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Designing Ensembles for
Climate PredictionPeter 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
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
1. Perturbed physics ensembles
Change inputs (parameters, initial conditions, …) to a single model
2. 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
is the variance matrix of θ in design S
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
1.Span the whole input space
2.Observe interactions
3.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
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
•This is defined in terms of a variance (σ2and 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
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
One at time (5) Five at a time
Sobol 10 +5
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)