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A Survey of Statistical Methods for Climate Extremes

A Survey of Statistical Methods for Climate Extremes. Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK. 9th International Meeting on Statistical Climatology, Cape Town, 26 May 2004. Overview. Climate extremes – Aims and issues – PRUDENCE project

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A Survey of Statistical Methods for Climate Extremes

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  1. A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International Meeting on Statistical Climatology, Cape Town, 26 May 2004

  2. Overview Climate extremes – Aims and issues – PRUDENCE project Extreme-value theory – Fundamental idea – Spatial modelling – Clustering Concluding remarks

  3. Description – Statistical properties Comparison – Space, time, model, obs Prediction – Space, time, magnitude Non-stationarity – Space, time Dependence – Space, time Data – Size, inhomogeneity Aims and Issues

  4. PRUDENCE European climate Control 1961–1990 Scenarios 2071–2100 10 high-resolution, limited domain regional GCMs 6 driving global GCMs

  5. Fundamental Idea Data sparsity requires efficient methods Extrapolation must be justified by theory Probability theory identifies appropriate models Example: X1 + … + Xn  Normal max{X1, …, Xn}  GEV

  6. Spatial Statistical Models Single-site models Conditioned independence: Y(s', t) Y(s, t) | (s) – Deterministically linked parameters – Stochastically linked parameters Residual dependence: Y(s', t) Y(s, t) | (s) – Multivariate extremes – Max-stable processes

  7. Generalised Extreme Value (GEV) Block maximum Mn = max{X1, …, Xn} for iid Xi Pr(Mnx)  G(x) = exp[–{1 +  (x – ) / }–1/ ] for large n

  8. Single-site Model Annual maximum Y(s, t) at site s in year t Assume Y(s, t) | (s) = ((s), (s), (s)) iid GEV((s)) for all t m-year return level satisfies G(ym(s) ; (s)) = 1 – 1 / m Daily max 2m air temperature (ºC) at 35 grid points over Switzerland from control run of HIRHAM in HadAM3H

  9. Temperature – Single-site Model    y100

  10. Generalised Pareto (GP) Points (i / n, Xi), 1 in, for which Xi exceeds a high threshold approximately follow a Poisson process Pr(Xi – u> x | Xi > u)  (1 +  x / u)–1/ for large u

  11. Deterministic Links Assume Y(s, t) | (s) = ((s), (s), (s)) iid GEV((s)) for all t Global model (s) = h(x(s) ; 0) for all s e.g. (s) = 0 + 1 ALT(s) Local model (s) = h(x(s) ; 0) for all sN(s0) Spline model (s) = h(x(s) ; 0) + (s) for all s

  12. (s) = 0 + 1ALT(s) 0 = 31.8ºC (0.2) 1 = –6.1ºC/km (0.1) p = 0.03 single site (y100) Temperature – Global Model  altitude (km) global (y100)

  13. Stochastic Links Model l((s)) = h(x(s) ; 0) + Z(s ; 1), random process Z Continuous Gaussian process, i.e. {Z(sj) : j = 1, …, J } ~ N(0, (1)), jk(1) = cov{Z(sj), Z(sk)} Discrete Markov random field, e.g. Z(s) | {Z(s') : s's} ~ N((s) +  (s, s'){Z(s') – (s)}, 2) s'N(s)

  14. Stochastic Links – Example Model (s) = 0 + 1 ALT(s) + Z(s | a , b , c) log (s) = log 0 + Z(s | a , b , c) (s) = 0 + Z(s | a , b , c) cov{Z*(sj), Z*(sk)} = a*2 exp[–{b* d(sj , sk)}c*] Independent, diffuse priors on a*, b*, c*, 0, 1, 0 and 0 Metropolis-Hastings with random-walk updates

  15. Temperature – Stochastic Links 0 1 global (y100) latent (y100)

  16. Multivariate Extremes Maxima Mnj = max{X1j, …, Xnj} for iid Xi = (Xi1, …, XiJ) Pr(Mnj xj for j = 1, …, J ) MEV for large n e.g. logisticPr(Mn1 x1, Mn2 x2) = exp{–(z1–1/ + z2–1/)} Model {Y(s, t) : s  N(s0)} | {,(s) : s  N(s0)} ~ MEV

  17. Assume Y(s, t)  Y(s', t) | Y(s0, t) for all s, s'  N(s0) and locally constant  single site (y100) Temperature – Multivariate Extremes  multivar (y100)

  18. Max-stable Processes Maxima Mn(s) = max{X1(s), …, Xn(s)} for iid {X(s) : sS} Pr{Mn(s) x(s) for sS}  max-stable for large n Model Y*(s, t) = max{rik(s, si) : i 1} where {(ri , si) : i 1} is a Poisson process on (0, ) S e.g. k(s, si)  exp{ – (s – si)' (1)–1 (s – si) / 2}

  19. Precipitation – Max-stable Process Estimate Pr{Y(sj , t)  y(sj) for j = 1, …, J } Max-stable model 0.16 Spatial independence 0.54 Realisation of Y*

  20. Clustering Extremes can cluster in stationary sequences X1, …, Xn Points i / n, 1 in, for which Xi exceeds a high threshold approximately follow a compound Poisson process

  21. Zurich Temperature (June – July) Extremal Index Pr(cluster size > 1) Threshold Percentile Threshold Percentile

  22. Review Linkage efficiency, continuous space, description, interpretation, bias, expense comparison Multivariate discrete space, model choice, description dimension limitation Max-stable continuous space, estimation, prediction model choice

  23. Future Directions Wider application of EV theory in climate science – combine with physical understanding – shortcomings of models, new applications Improved methods for non-identically distributed data – especially threshold methods with dependent data

  24. Further Information Climate Analysis Group www.met.rdg.ac.uk/cag/extremes NCAR www.esig.ucar.edu/extremevalues/extreme.html Alec Stephenson’s R software http://cran.r-project.org PRUDENCE http://prudence.dmi.dk ECA&D project www.knmi.nl/samenw/eca My personal web-site www.met.rdg.ac.uk/~sws02caf

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