Sliced rotated sphere packing designs

Space-filling designs are popular choices for computer experiments. A sliced design is a design that can be partitioned into several subdesigns. We propose a new type of sliced space-filling design called sliced rotated sphere packing designs. Their full designs and subdesigns are rotated sphere packing designs. They are constructed by rescaling, rotating, translating and extracting the points from a sliced lattice. We provide two fast algorithms to generate such designs. Furthermore, we propose a strategy to use sliced rotated sphere packing designs adaptively. Under this strategy, initial runs are uniformly distributed in the design space, follow-up runs are added by incorporating information gained from initial runs, and the combined design is space-filling for any local region. Examples are given to illustrate its potential application

Validating Sample Average Approximation Solutions with Negatively Dependent Batches

Sample-average approximations (SAA) are a practical means of finding approximate solutions of stochastic programming problems involving an extremely large (or infinite) number of scenarios. SAA can also be used to find estimates of a lower bound on the optimal objective value of the true problem which, when coupled with an upper bound, provides confidence intervals for the true optimal objective value and valuable information about the quality of the approximate solutions. Specifically, the lower bound can be estimated by solving multiple SAA problems (each obtained using a particular sampling method) and averaging the obtained objective values. State-of-the-art methods for lower-bound estimation generate batches of scenarios for the SAA problems independently. In this paper, we describe sampling methods that produce negatively dependent batches, thus reducing the variance of the sample-averaged lower bound estimator and increasing its usefulness in defining a confidence interval for the optimal objective value. We provide conditions under which the new sampling methods can reduce the variance of the lower bound estimator, and present computational results to verify that our scheme can reduce the variance significantly, by comparison with the traditional Latin hypercube approach

Data-Efficient Design and Analysis Methodologies for Computer and Physical Experiments

Data science for experimentation, including the rapidly growing area of the design and analysis of computer experiments, aims to use statistical approaches to collect and analyze (physical or virtual) experimental responses and facilitate decision-making. The cost for each run of an experiment can be expensive. This dissertation proposes novel data-efficient methodologies to tackle three different challenges in this field. The first two are regarding computer experiments, and the third one is regarding physical experiments. The first work aims to reconstruct the input-output relationship (surrogate model) given by the computer code via scattered evaluations with small sizes based on Gaussian process regression. Traditional isotropic Gaussian process models suffer from the curse of dimensionality when the input dimension is relatively high given limited data points. Gaussian process models with additive correlation functions are scalable to dimensionality, but they are more restrictive as they only work for additive functions. In the first work, we consider a projection pursuit model in which the nonparametric part is driven by an additive Gaussian process regression. We choose the dimension of the additive function higher than the original input dimension and call this strategy “dimension expansion”. We show that dimension expansion can help approximate more complex functions. A gradient descent algorithm is proposed for model training based on the maximum likelihood estimation. Simulation studies show that the proposed method outperforms the traditional Gaussian process models. The second work focuses on the designs of experiments (DoE) of multi-fidelity computer experiments with fixed budget. We consider the autoregressive Gaussian process model and the optimal nested design that maximizes the prediction accuracy subject to the budget constraint. An approximate solution is identified through the idea of multilevel approximation and recent error bounds of Gaussian process regression. The proposed (approximately) optimal designs admit a simple analytical form. We prove that, to achieve the same prediction accuracy, the proposed optimal multi-fidelity design requires much lower computational cost than any single-fidelity design in the asymptotic sense. The last work is proposed to model complex experiments when the distributions of training and testing input features are different (referred to as domain adaptation). In this work, we propose a novel transfer learning algorithm called Renewing Iterative Self-labeling Domain Adaptation (Re-ISDA) to tackle the domain adaptation problem. The learning problem is formulated as a dynamic programming model, and the latter is then solved by an efficient greedy algorithm by adding a renewing step to the original ISDA algorithm. This renewing step helps avoid a potential issue of the ISDA that the possible mis-labeled samples by a weak predictor in the initial stage of the iterative learning can cause serious harm to the subsequent learning process. Numerical studies show that the proposed method outperforms prevailing transfer learning methods. The proposed method also achieves high prediction accuracy for a cervical spine motion problem

Sliced Full Factorial-Based Latin Hypercube Designs as a Framework for a Batch Sequential Design Algorithm

The article of record as published may be found at http://dx.doi.org/10.1080/00401706.2015.1108233When fitting complex models, such as finite element or discrete event simulations, the experiment design should exhibit desirable properties of both projectivity and orthogonality. To reduce experimental effort, sequential design strategies allow experimenters to collect data only until some measure of prediction precision is reached. In this article, we present a batch sequential experiment design method that uses sliced full factorial-based Latin hypercube designs (sFFLHDs), which are an extension to the concept of sliced orthogonal array-based Latin hypercube designs (OALHDs). At all stages of the sequential design, good univariate stratification is achieved. The structure of the FFLHDs also tends to produce uniformity in higher dimensions, especially at certain stages of the design. We show that our batch sequential design approach has good sampling and fitting qualities through both empirical studies and theoretical arguments. Supplementary materials are available online.USMC-PMMIONR/NPS CRUSE