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The Gaussian process is an indispensable tool for spatial data analysts. The onset of the “big data” era, however, has lead to the traditional Gaussian process being computationally infeasible for modern spatial data. As such, various alternatives to the full Gaussian process that are more amenable to handling big spatial data have been proposed. These modern methods often exploit low-rank structures and/or multi-core and multi-threaded computing environments to facilitate computation. This study provides, first, an introductory overview of several methods for analyzing large spatial data. Second, this study describes the results of a predictive competition among the described methods as implemented by different groups with strong expertise in the methodology. Specifically, each research group was provided with two training datasets (one simulated and one observed) along with a set of prediction locations. Each group then wrote their own implementation of their method to produce predictions at the given location and each was subsequently run on a common computing environment. The methods were then compared in terms of various predictive diagnostics.

We provide a new approach to approximate emulation of large computer experiments. By focusing expressly on desirable properties of the predictive equations, we derive a family of local sequential design schemes that dynamically define the support of a Gaussian process predictor based on a local subset of the data. We further derive expressions for fast sequential updating of all needed quantities as the local designs are built up iteratively. Then we show how independent application of our local design strategy across the elements of a vast predictive grid facilitates a trivially parallel implementation. The end result is a global predictor able to take advantage of modern multicore architectures, providing a nonstationary modeling feature as a bonus. We demonstrate our method on two examples using designs with thousands of data points, and compare to the method of compactly supported covariances. Supplementary materials for this article are available online.

We investigate the merits of replication, and provide methods for optimal design (including replicates), with the goal of obtaining globally accurate emulation of noisy computer simulation experiments. We first show that replication can be beneficial from both design and computational perspectives, in the context of Gaussian process surrogate modeling. We then develop a lookahead-based sequential design scheme that can determine if a new run should be at an existing input location (i.e., replicate) or at a new one (explore). When paired with a newly developed heteroscedastic Gaussian process model, our dynamic design scheme facilitates learning of signal and noise relationships which can vary throughout the input space. We show that it does so efficiently, on both computational and statistical grounds. In addition to illustrative synthetic examples, we demonstrate performance on two challenging real-data simulation experiments, from inventory management and epidemiology. Supplementary materials for the article are available online.

Recent implementations of local approximate Gaussian process models have pushed computational boundaries for nonlinear, nonparametric prediction problems, particularly when deployed as emulators for computer experiments. Their flavor of spatially independent computation accommodates massive parallelization, meaning that they can handle designs two or more orders of magnitude larger than previously. However, accomplishing that feat can still require massive computational horsepower. Here we aim to ease that burden. We study how predictive variance is reduced as local designs are built up for prediction. We then observe how the exhaustive and discrete nature of an important search subroutine involved in building such local designs may be overly conservative. Rather, we suggest that searching the space radially, that is, continuously along rays emanating from the predictive location of interest, is a far thriftier alternative. Our empirical work demonstrates that ray-based search yields predictors with accuracy comparable to exhaustive search, but in a fraction of the time-for many problems bringing a supercomputer implementation back onto the desktop. Supplementary materials for this article are available online.

Games can be a powerful tool for learning about statistical methodology. Effective game design involves a fine balance between caricature and realism, to simultaneously illustrate salient concepts in a controlled setting and serve as a testament to real-world applicability. Striking that balance is particularly challenging in response surface and design domains, where real-world scenarios often play out over long time scales, during which theories are revised, model and inferential techniques are improved, and knowledge is updated. Here, I present a game, borrowing liberally from one first played over 40 years ago, which attempts to achieve that balance while reinforcing a cascade of topics in modern nonparametric response surfaces, sequential design, and optimization. The game embeds a blackbox simulation within a shiny app whose interface is designed to simulate a realistic information-availability setting, while offering a stimulating, competitive environment wherein students can try out new methodology, and ultimately appreciate its power and limitations. Interface, rules, timing with course material, and evaluation are described, along with a \"case study\" involving a cohort of students at Virginia Tech. Supplementary materials for this article are available online.

Motivated by a computer experiment for the design of a rocket booster, this article explores nonstationary modeling methodologies that couple stationary Gaussian processes with treed partitioning. Partitioning is a simple but effective method for dealing with nonstationarity. The methodological developments and statistical computing details that make this approach efficient are described in detail. In addition to providing an analysis of the rocket booster simulator, we show that our approach is effective in other arenas as well.

Epidemiological interventions aim to control the spread of infectious disease through various mechanisms, each carrying a different associated cost. We describe a flexible statistical framework for generating optimal epidemiological interventions that are designed to minimize the total expected cost of an emerging epidemic while simultaneously propagating uncertainty regarding the underlying disease model parameters through to the decision process. The strategies produced through this framework are adaptive: vaccination schedules are iteratively adjusted to reflect the anticipated trajectory of the epidemic given the current population state and updated parameter estimates. Using simulation studies based on a classic influenza outbreak, we demonstrate the advantages of adaptive interventions over non-adaptive ones, in terms of cost and resource efficiency, and robustness to model misspecification.

Ubiquitous automated data collection at an unprecedented scale is making available streaming, real-time information flows in a wide variety of settings, transforming both science and industry. Learning algorithms deployed in such contexts often rely on single-pass inference, where the data history is never revisited. Learning may also need to be temporally adaptive to remain up-to-date against unforeseen changes in the data generating mechanism. Online Bayesian inference remains challenged by such transient, evolving data streams. Nonparametric modeling techniques can prove particularly ill-suited, as the complexity of the model is allowed to increase with the sample size. In this work, we take steps to overcome these challenges by porting information theoretic heuristics, such as exponential forgetting and active learning, into a fully Bayesian framework. We showcase our methods by augmenting a modern non-parametric modeling framework, dynamic trees, and illustrate its performance on a number of practical examples. The end product is a powerful streaming regression and classification tool, whose performance compares favorably to the state-of-the-art.

Constrained blackbox optimization is a difficult problem, with most approaches coming from the mathematical programming literature. The statistical literature is sparse, especially in addressing problems with nontrivial constraints. This situation is unfortunate because statistical methods have many attractive properties: global scope, handling noisy objectives, sensitivity analysis, and so forth. To narrow that gap, we propose a combination of response surface modeling, expected improvement, and the augmented Lagrangian numerical optimization framework. This hybrid approach allows the statistical model to think globally and the augmented Lagrangian to act locally. We focus on problems where the constraints are the primary bottleneck, requiring expensive simulation to evaluate and substantial modeling effort to map out. In that context, our hybridization presents a simple yet effective solution that allows existing objective-oriented statistical approaches, like those based on Gaussian process surrogates and expected improvement heuristics, to be applied to the constrained setting with minor modification. This work is motivated by a challenging, real-data benchmark problem from hydrology where, even with a simple linear objective function, learning a nontrivial valid region complicates the search for a global minimum. Supplementary materials for this article are available online.

A single-index model (SIM) provides for parsimonious multidimensional nonlinear regression by combining parametric (linear) projection with univariate nonparametric (nonlinear) regression models. We show that a particular Gaussian process (GP) formulation is simple to work with and ideal as an emulator for some types of computer experiment as it can outperform the canonical separable GP regression model commonly used in this setting. Our contribution focuses on drastically simplifying, reinterpreting, and then generalizing a recently proposed fully Bayesian GP-SIM combination. Favorable performance is illustrated on synthetic data and a real-data computer experiment. Two R packages, both released on CRAN, have been augmented to facilitate inference under our proposed model(s).