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


Overview: 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.


Starting the program: Starting the program On the desktop, there is a folder andlt;GEM-SA tutorialandgt;, opening it will reveal two other folders: Inside the folder andlt;GEM-SA1.1andgt; is the program: Double-clicking this will start the program


Main window: Main window menu toolbar log window Sensitivity Analysis output grid


Generating input designs: 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


Generating input designs: 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


The design: 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


Creating/Editing a project: 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


Edit Project - Files: Edit Project - Files Names of input files Names of output files


Edit Project - Options: Edit Project - Options How many inputs? Edit input names


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


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


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


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


Input names: Input names By clicking the andlt;Names…andgt; 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.


Distributions for inputs: Distributions for inputs When we click the andlt;OKandgt; button, the following window opens. This windows allows us to specify our beliefs about the inputs.


A first run through: 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 andlt;Examples\Eg1andgt; to help save us time.


Monte Carlo method: 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


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


Prediction: 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


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


Cross validation: 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


A real example: 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.


The input names in order: 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)


Main effects plots: 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.


Producing main/joint effects plots for publication: 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


Limitations of GEM-SA: 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.


When it all goes wrong…: 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


Where to find the program: 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