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 inputs. Therefore, Gaussian process emulators are used to approximate the output of the code. To do this, the flow map of the system under study is emulated over a short time period. Then, it is used in an iterative way to predict the whole time series. A number of ways are proposed to take into account the uncertainty of inputs to the emulators, after fixed initial conditions, and the correlation between them through the time series. The methodology is illustrated with two examples: the highly non-linear dynamical systems described by the Lorenz and Van der Pol equations. In both cases, the predictive performance is relatively high and the measure of uncertainty provided by the method reflects the extent of predictability in each system.
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…
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 …
Obtaining accurate estimates of satellite drag coefficients in low Earth orbit is a crucial component in positioning and collision avoidance. Simulators can produce accurate estimates, but their computational expense is much too large for real-time application. A pilot study showed that Gaussian process (GP) surrogate models could accurately emulate simulations. However, cubic runtime for training GPs means that they could only be applied to a narrow range of input configurations to achieve the desired level of accuracy. In this paper we show how extensions to the local approximate Gaussian Pr…
This chapter provides a tutorial overview of first principles methods to describe the properties of matter at the ground state or equilibrium. It begins with a brief introduction to quantum and statistical mechanics for predicting the electronic structure and diverse static properties of of many-particle systems useful for practical applications. Pedagogical examples are given to illustrate the basic concepts and simple applications of quantum Monte Carlo and density functional theory -- two representative methods commonly used in the literature of first principles modeling. In addition, this …
Computer models, also known as simulators, can be computationally expensive to run, and for this reason statistical surrogates, known as emulators, are often used. Any statistical model, including an emulator, should be validated before being used, otherwise resulting decisions can be misguided. We discuss how current methods for validating Gaussian process emulators of deterministic models are insufficient for emulators of stochastic computer models and develop a framework for diagnosing problems in stochastic emulators. These diagnostics are based on independently validating the mean and var…
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…
University of South Carolina（南卡罗来纳大学）；University of North Carolina Charlotte（北卡罗来纳大学夏洛特分校）
Markov Chain Monte Carlo (MCMC) methods have a drawback when working with a target distribution or likelihood function that is computationally expensive to evaluate, specially when working with big data. This paper focuses on Metropolis-Hastings (MH) algorithm for unimodal distributions. Here, an enhanced MH algorithm is proposed that requires less number of expensive function evaluations, has shorter burn-in period, and uses a better proposal distribution. The main innovations include the use of Bayesian optimization to reach the high probability region quickly, emulating the target distribut…
Response probability distribution estimation of expensive computer simulators: A Bayesian active learning perspective using Gaussian process regression
We consider the problem of calibration and uncertainty analysis for activity-based transportation simulators. Activity-Based Models (ABMs) rely on statistical modeling of individual travelers' behavior to predict higher-order travel patterns in metropolitan areas. Input parameters are typically estimated from traveler surveys using maximum likelihood. We develop an approach that uses a Gaussian Process emulator to calibrate those parameters using traffic flow data. Our approach extends traditional emulators to handle the high-dimensional and non-stationary nature of transportation simulators. …
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…
Multi-fidelity Gaussian process is a common approach to address the extensive computationally demanding algorithms such as optimization, calibration and uncertainty quantification. Adaptive sampling for multi-fidelity Gaussian process is a changing task due to the fact that not only we seek to estimate the next sampling location of the design variable, but also the level of the simulator fidelity. This issue is often addressed by including the cost of the simulator as an another factor in the searching criterion in conjunction with the uncertainty reduction metric. In this work, we extent the …
Helsinki Institute for Information Technology（赫尔辛基信息技术研究所）；Aalto University（阿尔托大学）；University of Oslo（奥斯陆大学）；University of Helsinki（赫尔辛基大学）；University of Manchester（曼彻斯特大学）
In recent years, surrogate models have been successfully used in likelihood-free inference to decrease the number of simulator evaluations. The current state-of-the-art performance for this task has been achieved by Bayesian Optimization with Gaussian Processes (GPs). While this combination works well for unimodal target distributions, it is restricting the flexibility and applicability of Bayesian Optimization for accelerating likelihood-free inference more generally. We address this problem by proposing a Deep Gaussian Process (DGP) surrogate model that can handle more irregularly behaved ta…
The recent accelerated growth in the computing power has generated popularization of experimentation with dynamic computer models in various physical and engineering applications. Despite the extensive statistical research in computer experiments, most of the focus had been on the theoretical and algorithmic innovations for the design and analysis of computer models with scalar responses. In this paper, we propose a computationally efficient statistical emulator for a large-scale dynamic computer simulator (i.e., simulator which gives time series outputs). The main idea is to first find a good…
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…
An emulator is a fast-to-evaluate statistical approximation of a detailed mathematical model (simulator). When used in lieu of simulators, emulators can expedite tasks that require many repeated evaluations, such as sensitivity analyses, policy optimization, model calibration, and value-of-information analyses. Emulators are developed using the output of simulators at specific input values (design points). Developing an emulator that closely approximates the simulator can require many design points, which becomes computationally expensive. We describe a self-terminating active learning algorit…
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）
This paper proposes a statistical simulator for the engine knock based on the Mixture Density Network (MDN) and the accept-reject method. The proposed simulator can generate the random knock intensity signal corresponding to the input signal. The generated knock intensity has a consistent probability distribution with the real engine. Firstly, the statistical analysis is conducted with the experimental data. From the analysis results, some important assumptions on the statistical properties of the knock intensity are made. Regarding the knock intensity as a random variable on the discrete-time…
Carnegie Mellon University（卡内基梅隆大学）；University of Michigan（密歇根大学）
Autonomous vehicles are expected to navigate in complex traffic scenarios with multiple surrounding vehicles. The correlations between road users vary over time, the degree of which, in theory, could be infinitely large, thus posing a great challenge in modeling and predicting the driving environment. In this paper, we propose a method to model multi-vehicle interactions using a stochastic vector field model and apply non-parametric Bayesian learning to extract the underlying motion patterns from a large quantity of naturalistic traffic data. We then use this model to reproduce the high-dimens…
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 …
Tensor-variate Gaussian process regression for efficient emulation of complex systems: comparing regressor and covariance structures in outer product and parallel partial emulators
We provide a method to solve optimization problem when objective function is a complex stochastic simulator of an urban transportation system. To reach this goal, a Bayesian optimization framework is introduced. We show how the choice of prior and inference algorithm effect the outcome of our optimization procedure. We develop dimensionality reduction techniques that allow for our optimization techniques to be applicable for real-life problems. We develop a distributed, Gaussian Process Bayesian regression and active learning models that allow parallel execution of our algorithms and enable us…
University of Washington（华盛顿大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
Fitting a theoretical model to experimental data in a Bayesian manner using Markov chain Monte Carlo typically requires one to evaluate the model thousands (or millions) of times. When the model is a slow-to-compute physics simulation, Bayesian model fitting becomes infeasible. To remedy this, a second statistical model that predicts the simulation output -- an "emulator" -- can be used in lieu of the full simulation during model fitting. A typical emulator of choice is the Gaussian process (GP), a flexible, non-linear model that provides both a predictive mean and variance at each input point…
