Emulation, Uncertainty, and Sensitivity

Emulation, Uncertainty, and Sensitivity
Building an Emulator

UQ minitutorial– session 2
Outline Recipe for building an emulator – MUCM toolkit Screening – which simulator inputs matter Design – where to run the simulator Model structure – mean and covariance functions Estimation / inference – building the emulator Validation – making sure the emulator is OK Possible extensions Multiple outputs Dynamic simulators Bayes linear methods Summary of emulation UQ minitutorial– session 2

The ‘standard’ problem
The MUCM toolkit recipe

Step 0: Know your simulator
Before attempting to create an emulator it is important you understand your simulator What are the plausible input ranges? What constraints are there in input combinations? What is the output behaviour like? Ideally you may wish to elicit beliefs about the distributions of the inputs if these are not known At least ranges are needed for all inputs

Step 1: Screening – active inputs
All serious simulators require more than one input The norm is anything from a few to thousands All of the basic emulation theory in the toolkit assumes multiple inputs Large numbers of inputs pose computational problems Dimension reduction techniques have been developed Output typically depends principally on a few inputs Screening seeks to identify the most important inputs for a given output Most often the Morris method is used, which is a cheap sensitivity analysis approximation UQ12 mintutorial– session 2

Screening: the Morris method
Basic idea: develop a design that changes one input at a time, while filling space Morris designs based on a series of repeated trajectories which change only one input each step Example (3 inputs) Trajectory changes in one direction at a time Do several trajectories Rate of change in output in each leg: “Elementary effect” of that input UQ12 mintutorial– session 2

UQ12 minitutorial– session 2
Compute for each input: Average of the elementary effects (μ) Variance of the elementary effects (σ) Active inputs to retain: High variance indicates nonlinear effect or interaction High mean indicates strong linear effect Screen out inputs with low mean and variance non-linear effect non-linear effect no effect linear effect UQ12 minitutorial– session 2

UQ12 minitutorial – session 2
Step 2: Design To build an emulator, we use a set of simulator runs Our training data are y1 = f(x1), ..., yn = f(xn) Where x1, x2, ..., xn are n different points in the space of inputs This set of n points is a design A good design will provide us with maximum information about the simulator And hence an emulator that is as good as possible UQ12 minitutorial – session 2

There are many options for design and many issues
Design principles Design is all about choosing where to run the simulator to develop a good emulator We also need to consider validation and calibration There are many options for design and many issues In the absence of additional information space-filling designs are used But this is an area of ongoing research Grids are infeasible for all but trivial simulators

Latin hypercube designs
LHC designs Use n values for each input Combining randomly Here 50 x-values are combined randomly with 50 y-values Advantages Doesn’t necessarily require a large number of points Nothing lost if some inputs are inactive Disadvantages Random choice may not produce an even spread of points Need to generate many LHC designs and pick the best UQ12 minitutorial – session 2

Some more design choices
Various formulae and algorithms exist to generate space-filling designs for any number of inputs The Sobol sequence is often used Quick and convenient Not always good when some inputs are inactive Optimal designs maximise/minimise some criterion E.g. maximum entropy designs Can be hard to compute, often not massive gains Hybrid designs try to satisfy two criteria Space-filling but also having a few points closer together In order to estimate correlation lengths well UQ12 minitutorial – session 2

Step 3: Building the emulator
In deciding on the structure of the emulator we have some choices to make: The mean function The covariance function The prior specifications There are no universal solutions here So judgement and validation play an important role UQ12 minitutorial – session 2

The technical part (overview!)
The emulator is a Gaussian process The conditional distribution of the simulator output, y, given the input, x, and “hyperparameters” is multivariate normal The following initial choices are generally made: Mean function m(x) = h(x)Tβ, with h(x) typically [1,x] Covariance function σ2c(x,x’) = σ2exp-{(x-x’)TC(x-x’)} C is a diagonal matrix of inverse length scales 1/ δ2 The hyperparameters are (β, σ2, δ) The choices we make can be important The mean (black) and variance (grey) from a posterior GP (with Matern covariance) based on the red training data, and four realisations from this posterior. UQ12 minitutorial – session 2

The GP mean function We can use this to say what kind of shape we would expect the output to take as a function of the inputs Most simulator outputs exhibit some overall trend in response to varying a single input So we usually specify a linear mean function Slopes (positive or negative) are estimated from the training data The emulator mean smoothes the residuals after fitting the linear terms We can generalise to other kinds of mean function if we have a clear idea of how the simulator will behave The better the mean function the less the GP has to do UQ12 minitutorial – session 2

Example Simulator is solid line Dashed line is linear fit Blue arrows indicate fitted residuals Without the linear mean function, we’d have a horizontal (constant) fit and larger residuals leading to larger emulator uncertainty UQ12 minitutorial – session 2

The GP covariance function
The covariance function determines how ‘wiggly’ the response is to each input There’s a lot of flexibility here, but standard covariance functions have a parameter for each input these ‘correlation length’ parameters are also estimated from the training data but some care is needed For predicting output at untried x, correlation lengths are important they determine how much information comes from nearby training points and hence the emulator accuracy

Prior information enters through the form of the mean function
Prior distributions Prior information enters through the form of the mean function And to a lesser extent the covariance function But we can also supply prior information through the prior distributions for “hyperparameters” For slope/regression parameters and correlation lengths Also the overall variance parameter Putting in genuine prior information here generally improves emulator performance Compared with standard ‘non-informative’ priors e.g.

Step 4: Learning the emulator
We normally proceed using Bayesian inference Just how Bayesian depends on size of problem Ideally we would ‘integrate out’ all unknown parameters, but this can be difficult, requiring MCMC Details are on the toolkit, but in summary Typically one can integrate out the regression coefficients (β) and variance parameter (σ2) Optimise (maximum likelihood, or MAP) the covariance length scales (δ) Ignoring uncertainty in length scales can be a problem if they are not well identified Which is often the case UQ12 minitutorial – session 2

Prediction with the emulator
Once the (hyper)-parameters of the emulator have been learnt (or integrated out) one can use the emulator to predict at a new input what the simulator output would have been This is always a predictive distribution Example with 6 (left) and 12 (right) training points UQ12 minitutorial – session 2

Step 5: Validating the emulator
Validating the emulator is essential Full probabilistic assessment of fitness for purpose First examine the standardised residuals, with +/- 2 std intervals Visual assessment is often very helpful and provides diagnostic information More sophisticated diagnostics make use of correlation structure UQ12 minitutorial – session 2

What is validation? What does it mean to validate an emulator? Compare the emulator’s predictions with the simulator output Make a validation sample of runs at new input configurations The emulator mean is the best prediction and is always wrong But the emulator predicts uncertainty around that mean The emulator is valid if its expressions of uncertainty are correct Actual outputs should fall in 95% intervals 95% of the time No less and no more than 95% of the time Standardised residuals should have zero mean and unit variance See MUCM toolkit UQ12 minitutorial – session 2

Measures for validation
The Mahalanobis distance on a test set Accounts for the predictive covariance on the test set Follows an F-distribution so we can check the value is close to the theoretical one for a given test set size A useful diagnostic is the pivoted Cholesky decomposition of the predictive covariance Should look like a normal distribution Suggests non-stationary / poor predictive variance Suggests poor length scale / covariance function

Steps in building an emulator
Specify the Gaussian process model Select the prior distributions for the GP hyperparameters Choose a design for training and validation Fit the emulator to the simulator runs Validate and re-fit if needed UQ12 mintutorial – session 2

Simple Example: Energy Balance Model
Inputs: 18 initial surface temperatures +8 others Outputs: 18 final temperatures + 4 others We use the mean surface temperature as our output Vary solar constant

UQ12 mintutorial – session 2
Example UQ12 mintutorial – session 2

What types of simulator are amenable to emulation?
Extensions What types of simulator are amenable to emulation?

Many outputs Most simulators also produce multiple outputs For instance, a climate simulator may predict temperature on a grid, etc. Usually, for any given use of the simulator we are interested in just one output So we can just emulate that one, particularly if it is some combination of the others, e.g. mean global surface temperature But some problems require multi-output emulation Again, there are dimension reduction techniques All described in the MUCM toolkit UQ12 minitutorial – session 2

Multi-output emulators
When we need to emulate several simulator outputs, there are a number of available approaches Single output GP with added input(s) indexing the outputs For temperature outputs on a grid, make grid coordinates 2 additional inputs Independent GPs Multivariate GP Independent GPs for a linear transformation E.g. principal components Possibility for dimension reduction These are all documented in the MUCM toolkit UQ12 minitutorial – session 2

Dynamic emulation Many simulators predict a process evolving in time At each time-step the simulator updates the system state Often driven by external forcing variables at each time-step Climate models are usually dynamic in this sense We are interested in emulating the simulator’s time series of outputs The various forms of multi-output emulation can be used Or a dynamic emulator, emulating the single time-step And then iterating the emulator Also documented in the MUCM toolkit UQ12 minitutorial – session 2

Stochastic emulation Other simulators produce non-deterministic outputs Running a stochastic simulator twice with the same input x produces randomly different outputs Different emulation strategies arise depending on what aspect of the output is of interest Interest focuses on the mean Output has added noise Which we allow for when building the emulator Interest focuses on risk of exceeding a threshold Emulate the distribution and derive the risk Emulate the risk This is not yet covered in the MUCM toolkit UQ12 minitutorial – session 2

Bayes linear methods So far assumed a fully Bayesian framework But there is an alternative framework – Bayes linear methods Based only on first and second order moments Means, variances, covariances Avoids making assumptions about distributions Its predictions are also first and second order moments Means, variances, covariances but no distributions The toolkit contains theory and procedures for Bayes linear emulators UQ12 minitutorial – session 2

Bayes linear emulators
Much of the mathematics is very similar A Bayes linear emulator is not a GP but gives the same mean and variance predictions For given correlation lengths, mean function parameters Although these are handled differently But the emulator predictions no longer have distributions Compared with GP emulators Advantages – simpler and may be feasible for more complex problems Disadvantages – absence of distributions limits many of the uses of emulators Compromises made UQ12 minitutorial – session 2

Summary and Limitations
Why emulation is not a panacea

Some caveats on emulation
Not all simulators are suitable for emulation With very large numbers of (>50) outputs need specific emulators and large training sets For the problem you are solving are all outputs needed? For dynamic simulators with high dimensional state spaces there remain computational issues With discrete inputs and outputs Gaussian processes are not well suited But these issues are being addressed actively in research projects across the world including MUCM UQ12 minitutorial – session 2

Typical sequence of emulation
Define the problem you want to solve, identify the simulator Identify the inputs, define ranges and screen to select Design the training set and run the simulator Validate the emulator and if necessary refine Train the emulator using the training set and inference method Choose the emulator (mean and covariance) and define priors Use the emulator and if necessary refine Modify the simulator or refine it, maybe using observations UQ12 minitutorial – session 2

Summary Before you emulate know your simulator! Think carefully about the problem you really want to solve Emulation is a tool to solve interesting problems and not an aim in itself The more prior knowledge you bring the easier the task will be Choosing mean and covariance, eliciting priors Spend time on validation and refinement Building an emulator will help you understand your simulators … not replace them! UQ12 minitutorial – session 2

We have built an emulator. What now?
UQ12 minitutorial– session 2

UQ: Session 3 Using an emulator

Outline So we’ve built an emulator – what can we use it for? Prediction What would the simulator output y be at an untried input x? Uncertainty analysis Given uncertainty in x, what is the implied uncertainty in y? Sensitivity analysis Which inputs influence the output most? Which inputs are responsible for most output uncertainty? Calibration Given observation of the real system, how can we use that to learn about the best input values? UQ12 mintutorial – session 2 47

Prediction and UA

Prediction Prediction is simple because that’s precisely what the emulator does For any given x, the emulator mean E(f(x)) is an estimate The emulator variance var[f(x)] expresses uncertainty Known as code uncertainty Similarly, given x and some threshold c we can evaluate P[f(x) > c] UQ12 mintutorial – session 2 49

Uncertainty analysis If X has distribution g(x) then UA looks at the implied distribution of Y = f(X) How do we evaluate that? In Session 1 we used Monte Carlo for a simple nonlinear simulator Mean = 0.117 Median = 0.122 Std. dev. = 0.048 But all these are estimates Accuracy depends on the size of the Monte Carlo sample 95% interval for the mean is (0.116, 0.118) UQ12 mintutorial – session 2 50

UA with an emulator Consider the expected output M = E[Y] It is uncertain because of code uncertainty The emulator mean value for M is E[M] = ∫ E[f(x)] g(x) dx We can evaluate this by Monte Carlo Sample many values of x, evaluate the emulator mean E[f(x)] for each and average them This is already much faster than making many simulator runs to evaluate f(x) But we can often do the integral exactly UQ12 mintutorial – session 2 51

Why emulation is more efficient
Similarly we can evaluate var[M] This is code uncertainty and depends on the number of simulator runs to build the emulator We want to compute/estimate M sufficiently accurately, so we want var[M] sufficiently small Emulation is more efficient because we can typically achieve the desired accuracy using far fewer simulator runs to build the emulator than using traditional methods For the simple nonlinear model, using only 25 simulator runs to build the emulator, a 95% interval for M is (0.1173, ) Using the emulator we can also compute/estimate all those other quantities of interest, like var[Y] or P[Y > c] UQ12 mintutorial – session 2 52

Sensitivity analysis

Sensitivity analysis Which inputs affect the output most? This is a common question Sensitivity analysis (SA) attempts to address it There are various forms of SA The methods most frequently used are not the most helpful! UQ12 mintutorial – session 2 54

Recap – the nonlinear model
The simple nonlinear model of the first session y = sin(x1)/{1+exp(x1+x2)} Just two inputs Uncertainty analysis: Normal distributions on inputs Output mean = 0.117, median = 0.122 Std. dev. = 0.048 Which of these two inputs influences output most? And in what ways? UQ12 mintutorial – session 2 55

Local sensitivity analysis
To measure the sensitivity of y to input xi, compute the derivative of y with respect to xi Nonlinear model: At x1 = x2 = 0.5, the derivatives are wrt x1, 0.142; wrt x2, –0.094 How useful is this? Derivatives evaluated only at the central estimate Could be quite different at other points nearby Doesn’t capture interactions between inputs E.g. sensitivity of y to increasing both x1 and x2 could be greater or less than the sum of their individual sensitivities Not invariant to change of units UQ12 mintutorial – session 2 56

One-way SA Vary inputs one at a time from central estimate Nonlinear model: Vary x1 to 0.25, 0.75, output is 0.079, 0.152 Vary x2 to 0.25, 0.75, output is 0.154, 0.107 Is this more useful? Depends on how far we vary each input Relative sensitivities of different inputs change if we change the ranges But ranges are arbitrary Also fails to capture interactions Statisticians have known for decades that varying factors one at a time is bad experimental design! UQ12 mintutorial – session 2 57

Multi-way SA Vary factors two or more at a time Maybe statistical factorial design Full factorial designs require very many runs Can find interactions but hard to interpret Often just look for the biggest change of output among all runs Still dependent on how far we vary each input UQ12 mintutorial – session 2 58

Probabilistic SA Inputs varied according to their probability distributions As in Uncertainty Analysis (UA) Sensitivities still depend on ranges of distributions (variances), but these are now not necessarily arbitrary Gives an overall picture and can identify interactions UQ12 mintutorial – session 2 59

Variance decomposition
One way to characterise the sensitivity of the output to individual inputs is to compute how much of the UA variance is due to each input For the simple non-linear model, we have Input Contribution X1 80.30 % X2 16.77 % X1.X2 interaction 2.93 % UQ12 mintutorial – session 2 60

Main effects We can also plot the effect of varying one input averaged over the others Nonlinear model Averaging y = sin(x1)/{1+exp(x1+x2)} with respect to the uncertainty in x2, we can plot it as a function of x1 Similarly, we can plot it as a function of x2 averaged over uncertainty in x1 We can also plot interaction effects UQ12 mintutorial – session 2 61

Main effects in the simple nonlinear model
Red is main effect of x1 (averaged over x2) Blue is main effect of x2 (averaged over x1) UQ12 mintutorial – session 2 62

Joint effect in the simple nonlinear model
UQ12 mintutorial – session 2 63

A more complex example 5 inputs have appreciable influence, and account for 57% of the total UA variance Interactions account for 28% UQ12 mintutorial – session 2 64

Amplifying on variances
Main effect plots amplify on the information given in the variance decomposition The variance component associated with input xi is equal to the amount by which its main effect varies over the range of uncertainty in xi The 5 inputs with most influence are dashed UQ12 mintutorial – session 2 65

Demonstration with GEM-SA

Sensitivity – an example
UQ12 mintutorial – session 2

GEM-SA GEM-SA is a user-friendly piece of software that does many of the things we’ve been discussing Can create some kinds of design Fits an emulator to simulator output Computes uncertainty and sensitivity analyses It’s freely available It’s really useful for experimenting with relatively simple simulators But not always reliable UQ12 mintutorial – session 2 68

Example ForestETP vegetation simulator 7 input parameters 120 simulator runs Objective: conduct a variance-based sensitivity analysis To identify which uncertain inputs are driving the output uncertainty. UQ12 mintutorial – session 2 69

Exploratory scatter plots
UQ12 mintutorial – session 2 70

Looks like X6 is most important, and probably also X5 But these plots are hard to read because of the scatter Fit an emulator and carried out the uncertainty and sensitivity analyses UQ12 mintutorial – session 2 71

Variance of main effects
Main effects for each input. Input 6 has the greatest individual contribution to the variance Main effects sum to 66% of the total variance UQ12 mintutorial – session 2 72

Main effect plots UQ12 mintutorial – session 2 73

Main effect plots Fixing X6 = 18, this point shows the expected value of the output (obtained by averaging over all other inputs). Simply fixing all the other inputs at their central values and comparing X6=10 with X6=40 would underestimate the influence of this input (The thickness of the band shows emulator uncertainty) UQ12 mintutorial – session 2 74

Interactions and total effects
Main effects explain only 2/3 of the variance Model must contain interactions Any input can have small main effect, but large interaction effect, so overall still an ‘important’ input We can compute all pair-wise interaction effects 435 in total for a 30 input model – can take some time! Useful to know what to look for UQ12 mintutorial – session 2 75

For each input Xi Total effect = main effect for Xi + all interactions involving Xi Assumes independent inputs Main effects and total effects normalised by variance Total effect >> main effect implies interactions in the model Look for inputs with large total effects relative to main effects Investigate possible interactions involving those inputs UQ12 mintutorial – session 2 76

Total effects for inputs 4 and 7 much larger than their main effects. Implies presence of interactions. UQ12 mintutorial – session 2 77

Interaction effects Compute pair-wise joint effect variances All interactions between X4, X5, X6, X7 UQ12 mintutorial – session 2 78

Main and interaction effects
Note interactions involving inputs 4 and 7 Main effects and selected interactions now sum to almost 92% of the total variance UQ12 mintutorial – session 2 79

What have we learnt here?
Most important inputs are X4, X5, X6, X7 We can more or less ignore X1, X2, X3 Together these 3 account for < 10% of overall variance X6 is most important single input 36% of variance Has only minor interactions with other inputs X4, X5, X7 interact in complex ways But together account for over 50% of variance Main effect plots are useful Particularly for X6 But less so for the others! Need to look at how they interact UQ12 mintutorial – session 2 80

SA summary Why SA? For the model user: SA identifies which inputs it would be most useful to reduce uncertainty about For the model builder: main effect and interaction plots demonstrate how the simulator is behaving Sometimes surprisingly! UQ12 mintutorial – session 2 81

Calibration

Calibration Simulator users often want to tune the simulator using observations of the real system Adjust the input parameters so that the simulator output matches observations as well as possible Two very important points Calibration will reduce uncertainty about x but will not eliminate it It is necessary to understand how the simulator relates to reality Model discrepancy UQ12 mintutorial – session 2 83

Model discrepancy Simulator output y = f(x) will not equal the real system value z Even with best/correct inputs x Model discrepancy is the difference z – f(x) As discussed in Session 1, model discrepancy is due to Wrong or incomplete science Programming errors, rounding errors Inaccuracy in numerically solving systems of equations Ignoring model discrepancy leads to poor calibration Over-fitting of parameter estimates Over-confidence in the fitted values We’ll look at calibration in the next session UQ12 mintutorial – session 2 84

References For variance-based sensitivity analysis Toolkit has a “topic thread” on this subject GEM-SA Other software packages implementing many of the toolkit methods exist, but none are comprehensive or guaranteed to be stable UQ12 mintutorial – session 2 85

References !! UQ12 minitutorial – session 2

Emulation, Uncertainty, and Sensitivity

Similar presentations

Presentation on theme: "Emulation, Uncertainty, and Sensitivity"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Emulation, Uncertainty, and Sensitivity

Similar presentations

Presentation on theme: "Emulation, Uncertainty, and Sensitivity"— Presentation transcript:

Similar presentations

About project

Feedback