Nested Latin Hypercube Designs Research Articles

Nested Maximum Entropy Designs for Computer Experiments

Presently, computer experiments with multiple levels of accuracy are widely applied in science and engineering. This paper introduces a class of nested maximum entropy designs for such computer experiments. A multi-layer DETMAX algorithm is proposed to construct nested maximum entropy designs. Based on nested maximum entropy designs, we also propose an integer-programming procedure to specify the sample sizes in multi-fidelity computer experiments. Simulated annealing techniques are used to tackle complex optimization problems in the proposed methods. Illustrative examples show that the proposed nested entropy designs can yield better prediction results than nested Latin hypercube designs in the literature and that the proposed sample-size determination method is effective.

Iterative estimation of Sobol’ indices based on replicated designs

In the field of sensitivity analysis, Sobol’ indices are widely used to assess the importance of the inputs of a model to its output. Among the methods that estimate these indices, the replication procedure is noteworthy for its efficient cost. A practical problem is how many model evaluations must be performed to guarantee a sufficient precision on the Sobol’ estimates. The present paper tackles this issue by rendering the replication procedure iterative. The idea is to enable the addition of new model evaluations to progressively increase the accuracy of the estimates. These evaluations are done at points located in under-explored regions of the experimental designs, but preserving their characteristics. The key feature of this approach is the construction of nested space-filling designs. For the estimation of first-order indices, a nested Latin hypercube design is used. For the estimation of closed second-order indices, two constructions of a nested orthogonal array design are proposed. Regularity and uniformity properties of the nested designs are studied.

Uniform projection nested Latin hypercube designs

Computer experiments usually involve many factors, but only a few of them are active. In such a case, it is desirable to construct designs with good projection properties. Maximum projection designs and uniform projection designs have been developed for common experimental situations, however, there has been little study on constructing projection designs for high-accuracy computer experiments (HEs) and low-accuracy computer experiments (LEs) so far. In this paper, we propose a weighted uniform projection criterion, and construct uniform projection nested Latin hypercube designs to suit such computer experiment situations. We show that the obtained designs have good projection properties in all sub-dimensions, and we also discuss how to choose a proper value for the weight. Simulated examples are available to illustrate the effectiveness of the proposed designs.

A Minimum-Cost Modeling Method for Nonlinear Industrial Process Based on Multimodel Migration and Bayesian Model Averaging Method

With increasing drastic market competition, establishing an accurate and reliable performance prediction model for control and optimization at a minimum cost is a growing trend in industrial production. This article proposes a minimum-cost modeling method to develop the performance prediction model of a new nonlinear industrial process. The core idea of this approach is to migrate the useful information on multiple old and similar processes to develop a new process model. A multimodel migration strategy is proposed to migrate the useful information by combining the existing nonlinear process models and take full advantage of minimum data from the new nonlinear process. In order to obtain a set of optimal weights for combining the multiple old and similar process models, the Bayesian model averaging method is employed to estimate the contributions of each available old nonlinear process model to the new nonlinear process model. Moreover, a further experiment used nested Latin hypercube design (NLHD) to gather the necessary minimum data on the new nonlinear process for model migration. Finally, we apply the proposed minimum-cost modeling method to the new multistage centrifugal compressor in the combined cycle power plant, and the results show that the proposed method can develop an accurate compressor model at a minimal cost in terms of the amount of new process data. Note to Practitioners —Process optimal control and condition monitoring are vital for the stability and economic operation of industrial processes, and the basis of them is to quickly establish an accurate and reliable process performance prediction model. Traditional methods for developing process performance prediction models often require a large amount of complex calculations and rich process data, which is time- and cost-consuming. In particular, these methods focus only on the current process to be modeled, while ignoring the existing and similar process information, wasting process information. This article presents a minimum-cost modeling method for nonlinear industrial processes, which can make full use of information on multiple similar existing processes to assist the modeling of a new process to reduce the modeling cost of the new process. Specifically, a multimodel migration strategy including Bayesian model averaging is designed to migrate useful information from similar processes to the new process. The nested Latin hypercube design (NLHD) is employed to collect the necessary minimum data on the new nonlinear process. By applying the proposed approach to the industrial nonlinear process, it is possible to achieve the accurate performance prediction model with minimal new process data.

An output mapping variable fidelity metamodeling approach based on nested Latin hypercube design for complex engineering design optimization

Engineering design is usually a daunting optimization task which often involving time-consuming, even computation-prohibitive process. Variable fidelity metamodeling has been developed as an efficient approach to alleviate this issue, owing to its capacity of achieving accurate metamodels within limited sample size. An output mapping modeling method is proposed in this paper as an alternative variable fidelity metamodeling method in which the low fidelity outputs is directly mapped to the high fidelity outputs through least square support vector regression. Furthermore, a nested Latin hypercube design method is developed for the output mapping modeling, in which a simple sampling design is treated as a building block or DOE seed to be translated and propagated through the design space, resulting in a final sampling configuration. Effectiveness of the proposed method are demonstrated by several numerical functions and two engineering design problem, in which different sample sizes, predictive accuracies and robustness are considered.

A general construction for nested Latin hypercube designs

We propose a new construction for nested designs, called General Nested Latin Hypercube designs (GNLHs). Such designs contain nested Latin hypercube designs as special cases. Besides achieving maximum uniformity in one dimension, each layer of GNLHs is flexible in run sizes. Moreover, theoretical results and numerical simulations show that GNLHs perform well on the sampling variance.

Flexible Nested Latin Hypercube Designs for Computer Experiments

Multi-level and sequential computer experiments are commonly used to study complex systems in engineering and science. Suitable designs for such experiments require nested structures. Because a nested Latin hypercube design (NLHD) (Qian (2009)) can be constructed only when the run size in each larger design is a multiple of that in a smaller one, the use of NLHDs faces some limitations in practice. In this paper, we propose a random sampling procedure for constructing flexible NLHDs, which have no such a restriction. We also develop an efficient sequential algorithm to search for the optimal NLHDs with respect to some space-filling criteria. Simulation results show that our methods can yield flexible NLHDs with good space-filling properties.

Design for Sequential Follow-Up Experiments in Computer Emulations

ABSTRACTSequential experiments composed of initial experiments and follow-up experiments are widely adopted for economical computer emulations. Many kinds of Latin hypercube designs with good space-filling properties have been proposed for designing the initial computer experiments. However, little work based on Latin hypercubes has focused on the design of the follow-up experiments. Although some constructions of nested Latin hypercube designs can be adapted to sequential designs, the size of the follow-up experiments needs to be a multiple of that of the initial experiments. In this article, a general method for constructing sequential designs of flexible size is proposed, which allows the combined designs to have good one-dimensional space-filling properties. Moreover, the sampling properties and a type of central limit theorem are derived for these designs. Several improvements of these designs are made to achieve better space-filling properties. Simulations are carried out to verify the theoretical results. Supplementary materials for this article are available online.

Nested Latin Hypercube Designs with Sliced Structures

In computer experiments, space-filling designs with a sliced structure or nested structure have received much recent interest and been studied separately. However, it is likely that designs with both structures are needed in some situations, but there are no suitable designs so far. In this paper, we construct a special class of nested Latin hypercube designs with sliced structures, in such a design, a small sliced Latin hypercube design is nested within a large one. The construction method is easy to implement and the number of factors is flexible. Numerical simulations show the usefulness of the newly proposed designs.

Sequential Design and Analysis of High-Accuracy and Low-Accuracy Computer Codes

A growing trend in engineering and science is to use multiple computer codes with different levels of accuracy to study the same complex system. We propose a framework for sequential design and analysis of a pair of high-accuracy and low-accuracy computer codes. It first runs the two codes with a pair of nested Latin hypercube designs (NLHDs). Data from the initial experiment are used to fit a prediction model. If the accuracy of the fitted model is less than a prespecified threshold, the two codes are evaluated again with input values chosen in an elaborate fashion so that their expanded scenario sets still form a pair of NLHDs. The nested relationship between the two scenario sets makes it easier to model and calibrate the difference between the two sources. If necessary, this augmentation process can be repeated a number of times until the prediction model based on all available data has reasonable accuracy. The effectiveness of the proposed method is illustrated with several examples. Matlab codes are provided in the online supplement to this article.

Nested orthogonal array-based Latin hypercube designs

We propose two methods for constructing a new type of design, called a nested orthogonal array-based Latin hypercube design, intended for multi-fidelity computer experiments. Such designs are two nested space-filling designs in which the large design achieves stratification in both bivariate and univariate margins and the small design achieves stratification in univariate margins. These designs have better space-filling properties than nested Latin hypercube designs in which the large design possesses uniformity in univariate margins only. The first method expands an ordinary Latin hypercube design to a larger design that achieves uniformity in any one- or two-dimensional projection. The second method uses an orthogonal array with strength two to simultaneously construct a pair of nested orthogonal array-based Latin hypercube designs. Examples are given to illustrate the proposed methods. Sampling properties of the proposed designs are derived. Copyright 2011, Oxford University Press.

Nested Latin hypercube designs

We propose an approach to constructing nested Latin hypercube designs. Such designs are useful for conducting multiple computer experiments with different levels of accuracy. A nested Latin hypercube design with two layers is defined to be a special Latin hypercube design that contains a smaller Latin hypercube design as a subset. Our method is easy to implement and can accommodate any number of factors. We also extend this method to construct nested Latin hypercube designs with more than two layers. Illustrative examples are given. Some statistical properties of the constructed designs are derived.