building an emulator

Building an Emulator

EGU short course – session 2 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

The ‘standard’ problem

The MUCM toolkit recipewww.mucm.ac.uk

http://www.mucm.ac.uk/

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


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• Compute:

• average of the elementary effects (μ)

• variance of the elementary effects (σ)

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non-linear effect non

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Step 2: Design

• Design is all about choosing where to run the simulator to learn 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• grids are infeasible for all but trivial simulators


Training sample design

• To build an emulator, we use a set of simulator runs• our training data are y1 = f(x1), ..., yn = f(xn)

• 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


Latin hypercube designs

• LHC designs• use n values for each input• combining randomly

• 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


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


Step 3: Building the emulator

• In deciding on the structure of the emulator we have many 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


The technical part (overview!)

• The emulator is a Gaussian process with:• 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

• thus the conditional distribution of the simulator output, y, given the input, x and parameters (β, σ2, δ) is multivariate normal

• The choices we make can be important


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


Example

• Simulator issolid line

• Dashed line islinear fit

• Blue arrowsindicate fitted residuals

• Without the linear meanfunction, we’d have a horizontal (constant) fit• and larger residuals• leading to larger emulator uncertainty

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 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 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.

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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 parameters(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, but typically the mean function does most of the work


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


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


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 Bastos and O’Hagan preprint on MUCM website

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

Suggests non-stationary / poor predictive varianceSuggests poor length scale / covariance function

Should follow a t

distribution

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


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


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


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


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


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

Summary and Limitations

Why emulation is not always a silver bullet


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


Typical sequence of emulation

Define the problem youwant to solve, identify

the simulator

Identify the inputs,define ranges andscreen to select

Design the trainingset and run the

simulator

Choose the emulator (mean and covariance)

and define priors

Train the emulatorusing the training setand inference method

Use the emulatorand if necessary refine

Validate the emulatorand if necessary refine

Modify the simulatoror refine it, maybe using observations


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!

building an emulator

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