08:30
Emulating computer models with step-discontinuous outputs using Gaussian processes
Hossein Mohammadi | University of Exeter | United Kingdom
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Authors:
Hossein Mohammadi | University of Exeter | United Kingdom
Peter Challenor | University of Exeter | United Kingdom
Complex computer models (or simulators) are widely used in many applications for the study of real-world phenomena. Such models are based on complex mathematical equations, e.g. PDEs, that make them computationally expensive. One way to overcome computational time when performing analysis such as uncertainty quantification that requires very many simulation runs, is to approximate the simulator by a surrogate model that is fast. Gaussian process (GP) emulators are powerful probabilistic models which can be used to fit any smooth, continuous function. 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.
In this talk, we present several approaches for modelling step-discontinuities based on GPs. This includes adapting two “special” covariance functions and the transformation of input space, also known as warping. The two covariance functions are neural network and Gibbs kernels whose properties are demonstrated using several examples. In warping, the input space is transformed into a new space where a GP with a standard kernel, e.g. Matern family of kernels, is able to predict the discontinuous function well. The results show that the proposed methods have superior performance to GPs with standard covariance kernels in capturing sharp jumps in the “true” function.
08:50
Tree-based Gaussian Process with Many Qualitative Factors
Wei-Ann Lin | National Cheng Kung University | Taiwan
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Wei-Ann Lin | National Cheng Kung University | Taiwan
Chih-Li Sung | Michigan state University | United States
Ray-Bing Chen | National Cheng Kung University | Taiwan
In computer experiments, Gaussian process models are commonly used for emulation. However, when both qualitative and quantitative factors in the experiments, emulation using Gaussian process models becomes challenging. In particular, when many qualitative factors are in the experiments, existing methods in the literature become cumbersome due to curse of dimensionality. Motivated by the computer simulations for the design of a cooling system, we propose a new tree-based Gaussian process for emulating computer experiments with many qualitative and quantitative factors. The proposed method incorporates tree structures to model the qualitative factors, with Gaussian process models in the leaf nodes for modeling quantitative factors. Numerical simulations as well as a real example for the design of a cooling system show that the proposed enjoys good prediction accuracy while retaining the model interpretation.
09:10
Bayesian Variable Selection in Gaussian Process Models for Computer Experiments
Ray-Bing Chen | National Cheng Kung University | Taiwan
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Ray-Bing Chen | National Cheng Kung University | Taiwan
Fan Zhang | Arizona State University | United States
Ying Hung | Rutgers The State University of New Jersey | United States
Xinwei Deng | Virginia Tech | United States
In this work, we are interested in variable selection problems in Gaussian process models. Here, we not only focus on the mean regression function but also take the covariance structure into account. In this setting, a variable is called active if the corresponding regression coefficient is not zero or its hyperparameter in the correlation function is not zero. To accomplish our goal, we first treat the regression coefficient and the hyperparameter in the correlation function as a group, and an indicator is added into the model to denote the status of this group. A Bayesian selection approach is proposed, and the active variables are identified based on the posterior samples of the indicators. The performance of the proposed Bayesian selection approach is illustrated using simulations for real applications in Computer Experiments.
09:30
Sequential input adding in computer experiments
Thierry Gonon | Ecole centrale de Lyon | France
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Thierry Gonon | Ecole centrale de Lyon | France
Celine Helbert | Ecole centrale de Lyon | France
Christophette Blanchet | Ecole centrale de Lyon | France
Industrial numerical codes are complex as they involve lots of input variables. The first studies made on them generally focus on a small amount of important inputs. The rest of the inputs are fixed to nominal values. The following studies become more and more complex, involving some previously fixed variables. The classical approach of doing independent designs of experiments and metamodels for each study may be very greedy in simulations. An alternative solution is to update gradually the design and the metamodel based on the previous ones. In this talk, we present a recursive method which imbricates the new and the previous metamodel each time an input (or a group of inputs) is added. We propose a Gaussian process modelling. The complementary term between the previous and the new metamodel is assumed to be the realization of a Gaussian process. It is subject to be zero on the support of the previous metamodel. As the support is composed of an infinite number of points, specific mathematical tools are necessary. We also introduce a sequential sampling procedure to train the imbricated metamodels.
09:50
Alternative Latent Space Representations in Latent Variable Gaussian Process Modeling
Wei Chen | Northwestern University | United States
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Authors:
Siyu Tao | Northwestern University | United States
Daniel Apley | Northwestern University | United States
Matthew Plumlee | Northwestern University | United States
Wei Chen | Northwestern University | United States
Gaussian process (GP) models are extensively used for emulating expensive computer simulations. There is a need for extending the GP model to problems with both qualitative/categorical and quantitative/continuous variables. We recently proposed latent variable GP (LVGP) modeling approach which maps each qualitative variable to an underlying numerical latent variable and estimates the mapped value for each. The LVGP approach exploits that the categorical inputs’ effects in physics-based simulation are commonly due to some underlying quantitative variables that are generally unknown. LVs are used in LVGP models to mimic the underlying quantitative variables, and our empirical study has shown that low-dimensional (eg. 2D) LVs are typically sufficient to achieve very good modeling accuracy. However, the best form of LVs and their dimensionality are unknown. This talk will study alternative representations; specifically, we compare the previously used Cartesian space approach and a new hyperspherical space approach. We find that the previous approach is more suitable for models whose inter-level correlation matrices are high-rank. We also demonstrate the new approach is more flexible when the true inter-level correlation matrices are low-rank. Finally, we show that for specific modeling tasks, a model choice criterion can be used to select the most appropriate model and the associated latent space dimensionality.
10:10
Multifidelity Gaussian process metamodel for time-dependent outputs
Baptiste Kerleguer | CEA DAM | France
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Authors:
Baptiste Kerleguer | CEA DAM | France
Claire Cannamela | CEA DAM | France
Josselin Garnier | Centre de Mathematiques Appliquees, Ecole Polytechnique | France
Gaussian processes are widely used as surrogate models to study the outputs of interest of an expensive computer code. This work focuses on multi-fidelity computer codes, which are codes modeling the same phenomenon, but which can be hierarchically sorted according to their accuracy and numerical cost. To enrich the high-fidelity metamodel, we collect information on low-fidelity codes. We extend the multi-fidelity CoKriging method [1,2] when the output is a time series. Instead of reducing the dimension of the output by principal component analysis, the proposed approach uses a tensorized structure of the covariance functions. To show the efficiency of our method, we present some examples.
[1] M. C. KENNEDY and A. O'HAGAN, Predicting the output from a complex computer code when fast approximations are available, Biometrika, 2000, vol. 87, 1-13 (2000).
[2] L. LE GRATIET and J. GARNIER, Recursive co-kriging model for Design of Computer experiments with multiple levels of fidelity, Int. J. Uncertainty Quantification, vol. 4, 365-386 (2014).