SA which utilizes the SLD criterion. The study applies SRS, LHS and the nine OLHS strategies to MCSs of two synthetic test cases of SWI. The two test cases are the Henry problem and a two-dimensional radial representation of SWI in a circular island. The comparison demonstrates that the CLD-ESE strategy is the most efficient among the evaluated strategies. This paper also demonstrates how the space-filling characteristics of different OLHS designs change with variations in the input arguments of their optimization algorithms.

공간채움 조건을 만족하는 컴퓨터 실험 시나리오의 효율적 생성

In general, simulation models are effectively used in the field of engineering design. The experiment with simulation models to obtain optimal design parameters, however, is a time-consuming task and requires a lot of resources. Hence, meta-models representing the relationships between input variables and performance measures are exploited to efficiently determine the value of design parameters. To construct a meta-model, a number of simulation executions with sample scenarios are required. The number and quality of sample scenarios determine not only the level of efficiency in constructing the meta-model but also accuracy of the model. Space-filling condition is regarded to be an important condition for the quality of scenarios. This paper proposes sample scenario generation methods based on space-filling measures such as maxmin, Audze-Eglais, and centered L2-discrepancy. The performance of these scenario generation methods are evaluated through experiments.

An effective construction method for multi-level uniform designs

Uniform designs are widely used in various applications. However, it is computationally intractable to construct uniform designs, even for moderate number of runs, factors and levels. We establish a linear relationship between average squared centered L2-discrepancy and generalized wordlength pattern, and then based on it, we propose a general method for constructing uniform designs with arbitrary number of levels. The main idea is to choose a generalized minimum aberration design and then permute its levels. We propose a novel stochastic algorithm and obtain many new uniform designs that have smaller centered L2-discrepancies than the existing ones.

A note on lower bound of centered L 2-discrepancy on combined designs

This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered L2-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the initial design.

Lower bounds of various discrepancies on combined designs

The foldover is a useful technique in construction of two-level factorial designs. A foldover design is the follow-up experiment generated by reversing the sign(s) of one or more factors in the initial design. The full design obtained by joining the runs in the foldover design to those of the initial design is called the combined design. In this paper, some lower bounds of various discrepancies of combined designs, such as centered L2-discrepancy, symmetric L2-discrepancy and wrap-around L2-discrepancy, under a general foldover plan are obtained, which can be used as a benchmark for searching optimal foldover plans. Our results provide a theoretical justification for optimal foldover plans in terms of uniformity criterion.