"[B]egin upon the precept ... that the things we see are to be weighed in the scale with what we know" (Meredith, The Egoist, 1879, project Gutenberg)
Research Recent projects Donkeys! Publications Technical reports Statistics & probability Other science Economics & finance Books and book chapters Other Teaching Theory of Inference Miscellaneous Presentations Software Cabot Institute
Department of Mathematics,
University of Bristol,
University Walk,
Bristol BS8 1TW

tel: +44 (0)117 9287782
fax: +44 (0)117 9287999

Find me:
Room 4.12, left out of the lift on the fourth floor of the main Maths building.

Treat me:
Coffee and (vegan) cake at Cafe Kino, now on Stokes Croft.


Jonathan (Jonty) Rougier's homepage

Reader in Statistics

Summer School in risk and uncertainty in natural hazards

More details here.

Exciting new book!

   J.C. Rougier, R.S.J. Sparks, and L.J. Hill (eds), 2013, Risk and Uncertainty Assessment for Natural Hazards, Cambridge, UK: Cambridge University Press.

"This is an invaluable compendium for academic researchers and professionals working in the fields of natural hazards science, risk assessment and management and environmental science and will be of interest to anyone involved in natural hazards policy." I wrote that :)


My research concerns the assessment of uncertainty in complex systems, particularly environmental systems such as climate. Here is an
informal summary. The main issues I think about are:
  • How do we use scientific models? If they are to be used quantatively to predict the behaviour of a system, e.g. the climate, how do we represent the discrepancy between the model and the system?

  • Is there a general framework for combining model-evaluations, system observations, and expert judgements, that is applicable across different areas of science?

  • What is the role of probability in representing our uncertainty? What does a probability represent, and what are the limitations of the probability calculus? What is the best way to explain probabilistic concepts to non-statisticians?

  • How do we implement our inferential calculations with very large models, that may take weeks or even months to evaluate? What corners do we cut?

Some current and recent projects

  1. CREDIBLE: one of the two NERC-funded consortia on uncertainty and risk assessment for natural hazards; 2012-2015.

  2. RATES: NERC-funded project to quantify mass trends in Antarctica using a synthesis of model evaluations and several different datasets; 2011-2013.

  3. PalaeoQUMP: "PalaeoQUMP aims to constrain climate sensitivity [...] using a wider range of derived climate observations from the geological past (reconstructions from sediments and geomorphological changes for the Last Glacial Maximum and the mid-Holocene period), to evaluate climate model predictions generated using the same series of simulations as QUMP produced for the modern climate." Funded by NERC QUEST: 2007-2010, but we're still analysing the model runs!

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Technical reports

  1. J.C. Rougier (2011), Monte Carlo methods for Bayesian palaeoclimate reconstruction.

    Abstract. In palaeoclimate reconstruction, the natural modelling direction is forwards from climate to sensors to proxy measurements. Statistical methods can be used to invert this direction, making climate inferences from proxy measurements. Among these methods, the Bayesian method would seem to deal best with the substantial epistemic uncertainties about climate, and about its impact on sensors. The main challenge is to perform this inference efficiently within a simulation approach. This paper reviews the Importance Sampling approach to Bayesian palaeoclimate reconstruction, and then goes on to demonstrate the value of recent advances in Markov chain Monte Carlo (MCMC) inference.

    Available as a pdf file. This paper was written as an addendum to the SUPRAnet report Studying uncertainty in palaeoclimate reconstruction: A framework for research (still under development).

  2. J.C. Rougier (2011), A representation theorem for stochastic processes with separable covariance functions, and its implications for emulation. Substantially revised, again, 12 Oct 2011. As this paper gets longer, I hope it gets clearer.

    Abstract. Many applications require stochastic processes specified on two- or higher-dimensional domains; spatial or spatial-temporal modelling, for example. In these applications it is attractive, for conceptual simplicity and computational tractability, to propose a covariance function that is separable; e.g. the product of a covariance function in space and one in time. This paper presents a representation theorem for such a proposal, and shows that all processes with continuous separable covariance functions are second-order identical to the product of second-order uncorrelated processes. It discusses the implications of separable or nearly separable prior covariances for the statistical emulation of complicated functions such as computer codes, and critically reexamines the conventional wisdom concerning emulator structure, and size of design.

    Available as a pdf file.

  3. J.C. Rougier and M. Crucifix (2012), Uncertainty in climate science and climate policy.

    This is a book chapter for L. Lloyd and E. Winsberg, eds, Conceptual Issues in Climate Modeling, University of Chicago Press, forthcoming. Submitted draft chapter available as a pdf file.

  4. J.C. Rougier, A. Zammit-Mangion and N. Schoen (2013), Computation and visualisation for large-scale Gaussian updates.

    In geostatistics, and also in other applications in science and engineering, we are now performing updates on Gaussian process models with many thousands or even millions of components. These large-scale inferences involve computational challenges, because the updating equations cannot be solved as written, owing to the size and cost of the matrix operations. They also involve representational challenges, to account for judgements of heterogeneity concerning the underlying fields, and diverse sources of observations.

    Diagnostics are particularly valuable in this situation. We present a diagnostic and visualisation tool for large-scale Gaussian updates, the `medal plot'. This shows the updated uncertainty for each observation, and also summarises the sharing of information across observations, as a proxy for the sharing of information across the state vector. It allows us to `sanity-check' the code implementing the update, but it can also reveal unexpected features in our modelling. We discuss computational issues for large-scale updates, and we illustrate with an application to assess mass trends in the Antarctic Ice Sheet.

    UPDATED! Available at

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Statistics and Probability

A. Zammit Mangion, J.C. Rougier, N.W. Schoen, F. Lindgren, J.L. Bamber (2015). "Multivariate spatio-temporal modelling for assessing Antarctica's present-day contribution to sea-level rise." Forthcoming in Environmetrics.

J.C. Rougier and M. Goldstein (2014), "Climate Simulators and Climate Projections", Annual Review of Statistics and Its Application, 1, 103-123. Available online, doi:10.1146/annurev-statistics-022513-115652

A. Zammit Mangion, J.C. Rougier, J.L. Bamber and N.W. Schoen (2014), "Resolving the Antarctic contribution to sea-level rise: a hierarchical modelling framework", Environmetrics, 25(4), 245-264. Available online, doi:10.1002/env.2247

J.C. Rougier, M. Goldstein, and L. House (2013), "Second-order exchangeability analysis for multi-model ensembles", Journal of the American Statistical Association, 108, 852-863. Available online, doi:10.1080/01621459.2013.802963

I. Scheel, P.J. Green, and J.C. Rougier (2011), "A graphical diagnostic for identifying influential model choices in Bayesian hierarchical models", Scandinavian Journal of Statistics, 38(3), 529-550. doi:10.1111/j.1467-9469.2010.00717.x

J.C. Rougier (2010), Discussion of "A statistical analysis of multiple temperature proxies: Are reconstructions of surface temperatures over the last 1000 years reliable?" by McShane and Wyner, Annals of Applied Statistics, 5(1), 96-98. doi:10.1214/10-AOAS409

J.C. Rougier and M. Kern (2010), Predicting Snow Velocity in Large Chute Flows Under Different Environmental Conditions. Applied Statistics, 59(5), 737-760. doi:10.1111/j.1467-9876.2010.00717.x

J.C. Rougier, S. Guillas, A. Maute, A.D. Richmond (2009), Expert Knowledge and Multivariate Emulation: The Thermosphere-Ionosphere Electrodynamics General Circulation Model (TIE-GCM), Technometrics, 51(4), 414-424. doi:10.1198/TECH.2009.07123

M. Goldstein and J.C. Rougier (2009), Reified Bayesian Modelling and Inference for Physical Systems, Journal of Statistical Planning and Inference, 139(3), 1221-1239. doi:10.1016/j.jspi.2008.07.019 With discussion and rejoinder.

J.C. Rougier (2008), Efficient Emulators for Multivariate Deterministic Functions, Journal of Computational and Graphical Statistics, 17(4), 827-843. doi:10.1198/106186008X384032. R package OPE_0.8.tar.gz.

J.C. Rougier (2008), Discussion of 'Inferring Climate System Properties Using a Computer Model', by Sanso et al, Bayesian Analysis, 3(1), 45-56. DOI:10.1214/08-BA301B

M. Goldstein and J.C. Rougier (2006), Bayes Linear Calibrated Prediction for Complex Systems, Journal of the American Statistical Association, 101 (no. 475), 1132-1143.

M. Goldstein and J.C. Rougier (2004), Probabilistic Formulations for Transferring Inferences from Mathematical Models to Physical Systems, SIAM Journal on Scientific Computing, 26(2), 467-487. doi:/10.1137/S106482750342670X

I. MacPhee, J.C. Rougier and G. Pollard (2004), Server Advantage in Tennis Matches, Journal of Applied Probability, 41(4), 1182-1186.

J.C. Rougier and M. Goldstein (2001), A Bayesian Analysis of Fluid Flow in Pipelines, Applied Statistics, 50(1), 77-93.

P.S. Craig, M. Goldstein, J.C. Rougier and A.H. Seheult (2001), Bayesian Forecasting for Complex Systems Using Computer Simulators, Journal of the American Statistical Association, 96, 717-729.

Other science

A. Zammit Mangion et al (2015), A data-driven approach for assessing ice-sheet mass balance in space and time, Annals of Glaciology, forthcoming.

J.C. Rougier (2013), 'Intractable and unsolved': some thoughts on statistical data assimilation with uncertain static parameters, Phil. Trans R. Soc. A., 371, doi:10.1098/rsta.2012.0297

D.B. Stephenson, M. Collins, J.C. Rougier, and R.E. Chandler (2012), Statistical problems in the probabilistic prediction of climate change, Environmetrics, 23(5), 364-372. doi: 10.1002/env.2153

M. Collins, R.E. Chandler, P.M. Cox, J.M. Huthnance, J.C. Rougier, and D.B. Stephenson (2012), Quantifying future climate change, Nature Climate Change, 2, 403-409. doi:10.1038/nclimate1414

R.M. Gladstone, V. Lee, J.C. Rougier, A.J. Payne, H. Hellmer, A. Le Brocq, A. Shepherd, T.L. Edwards, J. Gregory, and S.L. Cornford (2012), Calibrated prediction of Pine Island Glacier retreat during the 21st and 22nd centuries with a coupled flowline model, Earth and Planetary Science Letters, 333-334, 191-199. doi:10.1016/j.epsl.2012.04.022

P.W. Fitzgerald, J.L. Bamber, J. Ridley and J.C. Rougier (2011), Exploration of parametric uncertainty in a Surface Mass Balance Model applied to the Greenland Ice Sheet, Journal of Geophysical Research, in press, doi:10.1029/2011JF002067

N.R. Edwards, D. Cameron, J.C. Rougier (2011), Precalibrating an intermediate complexity climate model, Climate Dynamics, 37, 1469-1482.

S. Guillas, J.C. Rougier, A. Maute, A.D. Richmond, and C.D. Linkletter (2009), Bayesian calibration of the Thermosphere-Ionosphere Electrodynamics General Circulation Model (TIE-GCM), Geoscientific Model Development, 2, 137-144. Available online.

M. Crucifix and J.C. Rougier (2009), On the use of simple dynamical systems for climate predictions: A Bayesian prediction of the next glacial inception. The European Physics Journal - Special Topics, 174(1), 11-31. DOI:10.1140/epjst/e2009-01087-5

J.C. Rougier, D.M.H. Sexton, J.M. Murphy, and D. Stainforth (2009), Analysing the climate sensitivity of the HadSM3 climate model using ensembles from different but related experiments. Journal of Climate, 22(13), 3540-3557. DOI:10.1175/2008JCLI2533.1

J.C. Rougier and D.M.H. Sexton (2007), Inference in Ensemble Experiments, Philosophical Transactions of the Royal Society, Series A, 365, 2133-2143. doi:10.1098/rsta.2007.2071

J.C. Rougier (2007), Probabilistic Inference for Future Climate Using an Ensemble of Climate Model Evaluations, Climatic Change, 81, 247-264. DOI:10.1007/s10584-006-9156-9

J.C Rougier (2005), Probabilistic Leak Detection in Pipelines Using the Mass Imbalance Approach. Journal of Hydraulic Research, 43(5), 556-566.

M. van Oijen, J.C. Rougier and R. Smith (2005), Bayesian Calibration of Process-Based Forest Models: Bridging the Gap Between Models and Data, Tree Physiology, 25, 915-927.

Economics and Finance

S.C. Parker and J.C. Rougier (2007), The Retirement Behaviour of the Self-Employed in Britain, Applied Economics, 39(6), 697-713.

P.R. Holmes and J.C. Rougier (2005), Trading Volume and Contract Rollover in Futures Contracts, Journal of Empirical Finance, 12(2), 317-338.

S.C. Parker and J.C. Rougier (2001), Measuring Social Mobility as Unpredictability, Economica, 68, 63-76.

B. Hillier and J.C. Rougier (1999), Real Business Cycles, Investment Finance and Multiple Equilibria, Journal of Economic Theory, 86, 100-22.

J.C. Rougier (1997), A Simple Necessary Condition for Negativity in the Almost Ideal Demand System with the Stone Price Index, Applied Economics Letters, 4, 97-9.

J.C. Rougier (1996), An Optimal Price Index for Stock Index Futures Contracts, Journal of Futures Markets, 16, 189-99.

J.C. Rougier (1993), The Impact of Margin-Traders on the Distribution of Daily Stock Returns: The London Stock Exchange, Applied Financial Economics, 3, 325-8.

Books and book chapters

J.C Rougier (2014), "Formal Bayes methods for model calibration with uncertainty", in K. Beven and J. Hall (eds), Applied Uncertainty Analysis for Flood Risk Management, Imperial College Press.

J.C. Rougier, R.S.J. Sparks, and L.J. Hill (eds), 2013, Risk and Uncertainty Assessment for Natural Hazards, Cambridge, UK: Cambridge University Press.

  • L.J. Hill, R.S.J. Sparks, and J.C. Rougier, Risk assessment and uncertainty in natural hazards, ch 1, pp 1-18.
  • J.C. Rougier, Quantifying hazard losses, ch 2, pp 19-39.
  • J.C. Rougier and K.J. Beven, Model and data limitations: the sources and implications of epistemic uncertainty, ch 3, pp 40-63.


K. Milner and J.C. Rougier (2014), "How to weigh a donkey in the Kenyan countryside", Significance, 11(4), 40-43.

J. Murphy, R. Clark, M. Collins, C. Jackson, M. Rodwell, J.C. Rougier, B. Sanderson, D. Sexton and T. Yokohata (2011), Perturbed parameter ensembles as a tool for sampling model uncertainties and making climate projections, Proceedings of ECMWF Workshop on Model Uncertainty, 20-24 June 2011, 183-208. Available online.

J.C. Rougier, T.L. Edwards, M. Collins and D.M.H. Sexton (2011), Low-noise projections of complex simulator output: A useful tool when checking for code errors, Proceedings of ECMWF Workshop on Model Uncertainty, 20-24 June 2011, 209-220. Available online.

J.C. Rougier and L. Chen (2010), Comment on the paper by Diggle et al., Journal of the Royal Statistical Society, Series C, 59(2), 216.

R. Chandler, J.C. Rougier, and M. Collins (2010), Climate change, Significance, 7(1), 9-12.

J.C. Rougier (2009), Notes on statistical modelling for complex systems, ver. 0.5, unpublished. Available as a pdf file. Please note the version number: this document is still evolving.

J.C. Rougier (2008), Climate change detection and attribution, ISBA bulletin, 15(4), 3-6. Available on-line.

J.C. Rougier (2008), Formal Bayes Methods for Model Calibration with Uncertainty, in K. Beven and J. Hall (eds), Applied Uncertainty Analysis for Flood Risk Management, Imperial College Press / World Scientific. Draft version available as a pdf file.

J.C. Rougier (2006), Comment on the paper by Haslett et al., Journal of the Royal Statistical Society, Series A, 169(3), 432-433.

J.C. Rougier (2005), Literate Programming for Creating and Maintaining Packages. R News, 5(1), 35-39.

J.C. Rougier (2004), Comment on the paper by Murphy et al. Nature did not want to publish this comment, but I think it says some useful things. Available as a pdf file.

J.C. Rougier (2001), Comment on the paper by Kennedy and O'Hagan, Journal of the Royal Statistical Society, Series B, 63, page 453.

J.C. Rougier (2001), What's the Point of `tensor'?, R News, 1(2), 26-27.

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Selected presentations (since 2009)

  • Modelling the eruption processes of a large number of similar but not identical volcanoes, OxWaSP Seminar on Applied Bayesian Analysis, Warwick Jan 2015. slides available. This is still work in progress.

  • Ensembles in weather forecasting and climate projection, Advances In Climate Theory, Brussels, Aug 2014.

  • Server advantages in tennis matches, Iain MacPhee day, Durham, April 2014. Slides available. This is a blackboard presentation of MacPhee et al, 2004.

  • "Intractable and unsolved": the challenge of data assimilation with uncertain static parameters, Intractability workshop, Bristol, April 2012. Slides available. This is really a talk about long-time-scale palaeoclimate reconstruction.

  • Nomograms for visualising relationships between three variables, UseR! 2011 invited talk. Slides available. Here is some beta code for producing the parallel scale nomograms used in the talk: parallelNomo.R. This is the picture you should get if you run the example: nomogramBMI.pdf.

    After the discussion at UseR! 2011 I appreciated that I had an extra degree of freedom, which I could use to set the righthand axis so that the typical donkey had a horizontal line on the nomogram. Then I put all the donkeys' lines on the plot, and finished it off with a couple of density functions, original and predicted. The results is not exactly 'less is more': donkDensity_19Aug.pdf.

    After seeing that line at the top of the 'not less is more' nomogram heading off to surprising lengths I created a parallel coordinate plot of the donkey data, using parallelCoord.R, which resulted in parallelCoord_26Aug.pdf. As Kate Milner noted, "that is an exceptionally long donkey" (128cm). And in fact she checked her notebooks and this length had been misrecorded: it was in fact 102cm. And what's the UseR! connection? it was over a beer at the mixer that David Nicolaides made me understand the value of this plot for screening.

  • Multivariate emulation for North American mid-Holocene temperature reconstructions, MPI-M, Hamburg, Mar 2011. Slides available. This is a longer version of the talk given at the Liverpool Marine Symposium, with a stronger focus on emulating large climate simulators (HadCM3, in our case). Note that the reconstruction at the end is purely illustrative.

  • Calibration and hypothesis testing for a model of avalanche behaviour, IHRR, Durham, Jan 2011. Slides available. Please note that this is a preliminary analysis and we are currently investigating a more sophisticated statistical model for the relationship between the HB parameters and the measurements.

  • Inference using large climate simulators: HadCM3 and North American Mid-Holocene temperature anomalies, Liverpool Marine Symposium, Jan 2011. Slides available. The slides for the talk at the MPI-M (Mar 2011) are more detailed.

  • Accounting for the limitations of quantitative models, SECaM, Exeter, Jan 2011. Slides available.

  • Stochastic dynamical modelling in Environmental Science, AMSTAT, Warwick, Oct 2010. See the slides for the Exeter Jan 2011 seminar.

  • What can pollen tell us about palaeo-climate?, ICAS, Leeds, Oct 2010. Slides available (more detailed).

  • Is Building Bigger and 'Better' Climate Change Models a Poor Investment?, CRASSH conference, Clare College Cambridge, 29 Sep 2010. text available, I was proposing the motion. Watch the debate.

  • Probabilistic frameworks, INI CLP workshop on probabilistic climate prediction, Exeter, Sep 2010. See the slide! Watch the presentation (multiple formats).

  • A new statistical framework for analysing multi-model ensembles, IMS 2010, Gothenburg, Aug 2010. Slides available.

  • A new statistical framework for analysing multi-model ensembles, 11th IMSC, Edinburgh, July 2010. Slides available.

  • What can pollen tell us about palaeo-biomes?, 11th IMSC, Edinburgh, July 2010. See the slides for the ICAS Oct 2010 seminar.

  • Assessing model limitations, Governing Through Uncertainty meeting, Bristol, May 2010. Slides available.

  • Complex systems: Accounting for model limitations, Research Students' Conference 2010, Warwick, April 2010. slides available.

  • Complex systems: Accounting for model limitations, St Andrews, Mar 2010. slides available.

  • Model limitations: Sequential data assimilation with uncertain static parameters, Edinburgh, Mar 2010. Slides available.

  • Uncertainty and risk in natural hazards, Nottingham, Nov 2009. Slides available.

  • Introduction to computer experiments, and the challenges of expensive models, RSS Statistical Computing Section, Bath, October 2009. Slides available.

  • Quantifying uncertainty in probability of exceedence (PE) curves, NERC/KTN workshop on Catastrophe modelling for natural hazard impact, Lloyd's, London, Oct 2009. Slides available and also R code for the pictures.

  • Emulator-based simulator calibration for high-dimensional data, JCGS session, JSM-2009, Washington DC, Aug 2009. Slides available.

  • The What, Why, and How of Multivariate Emulation, Spring Research Conference On Statistics in Industry and Technology, Vancouver, May 2009. Slides available.

  • Simple models for glacial cycles, Bath Institute for Complex Systems, Feb 2009. Slides available.


R packages

  • calibrater. A package of useful functions for calibrating a computer simulator. (Used to be 'calibrate'.)
  • paranomo. Draw parallel-scale nomograms. Also contains the 'donkeys' dataset (see Donkeys).
  • slice. General purpose adaptive slice sampling, also contains mvqqplot, a multivariate qqplot function.

Other resources


Joint work with Kate Milner.

Donkeys are extremely hard to weigh, but easy to measure with a measuring tape. Because weight is a important factor in health and in veterinary care, statistical models are used to predict weight on the basis of measured hearth girth and height. The Donkey Sanctuary sponsored Kate Milner to go to Kenya with a large weighing scales and collect data on donkeys. Together we are analysing the results, with the intention of producing an improved 'nomogram' (or similar) showing how to predict a Kenyan donkey's weight on the basis of its height and heart girth, and also age and condition.


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