Designing Ensembles for Climate Prediction Peter Challenor National Oceanography Centre

Why Ensembles for Climate Prediction? Not just a point estimate Uncertainty estimates as well Calibration of models against data Sensitivity analysis

Overview What is experimental design? Why should we be interested? Perturbed physics ensembles Space filling designs Some recent results Multimodel ensembles Conclusions

What is experimental design? Developed from agricultural experiments in the 1920’s How should you apply treatments to experimental plots in a field?

R.A. Fisher Greatest Statistician of the 20th Century Randomisation Block designs Latin squares Split plots …

Clinical Trials Randomisation Blind and Double Blind Trials Sequential Designs

Why should we worry about designing our experiments? Would you take medication that hadn’t been through a properly designed clinical trial? Would you set climate policy without a properly designed climate model experiment?

Computer Experiments (Climate model ensembles) Computer experiments are very different from either clinical trials or field experiments. In general we are using them to explore the properties of some computer simulator (model). This is usually the numerical solution of a system of PDE’s or ODE’s

Computer Experiments Mathematically we can write our computer simulator as where y is the output, x is the input and η(.) is the unknown mathematical function represented by the simulator y and x are often very high dimension y=\eta(x)

Computer Experiments Normally the purpose of our computer experiment is to make some inference about the model Estimate what the model does at inputs we haven’t run it at Optimise the model parameters w.r.to some data Make predictions

Types of ensemble Perturbed physics ensembles Change inputs (parameters, initial conditions, …) to a single model Multimodel ensembles Look at multiple models

What is a good experimental design? Make our inferences to the highest accuracy with the minimum cost

Optimal Design Fisher Information ≃ inverse of variance Maximise the information = minimising the variance D-optimal designs Minimise the determinant of the variance matrix A-optimal designs Minimise the trace of the variance matrix There are others (I, V, G, E optimality)

Bayesian Design Set up a D-optimal design by maximising the utility U(S)=\int_\theta log (det[\mathcal{I}(\theta|S)])p(\theta)d\theta \mathcal{I}(\theta|S) is the variance matrix of θ in design S

Fisher Information Matrix The Fisher Information matrix is an approximation to the inverse of the variance matrix. \mathcal{F}_{i,j}=\frac{\partial^2 L}{\partial x_i \partial x_j}

An Possible Design ‘Star’ design

No aphorism is more frequently repeated in connection with field trials experiments, than that we must ask Nature few questions or, ideally, one question, at a time. The writer is convinced that this view is wholly mistaken. Nature, he suggests, will best respond to a logical and carefully thought out questionnaire; indeed, if we ask her a single question, she will often refuse to answer until some other topic has been discussed. R.A. Fisher, 1926

What we need from the design of a climate model ensemble We want to Span the whole input space Observe interactions Minimise the number of simulator evaluations

Space Filling Designs Factorial Designs Latin Hypercubes Pseudo random sequences Sobol sequence

The Full Factorial We set each input (factor) at a set number of levels All combinations are included in the design n levels of m factors needs nm points This gets large quickly

An Example 52 factorial

Fractional Factorial Full factorials are expensive For large number of factors only 2 or 3 levels Can use fractional factorials (2 levels)

The Latin Hypercube Decide how many simulator runs you can afford Divide each input range into that number of intervals Allocate a point to each interval Randomly permute across each input

The Latin Hypercube

The Latin Hypercube We don’t have an algorithm for the optimal Latin hypercube What is a good Latin hypercube? Maximin Orthogonal designs Pragmatic designs

A Latin Hypercube

A maximin LHC

Are Factorials better than Latin Hypercubes

Low Discrepancy Sequences Alternative to Latin hypercubes Designed for multi-dimensional integrals Examples include Halton sequences, Niederreiter nets and Sobol sequences \delta^*(D)=\sup_B \left | \frac{I_B(D)}{N}-|B| \right | B= \left \{ \prod_{i=1}^{d} [0,u_i) : u_i \in (0,1] \right \}

Warning: Hard Maths

Discrepancy Discrepancy is defined as where is the number of points in B and \delta^*(D)=\sup_B \left | \frac{I_B(D)}{N}-|B| \right | B= \left \{ \prod_{i=1}^{d} [0,u_i) : u_i \in (0,1] \right \} I_B(D) |B| = \int_B dx

Low Discrepancy Sequences It is believed, but not proved, that minimum discrepancy sequences have the property Look for low values of Examples include Halton sequences, Niederreiter nets and Sobol sequences \delta^*(D) \leqslant C_d(\log n)^d + O(\log n)^{d-1} C_d

End of Hard Maths

Sobol Sequences A low discrepancy sequence A 2n-1 Sobol sequence is a Latin hypercube Some projections of multi-dimensional Sobol sequences are not ‘good’

Sobol Sequences

Sobol Sequences

Sequential Designs So far our designs have been one off We make a design and that dictates how we run the simulator We do not learn from the early runs An idea from clinical trials is to learn as we carry out the experiment

Sequential Design for Computer Experiments Perform an initial experiment (usually space filling) Add additional points to satisfy some criteria We might add additional points where our predictions of simulator output are most uncertain We might add additional points for optimisation

A D-optimal design for smoothness I’m fitting an emulator to a computer experiment Can we design an experiment to estimate the ‘smoothness’ parameters of the emulator optimally?

Emulators δ is a zero mean Gaussian process \eta(\theta) = \mu(\theta) + \delta(\theta) \mu(\theta) = A(\theta) \beta’ δ is a zero mean Gaussian process This is defined in terms of a variance (σ2 and a correlation function (C(x1,x2))

An Example of an Emulator

Zhu and Stein (2004) In the geostatistical context Zhu and Stein show that the Fisher information is approximately given by U(S)=\int_B log (det[\mathcal{I}(B|S)])p(B)dB \mathcal{I}_{i,j}(B|S) = \frac{1}{2} tr\left( \Sigma^{-1} \Sigma_i \Sigma^{-1} \Sigma_j\right) \Sigma_k(i,j) = \frac{\partial \Sigma(B)}{\partial B_k}-(x_{i,k} - x_{j,k})^2 \Sigma(x_i,x_j)

Bayesian Design Approximate the inverse of the covariance matrix by the Fisher information matrix Set up a D-optimal design by maximising the utility

5-point Sobol

10-point Sobol

Sobol 10 +5 One at time (5) Five at a time

Designing for Multiple Climate Models So far we have considered designs for single simulators How might we design for multiple models? The IPCC problem ‘Ensemble of opportunity’?

So What’s the Problem Common outputs between simulators Not common inputs An important area for research

Conclusions Designing model ensembles can make them more efficient make the experimenter think about the problem There are a variety of designs around Consult a statistician before you design the experiment Design of computer experiments is an active area of research (not only in climate/environmental sciences)