Rotated sphere packing designs

We propose a new class of space-filling designs called rotated sphere packing designs for computer experiments. The approach starts from the asymptotically optimal positioning of identical balls that covers the unit cube. Properly scaled, rotated, translated and extracted, such designs are excellent in maximin distance criterion, low in discrepancy, good in projective uniformity and thus useful in both prediction and numerical integration purposes. We provide a fast algorithm to construct such designs for any numbers of dimensions and points with R codes available online. Theoretical and numerical results are also provided

Improved Aircraft Environmental Impact Segmentation via Metric Learning

Accurate modeling of aircraft environmental impact is pivotal to the design of operational procedures and policies to mitigate negative aviation environmental impact. Aircraft environmental impact segmentation is a process which clusters aircraft types that have similar environmental impact characteristics based on a set of aircraft features. This practice helps model a large population of aircraft types with insufficient aircraft noise and performance models and contributes to better understanding of aviation environmental impact. Through measuring the similarity between aircraft types, distance metric is the kernel of aircraft segmentation. Traditional ways of aircraft segmentation use plain distance metrics and assign equal weight to all features in an unsupervised clustering process. In this work, we utilize weakly-supervised metric learning and partial information on aircraft fuel burn, emissions, and noise to learn weighted distance metrics for aircraft environmental impact segmentation. We show in a comprehensive case study that the tailored distance metrics can indeed make aircraft segmentation better reflect the actual environmental impact of aircraft. The metric learning approach can help refine a number of similar data-driven analytical studies in aviation.Comment: 32 pages, 11 figure

Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equations

Partial differential equations (PDEs) are widely used for the description of physical and engineering phenomena. Some key parameters involved in PDEs, which represent certain physical properties with important scientific interpretations, are difficult or even impossible to measure directly. Estimating these parameters from noisy and sparse experimental data of related physical quantities is an important task. Many methods for PDE parameter inference involve a large number of evaluations for numerical solutions to PDE through algorithms such as the finite element method, which can be time-consuming, especially for nonlinear PDEs. In this paper, we propose a novel method for the inference of unknown parameters in PDEs, called the PDE-Informed Gaussian Process (PIGP) based parameter inference method. Through modeling the PDE solution as a Gaussian process (GP), we derive the manifold constraints induced by the (linear) PDE structure such that, under the constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an augmentation method that transforms the nonlinear PDE into an equivalent PDE system linear in all derivatives, which our PIGP-based method can handle. The proposed method can be applied to a broad spectrum of nonlinear PDEs. The PIGP-based method can be applied to multi-dimensional PDE systems and PDE systems with unobserved components. Like conventional Bayesian approaches, the method can provide uncertainty quantification for both the unknown parameters and the PDE solution. The PIGP-based method also completely bypasses the numerical solver for PDEs. The proposed method is demonstrated through several application examples from different areas

Recent advances on the reduction and analysis of big and high-dimensional data

In an era with remarkable advancements in computer engineering, computational algorithms, and mathematical modeling, data scientists are inevitably faced with the challenge of working with big and high-dimensional data. For many problems, data reduction is a necessary first step; such reduction allows for storage and portability of big data, and enables the computation of expensive downstream quantities. The next step then involves the analysis of big data -- the use of such data for modeling, inference, and prediction. This thesis presents new methods for big data reduction and analysis, with a focus on solving real-world problems in statistics, machine learning and engineering.Ph.D

Contributions to quality improvement methodologies and computer experiments

This dissertation presents novel methodologies for five problem areas in modern quality improvement and computer experiments, i.e., selective assembly, robust design with computer experiments, multivariate quality control, model selection for split plot experiments, and construction of minimax designs. Selective assembly has traditionally been used to achieve tight specifications on the clearance of two mating parts. Chapter 1 proposes generalizations of the selective assembly method to assemblies with any number of components and any assembly response function, called generalized selective assembly (GSA). Two variants of GSA are considered: direct selective assembly (DSA) and fixed bin selective assembly (FBSA). In DSA and FBSA, the problem of matching a batch of N components of each type to give N assemblies that minimize quality cost is formulated as axial multi-index assignment and transportation problems respectively. Realistic examples are given to show that GSA can significantly improve the quality of assemblies. Chapter 2 proposes methods for robust design optimization with time consuming computer simulations. Gaussian process models are widely employed for modeling responses as a function of control and noise factors in computer experiments. In these experiments, robust design optimization is often based on average quadratic loss computed as if the posterior mean were the true response function, which can give misleading results. We propose optimization criteria derived by taking expectation of the average quadratic loss with respect to the posterior predictive process, and methods based on the Lugannani-Rice saddlepoint approximation for constructing accurate credible intervals for the average loss. These quantities allow response surface uncertainty to be taken into account in the optimization process. Chapter 3 proposes a Bayesian method for identifying mean shifts in multivariate normally distributed quality characteristics. Multivariate quality characteristics are often monitored using a few summary statistics. However, to determine the causes of an out-of-control signal, information about which means shifted and the directions of the shifts is often needed. We propose a Bayesian approach that gives this information. For each mean, an indicator variable that indicates whether the mean shifted upwards, shifted downwards, or remained unchanged is introduced. Default prior distributions are proposed. Mean shift identification is based on the modes of the posterior distributions of the indicators, which are determined via Gibbs sampling. Chapter 4 proposes a Bayesian method for model selection in fractionated split plot experiments. We employ a Bayesian hierarchical model that takes into account the split plot error structure. Expressions for computing the posterior model probability and other important posterior quantities that require evaluation of at most two uni-dimensional integrals are derived. A novel algorithm called combined global and local search is proposed to find models with high posterior probabilities and to estimate posterior model probabilities. The proposed method is illustrated with the analysis of three real robust design experiments. Simulation studies demonstrate that the method has good performance. The problem of choosing a design that is representative of a finite candidate set is an important problem in computer experiments. The minimax criterion measures the degree of representativeness because it is the maximum distance of a candidate point to the design. Chapter 5 proposes algorithms for finding minimax designs for finite design regions. We establish the relationship between minimax designs and the classical set covering location problem in operations research, which is a binary linear program. We prove that the set of minimax distances is the set of discontinuities of the function that maps the covering radius to the optimal objective function value, and optimal solutions at the discontinuities are minimax designs. These results are employed to design efficient procedures for finding globally optimal minimax and near-minimax designs.Ph.D