In many real-world applications we are interested in approximating costly functions that are analytically unknown, e.g. complex computer codes. An emulator provides a fast approximation of such functions relying on a limited number of evaluations. Gaussian processes (GPs) are commonplace emulators due to their statistical properties such as the ability to estimate their own uncertainty. GPs are essentially developed to fit smooth, continuous functions. However, the assumptions of continuity and smoothness is unwarranted in many situations. For example, in computer models where bifurcations or tipping points occur, the outputs can be discontinuous. This work examines the capacity of GPs for emulating step-discontinuous functions. Several approaches are proposed for this purpose. Two special covariance functions/kernels are adapted with the ability to model discontinuities. They are the neural network and Gibbs kernels whose properties are demonstrated using several examples. Another approach, which is called warping, is to transform the input space into a new space where a GP with a standard kernel, such as the Matern family, is able to predict the function well. The transformation is perform by a parametric map whose parameters are estimated by maximum likelihood. The results show that the proposed approaches have superior performance to GPs with standard kernels in capturing sharp jumps in the true function.
Computer models are widely used across a range of scientific disciplines to describe various complex physical systems, however to perform full uncertainty quantification we often need to employ emulators. An emulator is a fast statistical construct that mimics the slow to evaluate computer model, and greatly aids the vastly more computationally intensive uncertainty quantification calculations that an important scientific analysis often requires. We examine the problem of emulating computer models that possess multiple, partial discontinuities occurring at known non-linear location. We introdu…
A Gaussian process has been one of the important approaches for emulating computer simulations. However, the stationarity assumption for a Gaussian process and the intractability for large-scale dataset limit its availability in practice. In this article, we propose a clustered Gaussian process model which segments the input data into multiple clusters, in each of which a Gaussian process model is performed. The stochastic expectation-maximization is employed to efficiently fit the model. In our simulations as well as a real application to solar irradiance emulation, our proposed method had sm…
The dynamic emulation of non-linear deterministic computer codes where the output is a time series, possibly multivariate, is examined. Such computer models simulate the evolution of some real-world phenomenon over time, for example models of the climate or the functioning of the human brain. The models we are interested in are highly non-linear and exhibit tipping points, bifurcations and chaotic behaviour. However, each simulation run could be too time-consuming to perform analyses that require many runs, including quantifying the variation in model output with respect to changes in the inpu…
Emulators for Large-scale Computer Experiments with Quantitative and Qualitative Inputs
Queen’s University（女王大学）；Paul H. Chook Department of Information Systems and Statistics, Baruch College, The City University of New York（纽约市立大学巴鲁克学院保罗·H·丘克信息系统与统计系）
Emulation of complex computer simulations have become an effective tool in the exploration of the behaviour of the simulated processes. Agriculture is one such area where the simulation of crop growth, nutrition, soil condition and pollution could be invaluable in any land management decisions. In this paper, we study output from the EPIC simulation model to investigate the behaviour of crop yield in response to changes in inputs such as fertilizer levels, soil, steepness, and other environmental covariates. We build a model for crop yield around a non-linear Mitscherlich Baule growth model to…
Mathematical models of biological systems are beginning to be used for safety-critical applications, where large numbers of repeated model evaluations are required to perform uncertainty quantification and sensitivity analysis. Most of these models are nonlinear both in variables and parameters/inputs which has two consequences. First, analytic solutions are rarely available so repeated evaluation of these models by numerically solving differential equations incurs a significant computational burden. Second, many models undergo bifurcations in behaviour as parameters are varied. As a result, s…
Gaussian process (GP) based statistical surrogates are popular, inexpensive substitutes for emulating the outputs of expensive computer models that simulate real-world phenomena or complex systems. Here, we discuss the evolution of dynamic GP model - a computationally efficient statistical surrogate for a computer simulator with time series outputs. The main idea is to use a convolution of standard GP models, where the weights are guided by a singular value decomposition (SVD) of the response matrix over the time component. The dynamic GP model also adopts a localized modeling approach for bui…
Gaussian processes (GPs) are widely used as surrogate models for emulating computer code, which simulate complex physical phenomena. In many problems, additional boundary information (i.e., the behavior of the phenomena along input boundaries) is known beforehand, either from governing physics or scientific knowledge. While there has been recent work on incorporating boundary information within GPs, such models do not provide theoretical insights on improved convergence rates. To this end, we propose a new GP model, called BdryGP, for incorporating boundary information. We show that BdryGP not…
It is often desirable to build a statistical emulator of a complex computer simulator in order to perform analysis which would otherwise be computationally infeasible. We propose methodology to model multivariate output from a computer simulator taking into account output structure in the responses. The utility of this approach is demonstrated by applying it to a chemical and biological hazard prediction model. Predicting the hazard area that results from an accidental or deliberate chemical or biological release is imperative in civil and military planning and also in emergency response. The …
Complex computer codes are often too time expensive to be directly used to perform uncertainty propagation studies, global sensitivity analysis or to solve optimization problems. A well known and widely used method to circumvent this inconvenience consists in replacing the complex computer code by a reduced model, called a metamodel, or a response surface that represents the computer code and requires acceptable calculation time. One particular class of metamodels is studied: the Gaussian process model that is characterized by its mean and covariance functions. A specific estimation procedure …
In a number of cases, the Quantile Gaussian Process (QGP) has proven effective in emulating stochastic, univariate computer model output (Plumlee and Tuo, 2014). In this paper, we develop an approach that uses this emulation approach within a Bayesian model calibration framework to calibrate an agent-based model of an epidemic. In addition, this approach is extended to handle the multivariate nature of the model output, which gives a time series of the count of infected individuals. The basic modeling approach is adapted from Higdon et al. (2008), using a basis representation to capture the mu…
Gaussian process (GP) models are commonly used statistical metamodels for emulating expensive computer simulators. Fitting a GP model can be numerically unstable if any pair of design points in the input space are close together. Ranjan, Haynes, and Karsten (2011) proposed a computationally stable approach for fitting GP models to deterministic computer simulators. They used a genetic algorithm based approach that is robust but computationally intensive for maximizing the likelihood. This paper implements a slightly modified version of the model proposed by Ranjan et al. (2011), as the new R p…
Gaussian processes (GP) are widely used as a metamodel for emulating time-consuming computer codes. We focus on problems involving categorical inputs, with a potentially large number L of levels (typically several tens), partitioned in G << L groups of various sizes. Parsimonious covariance functions, or kernels, can then be defined by block covariance matrices T with constant covariances between pairs of blocks and within blocks. We study the positive definiteness of such matrices to encourage their practical use. The hierarchical group/level structure, equivalent to a nested Bayesian linear …
The graphics processing unit (GPU) has emerged as a powerful and cost effective processor for general performance computing. GPUs are capable of an order of magnitude more floating-point operations per second as compared to modern central processing units (CPUs), and thus provide a great deal of promise for computationally intensive statistical applications. Fitting complex statistical models with a large number of parameters and/or for large datasets is often very computationally expensive. In this study, we focus on Gaussian process (GP) models -- statistical models commonly used for emulati…
Calibration of expensive computer models with high-dimensional output fields can be approached via history matching. If the entire output field is matched, with patterns or correlations between locations or time points represented, calculating the distance metric between observational data and model output for a single input setting requires a time intensive inversion of a high-dimensional matrix. By using a low-dimensional basis representation rather than emulating each output individually, we define a metric in the reduced space that allows the implausibility for the field to be calculated e…
Generative Bayesian Computation as a Scalable Alternative to Gaussian Process Surrogates
Booth School of Business, University of Chicago（芝加哥大学商学院）；Department of Systems Engineering and Operations Research, George Mason University（乔治·马歇尔大学系统工程与运营管理系）
Gaussian Process (GP) models are popular statistical surrogates used for emulating computationally expensive computer simulators. The quality of a GP model fit can be assessed by a goodness of fit measure based on optimized likelihood. Finding the global maximum of the likelihood function for a GP model is typically very challenging as the likelihood surface often has multiple local optima, and an explicit expression for the gradient of the likelihood function is typically unavailable. Previous methods for optimizing the likelihood function (e.g. MacDonald et al. (2013)) have proven to be robu…
Surrogate models have become ubiquitous in science and engineering for their capability of emulating expensive computer codes, necessary to model and investigate complex phenomena. Bayesian emulators based on Gaussian processes adequately quantify the uncertainty that results from the cost of the original simulator, and thus the inability to evaluate it on the whole input space. However, it is common in the literature that only a partial Bayesian analysis is carried out, whereby the underlying hyper-parameters are estimated via gradient-free optimisation or genetic algorithms, to name a few me…
