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SAMSI Distinguished, October 2006Slide 1 Tony O’Hagan, University of Sheffield Managing Uncertainty in Complex Models

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2 Outline Uncertainty in models Quantifying uncertainty Example: dynamic global vegetation model Reducing uncertainty Example: atmospheric deposition model Methods Research directions Dynamic models MUCM

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3 Computer models In almost all fields of science, technology, industry and policy making, people use mechanistic models to describe complex real- world processes For understanding, prediction, control There is a growing realisation of the importance of uncertainty in model predictions Can we trust them? Without any quantification of output uncertainty, it’s easy to dismiss them

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4 Examples Climate prediction Molecular dynamics Nuclear waste disposal Oil fields Engineering design Hydrology

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5 Sources of uncertainty A computer model takes inputs x and produces outputs y = f(x) How might y differ from the true real-world value z that the model is supposed to predict? Error in inputs x Initial values, forcing inputs, model parameters Error in model structure or solution Wrong, inaccurate or incomplete science Bugs, solution errors

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6 Quantifying uncertainty The ideal is to provide a probability distribution p(z) for the true real-world value The centre of the distribution is a best estimate Its spread shows how much uncertainty about z is induced by uncertainties on the last slide How do we get this? Input uncertainty: characterise p(x), propagate through to p(y) Structural uncertainty: characterise p(z-y)

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7 Example: UK carbon flux in 2000 Vegetation model predicts carbon exchange from each of 700 pixels over England & Wales Principal output is Net Biosphere Production Accounting for uncertainty in inputs Soil properties Properties of different types of vegetation Aggregated to England & Wales total Allowing for correlations Estimate 7.61 Mt C Std deviation 0.61 Mt C

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8 Maps

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9 Sensitivity analysis Map shows proportion of overall uncertainty in each pixel that is due to uncertainty in the vegetation parameters As opposed to soil parameters Contribution of vegetation uncertainty is largest in grasslands/moorlands

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10 England & Wales aggregate PFT Plug-in estimate (Mt C) Mean (Mt C) Variance (Mt C 2 ) Grass Crop Deciduous Evergreen Covariances0.001 Total

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11 Reducing uncertainty To reduce uncertainty, get more information! Informal – more/better science Tighten p(x) through improved understanding Tighten p(z-y) through improved modelling or programming Formal – using real-world data Calibration – learn about model parameters Data assimilation – learn about the state variables Learn about structural error z-y Validation

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12 Example: Nuclear accident Radiation was released after an accident at the Tomsk-7 chemical plant in 1993 Data comprise measurements of the deposition of ruthenium 106 at 695 locations obtained by aerial survey after the release The computer code is a simple Gaussian plume model for atmospheric dispersion Two calibration parameters Total release of 106 Ru (source term) Deposition velocity

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13 Data

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14 A small sample (N=10 to 25) of the 695 data points was used to calibrate the model Then the remaining observations were predicted and RMS prediction error computed On a log scale, error of 0.7 corresponds to a factor of 2 Calibration Sample size N Best fit calibration Bayesian calibration

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15 So far, so good, but In principle, all this is straightforward In practice, there are many technical difficulties Formulating uncertainty on inputs Elicitation of expert judgements Propagating input uncertainty Modelling structural error Anything involving observational data! The last two are intricately linked And computation

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16 The problem of big models Tasks like uncertainty propagation and calibration require us to run the model many times Uncertainty propagation Implicitly, we need to run f(x) at all possible x Monte Carlo works by taking a sample of x from p(x) Typically needs thousands of model runs Calibration Traditionally this is done by searching the x space for good fits to the data This is impractical if the model takes more than a few seconds to run We need a more efficient technique

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17 Gaussian process representation More efficient approach First work in early 1980s (DACE) Consider the code as an unknown function f(.) becomes a random process We represent it as a Gaussian process (GP) Training runs Run model for sample of x values Condition GP on observed data Typically requires many fewer runs than MC And x values don’t need to be chosen randomly

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18 Emulation Analysis is completed by prior distributions for, and posterior estimation of, hyperparameters The posterior distribution is known as an emulator of the computer code Posterior mean estimates what the code would produce for any untried x (prediction) With uncertainty about that prediction given by posterior variance Correctly reproduces training data

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code runs Consider one input and one output Emulator estimate interpolates data Emulator uncertainty grows between data points

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code runs Adding another point changes estimate and reduces uncertainty

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code runs And so on

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22 Then what? Given enough training data points we can emulate any model accurately So that posterior variance is small “everywhere” Typically, this can be done with orders of magnitude fewer model runs than traditional methods Use the emulator to make inference about other things of interest E.g. uncertainty analysis, calibration Conceptually very straightforward in the Bayesian framework But of course can be computationally hard

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23 DACE and BACCO This has led to a wide ranging body of tools for inference about all kinds of uncertainties in computer models All based on building the GP emulator of the model from a set of training runs Early work concentrated on prediction – DACE Design and Analysis of Computer Experiments More recently, wider scope and emphasis on the Bayesian framework – BACCO Bayesian Analysis of Computer Code Output

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24 BACCO includes Uncertainty analysis Sensitivity analysis Calibration Data assimilation Model validation Optimisation Etc… All within a single coherent framework

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25 Research directions Models with heterogeneous local behaviour Regions of input space with rapid response, jumps High dimensional models Many inputs, outputs, data points Dynamic models Data assimilation Stochastic models Relationship between models and reality Model/emulator validation Multiple models Design of experiments Sequential design

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26 Dynamic models Initial state x 0 is updated recursively by the one-step model f 1 (x,c) Forcing inputs c t Interested in sequence x 1, x 2, … x T At least 4 approaches to emulating this XTXT X0X0 c1c1 f1f1 X1X1 c2c2 f1f1 X2X2 c3c3 f1f1 X T-2 c T-1 f1f1 X T-1 cTcT f1f1

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Treat time as input Emulate x t as the function f(x 0,t) = f t (x 0,c t ) = f 1 (x t-1,c t ) = f 1 (… f 1 (f 1 (x 0,c 1 ),c 2 )…,c t ) Easy to do Hard to get the temporal correlation structure right XTXT X0X0 c1c1 f1f1 X1X1 c2c2 f1f1 X2X2 c3c3 f1f1 X T-2 c T-1 f1f1 X T-1 cTcT f1f1

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Multivariate emulation Emulate the vector x = (x 1, x 2, … x T ) as the multi-output function f T (x 0 ) = (f 1 (x 0,c 1 ), f 2 (x 0,c 2 ), …, f T (x 0,c T )) Simple extension of univariate theory Restrictive covariance structure XTXT X0X0 c1c1 f1f1 X1X1 c2c2 f1f1 X2X2 c3c3 f1f1 X T-2 c T-1 f1f1 X T-1 cTcT f1f1

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Functional output emulation Treat x series as a functional output Identify features (e.g. principal components) to summarise function Same theory as for multivariate emulation, but lower dimension Loss of information, no longer reproduces training data XTXT X0X0 c1c1 f1f1 X1X1 c2c2 f1f1 X2X2 c3c3 f1f1 X T-2 c T-1 f1f1 X T-1 cTcT f1f1

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Recursive emulation Emulate the single step function f 1 (x,c) Iterate the emulator numerically May take longer than original model! Or approximate filtering algorithm May be inaccurate for large T XTXT X0X0 c1c1 f1f1 X1X1 c2c2 f1f1 X2X2 c3c3 f1f1 X T-2 c T-1 f1f1 X T-1 cTcT f1f1

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31 Some comparisons Multivariate emulation (approach 2) is better than treating time as an input (approach 1) Validates better Able to work with partial series Recursive emulation still under development Looking promising Only recursive emulation can realistically treat uncertainty in forcing inputs Only recursive emulation can extend T beyond training data

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32 MUCM Managing Uncertainty in Complex Models Large 4-year research grant 7 postdoctoral research assistants 4 PhD studentships Started in June 2006 Based in Sheffield and 4 other UK universities Objective: To develop BACCO methods into a robust technology … toolkit that is widely applicable across the spectrum of modelling applications case studies

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33 Thank you

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