1. 17 May 2007RSS Kent Local Group1 Quantifying uncertainty in the UK carbon flux Tony O’Hagan CTCD, Sheffield

2. 17 May 2007RSS Kent Local Group2 Outline Introduction Gaussian process emulation The England and Wales carbon flux 2000

3. 17 May 2007RSS Kent Local Group3 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

4. 17 May 2007RSS Kent Local Group4 Examples Climate prediction Molecular dynamics Nuclear waste disposal Oil fields Engineering design Hydrology

5. 17 May 2007RSS Kent Local Group5 Uncertainty analysis Consider just one source of uncertainty We have a computer model that produces output y = f (x) when given input x But for a particular application we do not know x precisely So X is a random variable, and so therefore is Y = f (X ) We are interested in the uncertainty distribution of Y How can we compute it?

6. 17 May 2007RSS Kent Local Group6 Monte Carlo The usual approach is Monte Carlo Sample values of x from its distribution Run the model for all these values to produce sample values y i = f (x i ) These are a sample from the uncertainty distribution of Y Neat but impractical if it takes minutes or hours to run the model We can then only make a small number of runs

7. 17 May 2007RSS Kent Local Group7 GP solution Treat f (.) as an unknown function with Gaussian process (GP) prior distribution Use available runs as observations without error, to derive posterior distribution (also GP) Make inference about the uncertainty distribution E.g. The mean of Y is the integral of f (x) with respect to the distribution of X Its posterior distribution is normal conditional on GP parameters

8. 17 May 2007RSS Kent Local Group8 Gaussian process emulation Principles of emulation The GP and how it works

9. 17 May 2007RSS Kent Local Group9 Emulation A computer model encodes a function, that takes inputs and produces outputs An emulator is a statistical approximation of that function Estimates what outputs would be obtained from given inputs With statistical measure of estimation error Given enough training data, estimation error variance can be made small

10. 17 May 2007RSS Kent Local Group10 So what? A good emulator estimates the model output accurately with small uncertainty and runs “instantly” So we can do uncertainty analysis etc fast and efficiently Conceptually, we use model runs to learn about the function then derive any desired properties of the model

11. 17 May 2007RSS Kent Local Group11 Gaussian process Simple regression models can be thought of as emulators But error estimates are invalid We use Gaussian process emulation Nonparametric, so can fit any function Error measures can be validated Analytically tractable, so can often do uncertainty analysis etc analytically Highly efficient when many inputs Reproduces training data correctly

12. 17 May 2007RSS Kent Local Group12 2 code runs Consider one input and one output Emulator estimate interpolates data Emulator uncertainty grows between data points

13. 17 May 2007RSS Kent Local Group13 3 code runs Adding another point changes estimate and reduces uncertainty

14. 17 May 2007RSS Kent Local Group14 5 code runs And so on

15. 17 May 2007RSS Kent Local Group15 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 This area is now known as BACCO Bayesian Analysis of Computer Code Output

16. 17 May 2007RSS Kent Local Group16 BACCO includes Uncertainty analysis Sensitivity analysis Calibration Data assimilation Model validation Optimisation Etc… All within a single coherent framework

17. 17 May 2007RSS Kent Local Group17 MUCM Managing Uncertainty in Complex Models Large 4-year research grant Started in June 2006 7 postdoctoral research assistants 4 PhD studentships Based in Sheffield, Durham, Aston, Southampton, LSE Objective: to develop BACCO methods into a robust technology that is widely applicable across the spectrum of modelling applications

18. 17 May 2007RSS Kent Local Group18 Example: UK carbon flux in 2000 Vegetation model predicts carbon exchange from each of 707 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 0.56 Mt C Analysis by Marc Kennedy and John Paul Gosling

19. 17 May 2007RSS Kent Local Group19 SDGVMd outputs for 2000

20. 17 May 2007RSS Kent Local Group20 Outline of analysis 1. Build emulators for each PFT at a sample of sites 2. Identify most important inputs 3. Define distributions to describe uncertainty in important inputs Analysis of soils data Elicitation of uncertainty in PFT parameters Need to consider correlations

21. 17 May 2007RSS Kent Local Group21 4. Carry out uncertainty analysis in each sampled site 5. Interpolate across all sites Mean corrections and standard deviations 6. Aggregate across sites and PFTs Allowing for correlations

22. 17 May 2007RSS Kent Local Group22 Sensitivity analysis for one pixel/PFT

23. 17 May 2007RSS Kent Local Group23 Elicitation Beliefs of expert (developer of SDGVMd) regarding plausible values of PFT parameters Important to allow for uncertainty about mix of species in a pixel and role of parameter in the model In the case of leaf life span for evergreens, this was more complex

24. 17 May 2007RSS Kent Local Group24 EvNl leaf life span

25. 17 May 2007RSS Kent Local Group25 Correlations PFT parameter in one pixel may differ from in another Because of variation in species mix Common uncertainty about average over all species induces correlation Elicit beliefs about average over whole UK EvNl joint distributions are mixtures of 25 components, with correlation both between and within years

26. 17 May 2007RSS Kent Local Group26 Mean NBP corrections

27. 17 May 2007RSS Kent Local Group27 NBP standard deviations

28. 17 May 2007RSS Kent Local Group28 Land cover (from LCM2000)

29. 17 May 2007RSS Kent Local Group29 Aggregate across 4 PFTs

30. 17 May 2007RSS Kent Local Group30 Sensitivity analysis Map shows proportion of overall uncertainty in each pixel that is due to uncertainty in the parameters of PFTs As opposed to soil parameters Contribution of PFT uncertainty largest in grasslands/moorlands

31. 17 May 2007RSS Kent Local Group31 England & Wales aggregate PFT Plug-in estimate (Mt C) Mean (Mt C) Variance (Mt C 2 ) Grass5.284.640.2689 Crop0.850.450.0338 Deciduous2.131.680.0128 Evergreen0.800.780.0005 Covariances0.0010 Total9.067.550.3170

32. 17 May 2007RSS Kent Local Group32 Conclusions Bayesian methods offer a powerful basis for computation of uncertainties in model predictions Analysis of E&W aggregate NBP in 2000 Good case study for uncertainty and sensitivity analyses But needs to take account of more sources of uncertainty Involved several technical extensions Has important implications for our understanding of C fluxes Policy implications