Matthew Plumlee

Fast Prediction of Deterministic Functions Using Sparse Grid Experimental Designs

Random field models have been widely employed to develop a predictor of an expensive function based on observations from an experiment. The traditional framework for developing a predictor with random field models can fail due to the computational burden it requires. This problem is often seen in cases where the input of the expensive function is high dimensional. While many previous works have focused on developing an approximative predictor to resolve these issues, this article investigates a different solution mechanism. We demonstrate that when a general set of designs is employed, the resulting predictor is quick to compute and has reasonable accuracy. The fast computation of the predictor is made possible through an algorithm proposed by this work. This article also demonstrates methods to quickly evaluate the likelihood of the observations and describes some fast maximum likelihood estimates for unknown parameters of the random field. The computational savings can be several orders of magnitude when the input is located in a high-dimensional space. Beyond the fast computation of the predictor, existing research has demonstrated that a subset of these designs generate predictors that are asymptotically efficient. This work details some empirical comparisons to the more common space-filling designs that verify the designs are competitive in terms of resulting prediction accuracy.

Bayesian Calibration of Inexact Computer Models

ABSTRACT Bayesian calibration is used to study computer models in the presence of both a calibration parameter and model bias. The parameter in the predominant methodology is left undefined. This results in an issue, where the posterior of the parameter is suboptimally broad. There has been no generally accepted alternatives to date. This article proposes using Bayesian calibration, where the prior distribution on the bias is orthogonal to the gradient of the computer model. Problems associated with Bayesian calibration are shown to be mitigated through analytic results in addition to examples. Supplementary materials for this article are available online.

Calibrating Functional Parameters in the Ion Channel Models of Cardiac Cells

ABSTRACT Computational modeling is a popular tool to understand a diverse set of complex systems. The output from a computational model depends on a set of parameters that are unknown to the designer, but a modeler can estimate them by collecting physical data. In the described study of the ion channels of ventricular myocytes, the parameter of interest is a function as opposed to a scalar or a set of scalars. This article develops a new modeling strategy to nonparametrically study the functional parameter using Bayesian inference with Gaussian process prior distributions. A new sampling scheme is devised to address this unique problem.

Building accurate emulators for stochastic simulations via Quantile Kriging

Computer simulation has increasingly become popular for analysis of systems that cannot be feasibly changed because of costs or scale. This work proposes a method to construct an emulator for stochastic simulations by performing a designed experiment on the simulator and developing an emulative distribution. Existing emulators have focused on estimation of the mean of the simulation output, but this work presents an emulator for the distribution of the output. This construction provides both an explicit distribution and a fast sampling scheme. Beyond the emulator description, this work demonstrates the emulator’s efficiency, that is, its convergence rate is the asymptotically optimal among all possible emulators using the same sample size (under certain conditions). An example of its practical use is demonstrated using a stochastic simulation of fracture mechanics. Supplementary materials for this article are available online.

Lifted Brownian Kriging Models

ABSTRACT Gaussian processes have become a standard framework for modeling deterministic computer simulations and producing predictions of the response surface. This article investigates a new covariance function that is shown to offer superior prediction compared to the more common covariances for computer simulations of real physical systems. This is demonstrated via a gamut of realistic examples. A simple, closed-form expression for the covariance is derived as a limiting form of a Brownian-like covariance model as it is extended to some hypothetical higher-dimensional input domain, and so we term it a lifted Brownian covariance. This covariance has connections with the multiquadric kernel. Through analysis of the kriging model, this article offers some theoretical comparisons between the proposed covariance model and existing covariance models. The major emphasis of the theory is explaining why the proposed covariance is superior to its traditional counterparts for many computer simulations of real physical systems. Supplementary materials for this article are available online.

Learning stochastic model discrepancy

The vast majority of stochastic simulation models are imperfect in that they fail to fully emulate the entirety of real dynamics. Despite this, these imperfect models are still useful in practice, so long as one knows how the model is inexact. This inexactness is measured by a discrepancy between the proposed stochastic model and a true stochastic distribution across multiple values of some decision variables. In this paper, we propose a method to learn the discrepancy of a stochastic simulation using data collected from the system of interest. Our approach is a novel Bayesian framework that addresses the requirements for estimation of probability measures.

Improving prediction from stochastic simulation via model discrepancy learning

Stochastic simulation is an indispensable tool in operations and management applications. However, simulation models are only approximations to reality, and often bear discrepancies with the generating processes of real output data. We investigate a framework to statistically learn these discrepancies under the presence of data on past implemented system configurations, which allows us to improve prediction using simulation models. We focus on the case of general continuous output data that generalizes previous work. Our approach utilizes (a combination of) regression analysis and optimization formulations constrained on suitable summary statistics. We demonstrate our approach with a numerical example.