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Project leader’s report MUCM Advisory Panel Meeting, November 2006

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Outline Background: uncertainty in models MUCM overview Putting the structures in place Specific progress

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Background: Uncertainty in Models

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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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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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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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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 7.55 Mt C Std deviation 0.57 Mt C

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Maps

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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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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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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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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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Data

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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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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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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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Gaussian process representation More efficient approach First work in early 1980s 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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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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2 code runs Consider one input and one output Emulator estimate interpolates data Emulator uncertainty grows between data points

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3 code runs Adding another point changes estimate and reduces uncertainty

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5 code runs And so on

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Then what? Given enough training data points we can emulate any model accurately So that posterior variance is small “everywhere” Typically, feasible with orders of magnitude fewer model runs than traditional methods Use the emulator to make inference about other things of interest Uncertainty analysis, sensitivity analysis, calibration, data assimilation, optimisation, … Conceptually very straightforward in the Bayesian framework But of course can be technically hard

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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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MUCM Overview

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MUCM in a nutshell Managing Uncertainty in Complex Models Four-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 Bayesian model uncertainty methods into a robust technology … toolkit, UML specifications that is widely applicable across the spectrum of modelling applications case studies

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Theme 1: High Dimensionality Tackling problems associated with dimensionality of inputs, outputs, parameters, and data WP 1.1 – Screening (PS) Identifying most important inputs/outputs WP 1.2 – Sparsity and Projection (RA) Dimension reduction using modern computational techniques WP 1.3 – Multiscale models (RA) Linking models and data at different resolutions Theme leader: Dan Cornford

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Theme 2: Using Observational Data Tackling problems associated with model structural error to link models to field data WP 2.1 – Linking Models to Reality (RA) Modelling structural error WP 2.2 – Diagnostics and Validation (PS) Criticising our statistical representations WP 2.3 – Calibration & Data Assimilation (RA) Extending calibration techniques, particularly to dynamic models Theme leader: Michael Goldstein

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Theme 3: Realising the Potential Turning theory into reliable, widely applicable techniques across a wide range of models WP 3.1 – Experimental Design (RA + PS) Designing input sets for running models, and planning observational studies WP 3.2 – The Toolkit (RA + PS) Distilling experience with methods into robust tools, relaxing constraints WP 3.3 – Case Studies (RA) Three substantial case studies Theme leader: Peter Challenor

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Organisation overview

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Organisation by theme 1. Cornford2. Goldstein3. Challenor 1.1 Boukouvalas Cornford (Challenor) 1.2 Maniyar Cornford (Wynn) 1.3 Cumming Goldstein (Rougier) 2.1 House Goldstein (O’Hagan) 2.2 Bastos O’Hagan (Rougier) 2.3 Bhattacharya Oakley (Cornford) 3.1 Maruri-Aguilar Wynn (Goldstein) Youssef Wynn (Oakley) 3.2 Gattiker Challenor (O’Hagan, Cornford) Stephenson Challenor (Oakley) 3.3 Gosling O’Hagan (Challenor) O’Hagan

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Organisation by committee The whole Team meets twice a year Presentations, reports and planning The Project Management Board meets four times a year Formal decision making, budgeting, personnel matters The Advisory Panel meets with the investigators twice a year Providing external support and advice

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The Team Investigators Challenor, Cornford, Goldstein, Oakley, O’Hagan, Rougier, Wynn Project manager Green RAs Bhattacharya, Cumming, Gattiker, Gosling, House, Maniyar, Maruri-Aguilar PSs Bastos, Bouskouvalas, Stephenson, Youssef

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The Board Project Management Board is the primary project management body Tony O’Hagan (Sheffield, Chair) Dan Cornford (Aston) Peter Challenor (Southampton) Michael Goldstein (Durham) Henry Wynn (LSE) Non-voting Jeremy Oakley (Sheffield) Jonty Rougier (Durham) Jo Green (Sheffield)

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The Panel Advisory Panel comprises modellers, model users and model uncertainty experts from a wide range of fields Industry Bob Parish, Hilmi Kurt-Elli, Clive Bowman Academia Ron Akehurst, Martin Dove, Keith Beven, Douglas Kell, Ian Woodward Research institutions Richard Haylock, Andrea Saltelli, Andy Hart, David Higdon, Mat Collins

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The Mentor Peter Green (Bristol) Appointed by EPSRC Liaise between project team and EPSRC Advise team

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Putting the Structures in Place

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General All RAs, PSs and Project Manager recruited Started at various times from 1 June to 1 October Need to replace Bhattacharya Website, wiki, lists, logo, templates created Reading list, glossary under development Monthly reporting established RAs set up reading club Links established with related projects Particularly with SAMSI programme in USA

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Project planning First draft of rolling workplans Descriptions and objectives Detailed plans and milestones for 12 months ahead With month-by-month detail for 6 months Outline plans and milestones for remainder of project Will be updated quarterly Milestones and deliverables carefully monitored Panel will receive plans from previous Board

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Financial management Handled at quarterly Board meetings Phased budget plan created for each institution RAs appointed initially for 3 years Fourth year funds retained in reserve

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Contacts with Panel members Introductory meetings held with most members An RA has been assigned to each To develop understanding of the models and the modelling area To act as link between other team members and Panel member Beginning to explore use of models Some models also sourced from other contacts

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Specific progress 1 Emulator fitting Study of methods to estimate roughness parameters Acquisition of existing packages Multiscale models Multiscale version of Daisyworld model created Non-homogeneous models Voronoi tessellation method improved Paper in preparation

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Specific progress 2 Design Study of aberration and relationship to kernel Paper in preparation Dynamic models Basic theory of dynamic emulation developed Toy dynamic model created and emulated Paper in preparation Hydrological model acquired

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