Gaussian process emulation of multiple outputs

Tony O’Hagan, MUCM, Sheffield

Outline Gaussian process emulators

Simulators and emulators GP modelling

Multiple outputs Covariance functions Independent emulators Transformations to independence Convolution Outputs as extra dimension(s) The multi-output (separable) emulator The dynamic emulator

Which works best? An example

Simulators and emulators A simulator is a model of a real process

Typically implemented as a computer code Think of it as a function taking inputs x and giving

outputs y y = f(x)

An emulator is a statistical representation of the function Expressing knowledge/beliefs about what the

output will be at any given input(s) Built using prior information and a training set of

model runs The GP emulator expresses f as a GP

Conditional on hyperparameters

GP modelling Mean function

Regression form h(x)Tβ Used to model broad shape of response Analogous to universal kriging

Covariance function Stationary Often use the Gaussian form σ2exp{-(x-x′) TD-2(x-x

′)} D is diagonal with correlation lengths on diagonal

Hyperparameters β, σ2 and D Uninformative priors

The emulator Then the emulator is the posterior distribution of f

After integrating out β and σ2, we have a t process conditional on D

Mean function made up of fitted regression hTβ* plus smooth interpolator of residuals

Covariance function conditioned on training data Reproduces training data exactly

Important to validate Using a validation sample of additional runs Check that emulator predicts these runs to within

stated accuracy No more and no less

Bastos and O’Hagan paper on MUCM website

Multiple outputs Now y is a vector, f is a vector function Training sample

Single training sample for all outputs Probably design for one output works for many

Mean function Modelling essentially as before, h i(x)Tβi for output i Probably more important now

Covariance function Much more complex because of correlations

between outputs Ignoring these can lead to poor emulation of

derived outputs

Covariance function Let fi(x) be i-th output Covariance function c((i,x), (j,x′)) = cov[fi (x), fj(x′)] Must be positive definite

Space of possible functions does not seem to be well explored

Two special cases Independence: c((i,x), (j,x′)) = 0 if i ≠ j

No correlation between outputs Separability: c((i,x), (j,x′)) = σij cx(x, x′)

Covariance matrix Σ between outputs, correlation cx between inputs

Same correlation function cx for all outputs

Independence Strong assumption, but ... If posterior variances are all small, correlations may

not matter How to achieve this?

Good mean functions and/or Large training sample

May not be possible in practice, but ... Consider transformation to achieve independence

Only linear transformations considered as far as I’m aware

z(x) = A y(x) y(x) = B z(x) c((i,x), (j,x′)) is linear mixture of functions for each z

Transformations to independence Principal components

Fit and subtract mean functions (using same h) for each y

Construct sample covariance matrix of residuals Find principal components A (or other

diagonalising transform) Transform and fit separate emulators to each z

Dimension reduction Don’t emulate all z Treat unemulated components as noise

Linear model of coregionalisation (LMC) Fit B (which need not be square) and

hyperparameters of each z simultaneously

Convolution Instead of transforming outputs for each x

separately, consider y(x) = ∫ k(x,x*) z(x*) dx* Kernel k

Homogeneous case k(x-x*) General case can model non-stationary y

But much more complex

Outputs as extra dimension(s) Outputs often correspond to points in some space

Time series outputs Outputs on a spatial or spatio-temporal grid

Add coordinates of the output space as inputs If output i has coordinates t then write fi(x) = f*(x,t)

Emulate f* as single output simulator In principle, places no restriction on covariance

function In practice, for single emulator we use restrictive

covariance functions Almost always assume separability -> separable y Standard functions like Gaussian correlation may not be

sensible in t space

The multi-output emulator Assume separability Allow general Σ Use same regression basis h(x) for all outputs Computationally simple

Joint distribution of points on multivariate GP have matrix normal form

Can integrate out β and Σ analytically

The dynamic emulator Many simulators produce time series output by

iterating Output yt is function of state vector st at time t Exogenous forcing inputs ut, fixed inputs (parameters) p Single time-step simulator f* st+1 = f*(st , ut+1 , p)

Emulate f* Correlation structure in time faithfully modelled Need to emulate accurately

Not much happening in single time step but need to capture fine detail

Iteration of emulator not straightforward! State vector may be very high-dimensional

Which to use? Big open question!

This workshop will hopefully give us lots of food for thought MUCM toolkit v3 scheduled to cover these issues

All methods impose restrictions on covariance function In practice if not in theory Which restrictions can we get away with in practice?

Dimension reduction is often important Outputs on grids can be very high dimensional Principal components-type transformations Outputs as extra input(s) Dynamic emulation Dynamics often driven by forcing

Example Conti and O’Hagan paper

On my website: http://tonyohagan.co.uk/pub.html Time series output from Sheffield Global

Dynamic Vegetation Model (SDGVM) Dynamic model on monthly timestep Large state vector, forced by rainfall, temperature,

sunlight 10 inputs

All others, including forcing, fixed 120 outputs

Monthly values of NBP for ten years

Multi-output emulator on left, outputs as input on rightFor fixed forcing, both seem to capture dynamics wellOutputs as input performs less well, due to more restrictive/unrealistic time series structure

Conclusions Draw your own!