Multi-output separable Gaussian process: Towards an efficient, fully Bayesian paradigm for uncertainty quantification

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractFor many real systems, several computer models may exist with different physics and predictive abilities.To achieve more accurate simulations/predictions, it is desirable for these models to be properly combined and calibrated. We propose the Bayesian calibration of computer model mixture method which relies on the idea of representing the real system output as a mixture of the available computer model outputs with unknown input dependent weight functions. The method builds a fully Bayesian predictive model as an emulator for the real system output by combining, weighting, and calibrating the available models in the Bayesian framework. Moreover, it fits a mixture of calibrated computer models that can be used by the domain scientist as a mean to combine the available computer models, in a flexible and principled manner, and perform reliable simulations. It can address realistic cases where one model may be more accurate than the others at different input values because the mixture weights, indicating the contribution of each model, are functions of the input. Inference on the calibration parameters can consider multiple computer models associated with different physics. The method does not require knowledge of the fidelity order of the models.We provide a technique able to mitigate the computational overhead due to the consideration of multiple computer models that is suitable to the mixture model framework. We implement the proposed method in a real world application involving the Weather Research and Forecasting large-scale climate model.

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“…An interesting extension would be to consider multi-output computer models by coupling the method with that of Bilionis et al [57].…”

Section: Conclusion and Extensionsmentioning

confidence: 99%

Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs

Karagiannis

Lin

2014

Journal of Computational Physics

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“…In fact, our examples imply that higher-order PC coefficients may be sparser in the spatial domain and adequately approximated by lower-degree polynomial series. The proposed method can be possibly extended to consider discontinuity by using binary tree partitioning (Chipman et al, 1998;Konomi et al, 2014) or capture smaller-scale variations, unexplained by the gPC part, by modeling the total truncation error term as a Gaussian process (O'Hagan, 1978;Bilionis et al, 2013). We believe that the former can lead to simpler gPC expansions while the latter can improve the estimates.…”

Section: Discussionmentioning

confidence: 99%

A Bayesian mixed shrinkage prior procedure for spatial–stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

Karagiannis

Konomi

Lin

2015

Journal of Computational Physics

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(2015) 'A Bayesian mixed shrinkage prior procedure for spatialstochastic basis selection and evaluation of gPC expansions : applications to elliptic SPDEs.', Journal of computational physics., 284 . pp. 528-546. Further information on publisher's website: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe propose a new fully Bayesian method to efficiently obtain the spectral representation of a spatial random field, which can conduct spatial-stochastic basis selection and evaluation of generalized Polynomial Chaos (gPC) expansions when the number of the available basis functions is significantly larger than the size of the training data-set. We develop a fully Bayesian stochastic procedure, called mixed shrinkage prior (MSP), which performs both basis selection and coefficient evaluation simultaneously. MSP involves assigning a prior probability on the gPC structure and assigning conjugate priors on the expansion coefficients that can be thought of as mixtures of Ridge-LASSO shrinkage priors, in augmented form. The method offers a number of advantages over existing compressive sensing methods in gPC literature, such that it recovers possible sparse structures in both stochastic and spatial domains while the resulted expansion can be re-used directly to economically obtain results at any spatial input values. Yet, it inherits all the advantages of Bayesian model uncertainty methods, e.g. accounts for uncertainty about basis significance and provides interval estimation through posterior distributions. A unique highlight of the MSP procedure is that it can address heterogeneous sparsity in the spatial domain for different random dimensions. Furthermore, it yields a compromise between Ridge and LASSO regressions, and hence combines a weak (l 2 -norm) and strong (l 1 -norm) shrinkage, in an adaptive, data-driven manner. We demonstrate the good performance of the proposed method, and compare it against other existing compressive sensing ones on elliptic stochastic partial differential equations.

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“…In contrast, here we are dealing with the problem of learning a multioutput function. Despite the fact that there is a wealth of methods for learning multioutput functions [30,31,32,33]; we chose a simple approach that treats each output dimension of f , independently. In particular, we assume that each of the components f r , r = 1, .…”

Section: Learning the Reduced Dynamicsmentioning

confidence: 99%

Data-driven model for solar irradiation based on satellite observations

2014

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We construct a data-driven model for solar irradiation based on satellite observations. The model yields probabilistic estimates of the irradiation field every thirty minutes starting from two consecutive satellite measurements. The probabilistic nature of the model captures prediction uncertainties and can therefore be used by solar energy producers to quantify the operation risks. The model is simple to implement and can make predictions in realtime with minimal computational resources. To deal with the high-dimensionality of the satellite data, we construct a reduced representation using factor analysis. Then, we model the dynamics of the reduced representation as a discrete (30-minute interval) dynamical system. In order to convey information about the movement of the irradiation field, the dynamical system has a two-step delay. The dynamics are represented in a nonlinear, nonparameteric way by a recursive Gaussian process. The predictions of the model are compared with observed satellite data as well as with a similar model that uses only ground observations at the prediction site. We conclude that using satellite data in an area including the prediction site signficantly improves the prediction compared with models using only ground observation site data.

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Multi-output separable Gaussian process: Towards an efficient, fully Bayesian paradigm for uncertainty quantification

Cited by 172 publications

References 32 publications

Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs

Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs

A Bayesian mixed shrinkage prior procedure for spatial–stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

Data-driven model for solar irradiation based on satellite observations

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