GEM-SA: a tutorial

1. GEM-SA: a tutorial John Paul Gosling University of Sheffield

2. Overview • GEM-SA: Gaussian Emulation Machine for Sensitivity Analysis • It’s a Windows based program that has a graphical interface created by Marc Kennedy during his time in CTCD • It does emulation for prediction, uncertainty analysis and sensitivity analysis • It also has a facility to create experimental designs for the analysis of computer models.

3. Starting the program • On the desktop, there is a folder <GEM-SA tutorial>, opening it will reveal two other folders: • Inside the folder <GEM-SA1.1> is the program: • Double-clicking this will start the program

4. Main window menu toolbar Sensitivity Analysis output grid log window

5. Generating input designs • There are two designs available: LP-TAU and Maximin Latin Hypercube. Both have good space filling properties. Press this button to create a file of inputs for your computer model

6. Generating input designs • Then we specify ranges over which the input will be of interest • These must cover your beliefs about the range of each input

7. The design • Here’s a 50-point LP-TAU design for three inputs • You’ll also find they’ve been written to the file you specified (LP_TAU50.txt) in GEM-SA’s working directory

8. Creating/Editing a project • Now, we’ll run through some of the options available to us for emulator building. • We can create a new project or edit an existing project by selecting the appropriate item from the project menu. • Or we can use these toolbar buttons. New Edit

9. Edit Project - Files Names of input files Names of output files

10. Edit Project - Options Edit input names How many inputs?

11. Edit Project - Options What should be calculated, and how? Which joint effects should be calculated?

12. Edit Project - Options What prior mean for the output? Are the inputs uncertain?

13. Edit Project - Options What kind of predictions and cross validation?

14. Edit Project - Simulations MCMC control parameters Number of realisations for prediction and ME/JE How many points used to calculate main effects, joint effects

15. Input names • By clicking the <Names…> button, a window opens that allows us to name each of the inputs. • This can be handy when viewing the variance decomposition results and main effects plots.

16. Distributions for inputs • When we click the <OK> button, the following window opens. • This windows allows us to specify our beliefs about the inputs.

17. A first run through • Consider the simple nonlinear model we saw earlier y = sin(x1)/{1+exp(x1+x2)} • We have 2 inputs, x1 and x2, and we assume they both must be valued in the range [0,1]. • 20 points will give us a decent coverage of the unit square that is the input space here. • Two files have already been saved in the folder <Examples\Eg1> to help save us time.

18. Monte Carlo method • Here’s the result of a Monte Carlo analysis using 30 input pairs. • Mean = 0.139, median = 0.142 • Std. dev. = 0.053 • Variance = 0.0028

19. Monte Carlo method • Here’s the result of a Monte Carlo analysis using 10,000 input pairs. • Mean = 0.114, median = 0.115 • Std. dev. = 0.054 • Variance = 0.0029

20. Prediction • Predictions can be • Correlated realisations of outputs at the prediction inputs • Similar to main effect outputs • Marginal means and variances of outputs at the prediction inputs • Faster to compute, especially with many prediction points • Easy to interpret

21. A plot of the predictions • Here is the prediction output files plotted with the real function with x2 fixed at 0.5.

22. Cross validation • Choice of none, leave-one-out or leave final 20% out • Leave-one-out • Hyperparameters use all data and are then fixed when prediction is carried out for each omitted point • Leave final 20% out • Hyperparameters are estimated using the reduced data subset

23. A real example • A dynamic vegetation model is being used to predict the NBP of deciduous broadleaf woodland in the vicinity of Whitby, North Yorkshire. • The scientists are uncertain about ten inputs of the model and want to know how this uncertainty affects the NBP output of the model – Monte Carlo methods are out of the question as the model is too complex. • When they used their best guesses for these inputs, the model returned a NBP of 146.4gC/m2.

24. The input names in order • Maximum age (years) N(200,625) • Water potential (M Pa) N(3,0.25) • Leaf life span (days) N(190,1600) • Leaf mortality index N(0.005,6.25e-6) • Bud burst limit (degree days) N(135,6.25) • Seeding density (m2) N(0.1,0.0001) • Soil sand (%) N(43.27,222.12) • Soil clay (%) N(22.36,49.21) • log(stem growth rate) N(-5.116,0.041209) • Bulk density N(1.214,0.0325)

25. Main effects plots • The plug-in estimate of the NBP is far away from our mean for NBP as the main effect plot for bulk density is concave around it’s expected value of 1.214.

26. Producing main/joint effects plots for publication • In the files section of the edit project window, there are two fields that allow the user to specify where the main/joint effects data should be written. • These files can be used to produce graphs like the one I showed earlier. • The main effects file is structured as follows: • There are a number of blocks of function realisations – one for each input. • These are controlled by

27. Limitations of GEM-SA • In theory, the methods used by GEM-SA are limitless; however, the program itself isn’t. • It can handle up to 30 inputs and 400 training data. • Also, the distributions that are used to express our uncertainty about the inputs are limited to uniform or normal.

28. When it all goes wrong… • How do we know when the emulator is not working? • Large roughness parameters • Especially ones hitting the limit of 99 • Large emulation variance on UA mean • Poor CV standardised prediction error • Especially when some are extremely large • In such cases, see if a larger training set helps • Other ideas like transforming output scale

29. Where to find the program • GEM-SA is available on the web along with tutorial slides from a longer course and further example data sets. • Links to it can be found on my website where there is also a technical report explaining the perils of using the “plug-in” approach: j-p-gosling.staff.shef.ac.uk